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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nGiven the abundance of hydrogen in the universe, the Lyman $\\alpha$ (Ly$\\alpha$) line is an important component of radiation fields in a wide range of astrophysical settings. \\lya radiation transport is an active area of research in the study of planets, stars, galaxies, and cosmology \\citep{2019SAAS...46....1D}. An example application motivating our work is the role of \\lya in planetary atmospheres. The outer layers of the atmosphere are central to a planet's evolution, since they can shelter the lower atmosphere from high energy radiation as well as regulate the escape of gas into space.  There are two sources of Ly$\\alpha$: the star, and recombinations in the planet's atmosphere. \\lya may ionize atoms and dissociate molecules, as well as exert pressure forces that drive an outflow \\citep{2018A&A...620A.147B}. \\lya can also excite H atoms to the 2p state, creating a population of Balmer-line absorbers that can be observed via transmission spectroscopy \\citep{2017ApJ...851..150H, 2021ApJ...907L..47Y}. Due to the low gas densities in the upper atmosphere, collisional de-excitation and broadening are of secondary importance and \\lya may undergo ``resonant scattering''.\n\nHubble Space Telescope (HST) observations with the STIS have found large \\lya transit depths around a handful of exoplanets \\citep{2003Natur.422..143V, 2012A&A...543L...4L, 2012A&A...547A..18E, 2015Natur.522..459E,  2017A&A...597A..26B, 2017A&A...599L...3B, 2017A&A...602A.106B, 2018A&A...620A.147B, 2019AJ....158...50W, 2019EPSC...13.1928L, 2020ApJ...888L..21G,2021arXiv210309864B}. These observations have revealed a population of atoms extending out to distances of order a few planetary radii or more for several planets around bright, nearby stars, motivating a study of the physics of \\lya interactions with the H atom population. The transition from the atomic to the molecular layer in these hot upper atmospheres may take place at pressures of order ${\\sim}\\ 10\\ \\mu$bar (see discussion in \\citealt{2017ApJ...851..150H} for details). This suggests the presence of a thick layer of atomic H which can have a line center optical depth of $\\tau_0\\ {\\sim}\\ 10^8$. A careful treatment of resonant scattering is necessary in order to construct accurate models of H atom excitation, heating, and radiative forces. \n\nDue to the technical challenge of including resonant scattering, the fully three-dimensional geometry, and the presence of an outflow, numerical simulations may be required to fully understand the dynamics of these irradiated exoplanet atmospheres. The large optical depths at \\lya line center impose a steep computational cost for solving radiative transfer with Monte Carlo methods directly coupled with fluid dynamics \\citep{2017MNRAS.464.2963S}. The number of scatterings a photon undergoes is proportional to the line center optical depth, $\\tau_0$, of the domain \\citep{1972ApJ...174..439A}. Near the base of the atomic layer, the line center optical depth is ${\\sim}10^8$, so most of the time is spent following photons in these cells. A method that can accurately characterize transfer through these zones without following every photon scattering has the potential to greatly accelerate the calculation (see e.g. \\citealt{1968ApJ...153..783A,2002ApJ...567..922A}).\n\nApproximate analytic solutions for resonant scattering exist in certain limits. \\citet{1973MNRAS.162...43H} showed that when most of the radiation is in the damping wings, the transfer equation reduces to the Poisson equation. However, their solution uses an ansatz to handle the boundary condition. To our knowledge, the errors introduced by this treatment have never been quantified. They attempt a separation of variables as $J(\\tau,\\sigma) = \\theta(\\tau) j(\\sigma)$ in spatial variable $\\tau$ and frequency variable $\\sigma$ (their Equations 16 and 23). The solutions for the eigenfunctions $\\theta(\\tau)$ and $j(\\sigma)$ then depend explicitly on the separation constant $\\lambda$. In order to satisfy the boundary conditions, the separation constant is shown to satisfy an eigenvalue equation of the form\n\\begin{eqnarray}\n\\lambda \\tan(\\lambda B) & = & \\frac{3}{2} \\phi \\Delta,\n\\label{eq:evalue}\n\\end{eqnarray}\nwhere $2B$ is the slab optical depth at line center, $\\phi$ is the line profile, and $\\Delta$ is the Doppler width. The key point is that the line profile depends on one of the coordinates: frequency. This causes the eigenvalues of the separation constant to be frequency-dependent. Thus, the separation ``constant'' is not constant, and the function does not satisfy the Poisson equation since the frequency derivatives will act on the separation ``constant\", giving extra terms. In the limit of large optical depth $B$, they approximate the eigenvalues as $\\lambda_n B \\simeq \\pi (n-1\/2)$, which gives zero mean intensity at the surface. Their Equation 34 subsequently allows $\\lambda$ to have a small deviation from the above expression, which is explicitly frequency dependent. This allows a nonzero intensity at the surface, but at the cost of rendering the separation of variables assumption invalid. Our treatment, using the correct boundary condition, quantifies the errors in this ansatz.\n\nSeveral other works have followed \\citet{1973MNRAS.162...43H}. \\citet{1990ApJ...350..216N} extends the solution to media of intermediate optical depth, including the effects of scattering in the Doppler core of the line. \\citet{2006ApJ...649...14D} generalize the same problem to spherical geometry, as is used here. \\cite{2020MNRAS.497.3925L} generalize both the slab and sphere solutions to arbitrary power-law density and emissivity profiles. Each of these works, and several others \\citep{2020ApJS..250....9S, 2021MNRAS.504...89T}, use either the same surface boundary condition and ansatz as \\citet{1973MNRAS.162...43H}, or use a solution that does not handle the frequency-dependence of the boundary condition. Our novel steady-state solution involves a frequency-dependent correction to the solution that fixes an observed excess at the spectral peaks as compared with Monte Carlo, which is present in many of the works cited above.\n\nThe motivation for including time-dependence in the transfer equation is to characterize the distribution of photon escape times, which is needed to calculate the radiation moments in the Monte Carlo simulation. Additionally, steady-state solutions to this problem are not always sufficient to describe all the physics of \\lya transport. Time-variable, optically-thick environments necessitate a time-dependent solution to include the dynamic effects of \\lya transfer. These include the optical afterglow of gamma-ray bursts \\citep{2010ApJ...716..604R} and \\lya sources redshifted by cosmological expansion \\citep{2011MNRAS.418..853X}, among others.\n\n\\section{STEADY-STATE SOLUTION}\n\\label{sec:steadystate}\n\nConsider a sphere of radius $R$ with uniform density $n_{\\rm sc}$, luminosity $L$, and line-center optical depth $\\tau_0$, containing a point source of photons. We aim to find the intensity within the sphere as a function of radius and photon frequency. The point source is assumed to be a delta function in space and photon frequency. Photons of frequency $\\nu$ near the line center frequency $\\nu_0$ are considered. The photon frequency of the source is $\\nu_s$. The Doppler width is $\\Delta = \\nu_0 v_{\\rm th}\/c$, where $v_{\\rm th}=\\sqrt{2k_{\\rm B}T\/m_{\\rm H}}$ is the thermal speed of hydrogen atoms of mass $m_{\\rm H}$ and temperature $T$, and $c$ is the speed of light. The photon frequency in Doppler units is $x = (\\nu-\\nu_0)\/\\Delta$, and $x_s = (\\nu_s - \\nu_0)\/\\Delta$ is the corresponding source frequency. For upper-state de-excitation rate $\\Gamma$, the ratio of natural to Doppler broadening is $a=\\Gamma\/(4\\pi \\Delta)$. For the \\lya transition and T=$10^4$ K, $a = 4.72\\times 10^{-4}$. $\\mathcal{H}(x,a)$ is the Voigt function, and the Voigt line profile is $\\phi = \\mathcal{H}(x,a)\/(\\sqrt{\\pi} \\Delta)$, which is normalized as $\\int d\\nu\\, \\phi(\\nu) = 1$. The line center optical depth is $\\tau_0 = kR\/(\\sqrt{\\pi}\\Delta)$, where $k = n_{\\rm sc} \\pi e^2 f\/(m_e c)$. Here, $e$ and $m_e$ are the charge and mass of the electron, and $f$ is the oscillator strength of the transition, which is 0.4162 for \\lya \\citep{1986rpa..book.....R}.\n\nAppendix \\ref{app:rteqn_derivation} contains a derivation of the transfer equation for convenience. Starting with the full transfer equation, Equation (\\ref{eq:finaleqn}), ignoring photon destruction and including a photon emission term given by Equation (\\ref{eq:jem}), the steady-state transfer equation is\n\\begin{eqnarray}\n\\nabla^2 J + \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2 J}{\\partial \\sigma^2} & = & \n- \\frac{ \\sqrt{6} kL}{4\\pi \\Delta^2} \\delta^3(\\vec{x} - \\vec{x}_s) \\delta (\\sigma - \\sigma_{\\rm s}).\n\\label{eq:rt_no_destr}\n\\end{eqnarray}\nwhere $J$ is the mean intensity, the spatial variable is $\\vec{x}$, and $\\vec{x}_s$ is the position of the source. We will consider only the case where $\\vec{x}_s=0$. Following \\citet{1973MNRAS.162...43H}, we have used a change of variables in photon frequency from $x$ to $\\sigma$,\n\\begin{eqnarray} \\label{eq:int_change_of_variables}\n\\sigma(x) = \\sqrt{\\frac{2}{3}}\\int_0^x \\frac{dx}{\\phi(x) \\Delta} \\approx \\sqrt{\\frac{2}{3}}\\frac{\\pi}{a}\\frac{x^3}{3}, \n\\end{eqnarray}\nwhere the approximation is applicable in the damping wing. From Equation (\\ref{eq:app:line_profile_wing}), the line profile is then approximately \n\\begin{eqnarray} \\label{eq:line_profile_approx}\n\\phi \\approx \\frac{a}{\\pi x^2 \\Delta} \\approx \\frac{1}{3 \\Delta}\\left(\\frac{2a}{\\pi}\\right)^{1\/3}|\\sigma|^{-2\/3}.\n\\end{eqnarray}\nIn Equation (\\ref{eq:rt_no_destr}), $\\sigma_s \\equiv \\sigma(x_s)$ is the photon frequency of the source. $\\sigma_s$ is interchangeable with $x_s$ and $\\nu_s$ in Doppler widths or Hz, respectively. Balancing the two terms on the left-hand side of Equation (\\ref{eq:rt_no_destr}) gives $\\sigma \\ {\\sim}\\ \\tau_0$, or $x_{\\rm peak}\\ {\\sim}\\ (a\\tau_0)^{1\/3}$. The boundary condition of no incoming intensity at the surface \\citep{1986rpa..book.....R} is\n\\begin{eqnarray}\nJ & = & \\sqrt{3} H\n\\label{eq:bc}\n\\end{eqnarray}\nat $r=R$. \n\nA solution for the mean intensity $J_d$ which is divergent at the origin\nand $\\sigma=\\sigma_s$ and is zero at infinity is presented in \\citet{1990ApJ...350..216N}. Here it is extended to spherical geometry and generalized to allow emission frequencies away from line center: \n\\begin{eqnarray}\nJ_{\\rm d} & = & \n\\left(\\frac{\\sqrt{6}k^2L}{16\\pi^3 \\Delta^3}\\right)\\left(\\frac{1}{(kr\/\\Delta)^2 + (\\sigma - \\sigma_{\\rm s})^2}\\right)\n\\label{eq:Jd}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH_{\\rm d} & = & - \\frac{1}{3k\\phi} \\frac{\\partial J_d}{\\partial r}\n=  \\left( \\frac{1}{3k\\phi} \\right) \n\\left( \\frac{ \\sqrt{6}k^3L }{ 8\\pi^3 \\Delta^4} \\right)\n\\left( \\frac{k r\/\\Delta}{ \\left[ (kr\/\\Delta)^2 + (\\sigma-\\sigma_{\\rm s})^2 \\right]^2 } \\right).\n\\label{eq:Hd}\n\\end{eqnarray}\nThis solution is useful as a simple analytic formula. However, it is not a good approximation to the true solution, as it is too large at $r=R$ by a factor of $J_{\\rm d}(R,\\sigma)\/ H_{\\rm d}(R,\\sigma) \\sim a\\tau_0\/x^2 \\sim (a\\tau_0)^{1\/3} \\gg 1$ and does not adhere to the correct boundary condition. This solution is included in Figure \\ref{fig:sol_mc_residual_0} for illustration.\n\nA better approximation to the true solution has been derived by \\citet{2006ApJ...649...14D}, who generalized the closed-form solution in slab geometry found in \\citet{1973MNRAS.162...43H}. It satisfies a $J=0$ boundary condition at $r=R$. Again, we generalize their solution to allow emission at frequency $\\sigma_{\\rm s}$ away from line center. The result can be written as a sum over spatial modes,\n\\begin{eqnarray} \\label{eq:J0_sum}\nJ_0 = \\frac{\\sqrt{6}L}{16\\pi \\Delta} \\frac{1}{R^2}\\sum_{n=1}^{\\infty}n\\frac{\\sin{\\kappa_n r}}{\\kappa_n r}\\exp{\\left(\\frac{-\\kappa_n \\Delta}{k}|\\sigma - \\sigma_s|\\right)},\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:H0_sum}\nH_0 = - \\frac{1}{3k\\phi} \\frac{\\partial J_0}{\\partial r} = -\\frac{1}{3k\\phi}\\frac{\\sqrt{6}L}{16\\pi\\Delta} \\frac{1}{R^2}\\sum_{n=1}^{\\infty}n\\left(\\frac{\\cos{\\kappa_n r}}{r} - \\frac{\\sin{\\kappa_n r}}{\\kappa_n r^2}\\right)\\exp{\\left(\\frac{-\\kappa_n \\Delta}{k}|\\sigma - \\sigma_s|\\right)},\n\\end{eqnarray}\nwhere $\\kappa_n=n\\pi\/R$. These can be summed to give the closed form expressions\n\\begin{eqnarray}\nJ_0 & = & \\frac{\\sqrt{6}L}{32\\pi^2 \\Delta}\n\\frac{1}{Rr}\n\\left( \n\\frac{ \\sin(\\pi r\/R) }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R)}\n\\right)\n\\label{eq:J0}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nH_0 & = &\\frac{1}{3k\\phi}\n\\frac{\\sqrt{6}L}{32\\pi^2 \\Delta}\n\\frac{1}{Rr^2}\n\\left( \n\\frac{ \\sin(\\pi r\/R) }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R)}\n\\right. \\nonumber \\\\ & & \\left. - \\left( \\frac{\\pi r}{R} \\right)\n\\frac{ \\cos(\\pi r\/R) }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R)}\n+ \\left( \\frac{\\pi r}{R} \\right)\n\\frac{ \\sin^2(\\pi r\/R) }{ \\left[ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R) \\right]^2 }\n\\right).\n\\label{eq:H0}\n\\end{eqnarray}\nThese solutions agree with Equations (\\ref{eq:Jd}) and (\\ref{eq:Hd}) when the arguments of the trigonometric and hyperbolic functions are small. Again $J_0 \\gg H_0$, except near $r=R$, where it goes to zero. The flux at $r=R$ can be written\n\\begin{eqnarray}\n\\nonumber\nH_0(R, \\sigma) & = & - \\frac{1}{3k\\phi}\n\\frac{\\sqrt{6}L}{16\\pi \\Delta}\n\\frac{1}{R^3}\n\\sum_{n=1}^{\\infty} \nn (-1)^n \\exp{\\left(\\frac{-\\kappa_n \\Delta}{k}|\\sigma - \\sigma_s|\\right)}\\\\\n& = &  \\frac{1}{3k\\phi}\n\\frac{\\sqrt{6}L}{32\\pi \\Delta}\n\\frac{1}{R^3}\n\\left( \n\\frac{ 1 }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] +1 }\n\\right).\n\\label{eq:H0surf}\n\\end{eqnarray}\nEquation (\\ref{eq:H0}) will be shown to be a better approximation to the solution than Equation (\\ref{eq:Hd}). It is still valid near the delta function at $r=0$, but is also a better approximation at $r=R$. $J_0$ decreases exponentially, rather than as a power-law in frequency as it does for $J_d$, giving a much smaller flux in the line wings as compared to the divergent solution. \n\nIn order to enforce the boundary conditions, a different solution method is attempted here, namely a continuous Fourier expansion in the frequency variable $\\sigma$. The solution of this problem is split into two pieces: $J_0$ which includes the delta function source and satisfies $J=0$ at $r=R$, and $J_{\\rm bc}$ which allows the boundary condition $J=\\sqrt{3}H$ to be satisfied at $r=R$. The total solution is\n\\begin{eqnarray}\nJ(r,\\sigma) & = & J_0(r,\\sigma) + J_{\\rm bc}(r,\\sigma)\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:totalflux}\nH(r,\\sigma) & = & H_0(r,\\sigma) + H_{\\rm bc}(r,\\sigma).\n\\end{eqnarray}\nThe additional term $J_{\\rm bc}$ must then be a solution of the homogeneous equation\n\\begin{eqnarray} \\label{eq:diffeq}\n\\frac{\\partial^2J_{\\rm bc}}{\\partial r^2} + \\frac{2}{r} \\frac{\\partial J_{\\rm bc}}{\\partial r}\n+ \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2 J_{\\rm bc}}{\\partial \\sigma^2} &= & 0\n\\end{eqnarray}\nwith no delta function source term, and it must allow the boundary conditions to be satisfied at the surface. Since $J_0(R,\n\\sigma)=0$, the surface boundary condition becomes\n\\begin{eqnarray}\nJ_{\\rm bc}(R,\\sigma) - \\sqrt{3} H_{\\rm bc}(R,\\sigma) & = \n\\sqrt{3} H_0(R,\\sigma).\n\\label{eq:bc2}\n\\end{eqnarray}\nInserting a frequency dependence $J_{\\rm bc} \\propto e^{is\\sigma}$, for ``wavenumber\" $s$, gives the equation for modified spherical Bessel functions of the first kind, $i_0(z)=\\sinh(z)\/z$ for the radial dependence. The solution can then be represented as\n\\begin{eqnarray}\nJ_{\\rm bc}(r,\\sigma) & = & \n\\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} e^{is\\sigma} A(s) \n\\frac{i_0(krs\/\\Delta)}{i_0(kRs\/\\Delta)},\n\\label{eq:Jbc}\n\\end{eqnarray}\nwhere $A(s)$ is the Fourier amplitude. Inserting Equation (\\ref{eq:Jbc}) into Equation (\\ref{eq:bc2}) leads to the following equation for the Fourier amplitudes,\n\\begin{eqnarray}\n\\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} e^{is\\sigma} A(s)\n\\left[ 1 + \\left( \\frac{s}{\\sqrt{3} \\Delta \\phi} \\right) \\left( \\frac{i_0^\\prime(kRs\/\\Delta)}{i_0(kRs\/\\Delta)} \\right) \\right]\n& = & \\sqrt{3} H_0(R,\\sigma).\n\\label{eq:bc3}\n\\end{eqnarray}\nDiscretization of Equation (\\ref{eq:bc3}) for frequency variables $\\sigma_i$ and wavenumbers $s_j$\nleads to a set of coupled linear equations for the $A(s_j)$. We use equally-spaced points $\\delta \\sigma = 2\\sigma_{\\rm max}\/(N-1)$ and $\\delta s = 2\\pi\/(N\\delta \\sigma)$, where $N$ is the number of points for each grid. The maximum frequency is set as $\\sigma_{\\rm max} = {\\rm constant} \\times \\tau_0$, for a large enough constant that the end of the frequency grid is at such small intensities that it does not affect the solution except close to the boundaries. The number of points was increased until the solution was well-resolved near line center, and only became inaccurate close to the boundaries. We found that values of $N=4097$ and $\\sigma_{\\rm max} = 60 \\tau_0$ were sufficient. Given the Fourier amplitudes $A(s)$, $J_{\\rm bc}$ is computed using Equation (\\ref{eq:Jbc}), and the flux is given by\n\\begin{eqnarray}\nH_{\\rm bc}(r,\\sigma) & = & -\\frac{1}{3k\\phi}\n\\frac{\\partial J_{\\rm bc}(r,\\sigma)}{\\partial r}\n= -\\frac{1}{3k\\phi}\n\\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} e^{is\\sigma} A(s) \n\\left( \\frac{ks}{\\Delta} \\right) \n\\left( \\frac{i_0^\\prime(krs\/\\Delta)}{i_0(kRs\/\\Delta)} \\right).\n\\label{eq:Hbc}\n\\end{eqnarray}\nThe Bessel functions are finite at the center and rise steeply toward the surface when $kRs\/\\Delta \\gg 1$. \n\n\\subsection{Scaling with Line Center Optical Depth $\\tau_0$}\n\nWe now estimate the scaling of $H_{\\rm bc}$ with $\\tau_0$. In the limit $J_{\\rm bc} \\gg H_{\\rm bc}$, we find that $J_{\\rm bc} \\approx \\sqrt{3} H_0$ from Equation (\\ref{eq:bc2}). We  estimate $H_{\\rm bc}$ from $J_{\\rm bc}$ using Equation (\\ref{eq:Hbc}) as\n\\begin{eqnarray}\nH_{\\rm bc}(R, \\sigma) \\approx \\frac{1}{\\sqrt{3}k\\phi}\\frac{ks}{\\Delta}H_0\\ {\\sim}\\ H_0 s \\frac{x^2}{a}\\ {\\sim}\\ H_0 \\frac{1}{\\tau_0}\\frac{(a\\tau_0)^{2\/3}}{a}\\ {\\sim}\\ H_0 (a\\tau_0)^{-1\/3},\n\\end{eqnarray}\nwhere we have used $s\\ {\\sim}\\ 1\/\\sigma\\ {\\sim}\\ 1\/\\tau_0$ so that\n\\begin{eqnarray} \\label{eq:hbc_scaling}\n\\frac{H_{\\rm bc}(R, \\sigma)}{H_0(R, \\sigma)} \\propto (a\\tau_0)^{-1\/3}.\n\\end{eqnarray}\nAt large $\\tau_0$, it is expected that the correction term will be small, but it will become increasingly important as $\\tau_0$ decreases. Our solution of the transfer equation is only valid when the peaks of the spectral energy distribution lie well outside of the Doppler core, i.e., for large $\\tau_0$. The value of $x$ at which the Doppler and Lorentzian components of the line profile are equal is $x_{\\rm cw}=3.3$. Setting $x_{\\rm cw} = x_{\\rm peak}$ and solving for $\\tau_0$ gives the value at which the peak of the spectrum falls at the Doppler core boundary, which is $\\tau_{\\rm cp} \\approx 10^5$ (``core-peak'' optical depth). Hence $H_{\\rm bc}\/H_0 {\\sim} (\\tau_{\\rm cp}\/\\tau_0)^{1\/3}$ is large at $\\tau_0 \\leq \\tau_{\\rm cp}$ and decreases relatively slowly as $\\tau_0$ increases. Additionally, the optical depth at $x_{\\rm peak}$ is proportional to $(a\\tau_0)^{1\/3}$, so photons here become optically thin when $a\\tau_0 {\\sim} 1$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{final_residual.pdf}\n    \\caption{Spectrum $P(x)$ vs. frequency $x$ for $\\tau_0 = 10^7$ with $x_s = 0$. At $T=10^4$ K, this corresponds to $a\\tau_0=4.7 \\times 10^3$. The legend to the right describes each line style. For reference, $\\rm H_{0}$: fiducial solution, $\\rm H_{d}$: divergent solution, $\\rm H_{bc}$: our boundary condition correction to the fiducial solution. The top panel is linear scale, the middle panel is log scale with $|\\rm H_{bc}|$ shown instead of $\\rm H_{bc}$, and the bottom panel is the residual of each solution with Monte Carlo.} \n    \\label{fig:sol_mc_residual_0}\n\\end{figure}\n\n\\subsection{Comparison to Monte Carlo}\nThe Monte Carlo method is used to solve the transfer equation numerically in order to compare the analytic approximation to an ``exact'' solution. This method is valid at all $\\tau_0$, being restricted only by the computational demand, which grows proportionally to the number of photons used and $\\tau_0$. For each simulation, a total of ${\\sim}10^6$ photon packets are initialized at a monochromatic source frequency $x_s$ and are allowed to propagate through the sphere until escaping, at which point their positions, outgoing angles, and escape frequencies are tabulated to obtain the spectrum at the surface of the spherical simulation domain. A constant temperature of $T=10^4\\ \\rm K$ is set for the gas. Frequency redistribution is calculated at each scattering. In the comparisons shown in this section, the raw photon data is binned in frequency to obtain spectra. Further details of the Monte Carlo implementation are discussed in \\citet{2017ApJ...851..150H}.\n\nWe now compare each of the previously-discussed solutions for surface flux to the Monte Carlo results. The spectrum $P(x)$ is defined as the specific luminosity at the surface divided by the source luminosity, or\n\\begin{eqnarray} \\label{eq:prob_spectrum}\nP(x) = \\frac{16\\pi^2R^2H(R, x)\\Delta}{L}.\n\\end{eqnarray}\nThis is normalized so that $\\int P(x)dx = 1$. Since $H(R, x)$ is per $d\\nu$, a factor of $\\Delta$ gives the expression the correct units. \n\nIn Figure \\ref{fig:sol_mc_residual_0}, the Monte Carlo spectrum is shown along with that of the solutions $H_{\\rm d}$, $H_0$, and $H_0 + H_{\\rm bc}$ for an optical depth of $\\tau_0 = 10^7$ and photons emitted at line center $\\rm x_s = 0$.  Note that the errorbars shown on the Monte Carlo data points are proportional to $\\sqrt{N}$, with $N$ being the photon count in each frequency bin, since the photons are all equally weighted. The $H_{\\rm bc}$ term is negative at the peak of the spectrum and positive in the line wing such that, when added to $H_0$, it corrects for the apparent excess of flux in the peaks of the spectrum. The solution with the correct frequency-dependent boundary condition enforced, $H_0 + H_{\\rm bc}$, has lower residuals to Monte Carlo results than the other solutions, especially in the line wing. The boundary term corrects the deficit of $H_0$ in the line wings, further improving agreement with the numerical result. The residuals to the $H_0$ solution are a close match to the $H_{\\rm bc}$ term, since the Monte Carlo represents the ``true'' solution, $H$, and $H_{\\rm bc} = H - H_0$. It is evident that the divergent solution $H_{\\rm d}$ fails in the line wings. Also note that the ``V'' shape of the solution in the line core is due to the low number of points plotted, as the analytic solutions are not valid in this frequency regime since they utilize the damping wing approximation of the Voigt line profile.\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{tau_threepanel.pdf}\n    \\caption{The same as Figure \\ref{fig:sol_mc_residual_0}, but for $\\tau_0 = 10^5$ (top panel), $10^6$ (middle panel), and $10^7$ (bottom panel). The $x$ and $y$ axes are scaled by $(a\\tau_0)^{1\/3}$.}\n    \\label{fig:sol_mc_tau}\n\\end{figure}\n\nThe size of $H_{\\rm bc}$ is dependent on $\\tau_0$. $H_{\\rm bc}$ is significant even at $\\tau_0 {\\sim} 10^7$ where the $H_0$ solution is expected to perform well, i.e., photons are pushed further out into the wing where the simplifying assumptions made in the derivation of the differential equation are a better approximation. \n\nIn Figure \\ref{fig:sol_mc_tau} we show the solutions alongside Monte Carlo, now for three different optical depths $\\tau_0=10^5, 10^6$, and $10^7$. From Equation (\\ref{eq:hbc_scaling}), the size of the term $H_{\\rm bc}$ should become smaller with larger optical depths, following a $(a\\tau_0)^{-1\/3}$ scaling. Indeed, agreement between Equation (\\ref{eq:H0surf}) and the Monte Carlo points in Figure \\ref{fig:sol_mc_tau} improves as $\\tau_0$ increases, with $H_{\\rm bc}$ providing a fractionally smaller correction to $H_0$. One factor of $(a\\tau_0)^{1\/3}$ has been scaled out of the x-axis such that the peaks of the distributions are horizontally aligned. This scaling has also been applied to the y-axis to preserve normalization of the escape probability. At lower $\\tau_0$, the scattering of photons within the Doppler core of the line becomes important, but our analytic solution does not include this effect. The effects of line core scattering can be seen in the Monte Carlo data for $\\tau_0=10^5$ and, to a lesser extent, $\n\\tau_0=10^6$.\n \n \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{xinit_threepanel.pdf}\n    \\caption{The same as Figure \\ref{fig:sol_mc_residual_0}, but for $\\tau_0=10^7$ and $\\rm x_s = 0$ (top panel), $6$ (middle panel), and $12$ (bottom panel). The optical depth at each of these source frequencies is $\\tau_s = 10^7, 77,$ and $19$, respectively.} \n    \\label{fig:sol_mc_xinit}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{xinit_threepanel_tau1e6.pdf}\n    \\caption{The same as Figure \\ref{fig:sol_mc_xinit}, but at a lower optical depth $\\tau_0 = 10^6$. The shift $x_s$ is a much larger fraction of the distance to the spectral peak $(a\\tau_0)^{1\/3}$, and thus the asymmetry in the spectrum is much larger. The optical depth at the source frequency is $\\tau_s = 10^6$, $7.7$, and $1.9$ for $x_s=0, 6,$ and $12$, respectively. } \n    \\label{fig:sol_mc_xinit_lowtau}\n\\end{figure}\n\nNext, we show $P(x)$ for $x_s \\neq 0$. Photons initialized further out in the line wing have larger mean free paths. The larger spatial diffusion implies greater escape probability for these photons. In the limit that $|\\rm x_s|$ becomes large, the distribution becomes a delta function at $x_s$ as all photons escape the sphere without scattering. \n\nFigure \\ref{fig:sol_mc_xinit} shows calculations performed for $x_s = 0, 6$, and $12$ and $\\tau_0=10^7$. The asymmetry of the spectrum is slight for $\\rm x_s = 6$ where $\\tau(x_s)=77$, but is larger for $\\rm x_s=12$ outside the line core where $\\tau(x_s)=19$. It is seen here that the difference between the Monte Carlo data and $H_0$ becomes larger as $\\rm x_s$ increases. Thus, for large $|x_s|$, inclusion of $H_{\\rm bc}$ is more important.\n\nFigure \\ref{fig:sol_mc_xinit_lowtau} shows emission away from line center at the same values of $x_s$ as in Figure \\ref{fig:sol_mc_xinit}, but for $\\tau_0=10^6$ rather than $10^7$. The difference between the left and right side of the escaping spectrum is now substantial since $(a\\tau_0)^{-1\/3}$ increased by a factor of ${\\sim}2$. It is clear from the figure that as $x_s$ extends further into the wing, the spectrum becomes more strongly peaked in frequency. Additionally, since the sphere is increasingly optically thin in the wing, we expect there to be a stronger disagreement with the Monte Carlo as the analytic solution assumed large optical depths.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{xinit.pdf}\n    \\caption{$P_{\\rm bc}(x) = 16\\pi^2R^2H_{\\rm bc}(R, x)\\Delta\/L$ (Equation (\\ref{eq:prob_spectrum}) with $H \\rightarrow H_{\\rm bc}$) vs. frequency $x$. Each panel shows $\\sigma_s$ further from line center, labeled by $\\sigma_s=0, \\tau_0, 2\\tau_0$. The solution is shown at a range of line center optical depths between $\\tau_0=10^5$ and $10^9$ with solid lines. The location of the core-wing boundary $x_{cw}$ for each $\\tau_0$ is shown with dashed vertical lines which match the color of the corresponding solution. The axes are scaled to show that the size of the correction factor agrees with the predicted $(a\\tau_0)^{-1\/3}$ scaling.}\n    \\label{fig:xinit}\n\\end{figure}\n\nFigure \\ref{fig:xinit} shows how the correction $P_{\\rm bc}(x)$, Equation (\\ref{eq:prob_spectrum}) with $H \\rightarrow H_{\\rm bc}$, scales with both $\\tau_0$ and $x_s$. In this figure, $\\sigma_s$ is shifted by integer multiples of $\\tau_0$ (Equation \\ref{eq:change_of_variables}) in each panel such that the source falls near the peak of the spectrum for each $\\tau_0$. For clarity, only the $x > 0$ side of the spectrum is shown. From Equation (\\ref{eq:hbc_scaling}), it is expected that the fractional size of $H_{\\rm bc}$ relative to $H_0$ should become smaller with larger optical depths following $(a\\tau_0)^{-1\/3}$. This factor has been scaled out of the figure such that solutions for different $\\tau_0$ and the same $\\sigma_s$ should show close agreement in scale on the figure's vertical axis if the relation holds. Indeed, the scaled solutions converge as $\\tau_0$ becomes larger, indicating agreement with the $(a\\tau_0)^{-1\/3}$ scaling. The remaining discrepancy present in the vertical axis for fixed $\\sigma_s$ results from $x_{\\rm peak}$ becoming close to $x_{\\rm cw}$; at lower $\\tau_0$, this causes the line profile approximation in the wing, Equation (\\ref{eq:app:line_profile_wing}), to break down. From this, we conclude that the errors introduced by the incorrect separation of variables in \\citet{1973MNRAS.162...43H}, \\citet{1990ApJ...350..216N}, \\citet{2006ApJ...649...14D} and others are indeed proportional to $(a\\tau_0)^{-1\/3}$.\n\n\\section{Time-Dependent Diffusion}\n\\label{sec:time_dependent}\n\nIn order to understand how long it takes for the photons to escape the uniform sphere of gas we must reintroduce the time dependence of the diffusion equation, which was ignored in the steady-state calculations in Section \\ref{sec:steadystate}. To obtain the radiative intensity $I=dE\/(dAdtd\\Omega d\\nu)$ on timescales comparable to the light-crossing time $t_{\\rm lc} = R\/c$, the time-dependent response to a delta function impulse is found. This allows the distribution of photon escape times (the ``wait time distribution'') to be characterized. For simplicity, a $J=0$ boundary condition will be used in the following derivations, which is a rough approximation for $a\\tau_0 \\gg 1$. \n\n\\subsection{Derivation of the time-dependent solution}\n\\label{subsec:time_dependent:background}\n\nThe emissivity for an impulsive source with energy E, source position $\\vec{x}_s$, and frequency $\\nu_s$ is derived in Appendix \\ref{app:rteqn_derivation}. Considering a photon source at $\\vec{x}_s=0$, we have (Equation \\ref{eq:jem})\n\\begin{eqnarray}\nj_{\\rm em} & = & \\frac{E}{4\\pi} \\delta^3(\\vec{x}) \\delta(\\nu-\\nu_{\\rm s})\\delta (t) ,\n\\label{eq:jem2}\n\\end{eqnarray} \nThe resulting equation for $J(r,\\sigma,t)$ is\n\\begin{eqnarray}\n \\frac{-3k\\phi}{c} \\frac{\\partial J}{\\partial t} + \\nabla^2 J + \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2 J}{\\partial \\sigma^2}\n& = & - \\frac{\\sqrt{6} kE}{4\\pi \\Delta^2} \\delta^3(\\vec{x}) \\delta (\\sigma - \\sigma_s ) \\delta (t).\n\\label{eq:diffusion_eqn}\n\\end{eqnarray}\nWe employ an expansion in terms of spherical Bessel functions in $r$ and Fourier transform in time. The zeroth spherical Bessel function is $j_0(x) = \\sin{x}\/x$. The expansion for $J(r, \\sigma, t)$ is then\n\\begin{eqnarray}\n\\label{eq:jrsigmat_expansion}\nJ(r, \\sigma, t) = \\sum_{n=1}^{\\infty} \\int_{-\\infty}^\\infty \\frac{d\\omega}{2\\pi} e^{-i\\omega t} j_0\\left(\\kappa_n r\\right) J(n, \\sigma, \\omega),\n\\end{eqnarray}\nwith\n\\begin{eqnarray} \\label{eq:jnsigmaomega}\nJ(n, \\sigma, \\omega) = \\frac{2\\kappa_n^2}{R} \\int_0^R dr\\ r^2 j_0(\\kappa_n r) \\int_{-\\infty}^\\infty dt\\ e^{i\\omega t} J(r, \\sigma, t).\n\\end{eqnarray}\nHere, $\\kappa_n = n\\pi\/R$ and $\\omega$ describes the time-dependence of $J$. Though it is written as a function of the photon frequency variable $\\sigma$, $J(r, \\sigma, t)$ is the specific mean intensity $dE\/(dA dt d\\nu)$ and is a distribution in $\\nu$. The Fourier coefficient $J(n, \\sigma, \\omega)$ has units $dE\/(dA d\\nu)$. Using Equation (\\ref{eq:jnsigmaomega}), we obtain\n\\begin{eqnarray} \\label{eq:diffusion_plugged_in}\n \\left( \\frac{3k\\phi}{c}i\\omega  -   \\kappa_n^2 \\right) J(n,\\sigma,\\omega)  &+& \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2J(n,\\sigma,\\omega)}{\\partial\\sigma^2} = -\\frac{2\\kappa_n^2}{R} \\frac{\\sqrt{6}}{4\\pi} \\frac{kE}{\\Delta^2} \\frac{1}{4\\pi} \\delta(\\sigma - \\sigma_s).\n\\end{eqnarray}\nAt $\\sigma=\\sigma_s$, continuity must be enforced,\n\\begin{eqnarray} \\label{eq:matching_condition_1}\nJ(n, \\sigma^-, \\omega) = J(n, \\sigma^+, \\omega),\n\\end{eqnarray}\nand the discontinuity in $dJ\/d\\sigma$ due to the source is\n\\begin{eqnarray} \\label{eq:matching_condition_2}\n\\frac{\\partial J(n, \\sigma^+, \\omega)}{\\partial \\sigma} - \\frac{\\partial J(n, \\sigma^-, \\omega)}{\\partial \\sigma} & = & \n- \\frac{\\sqrt{6}}{8} n^2 \\frac{E}{kR^3}.\n\\end{eqnarray}\nAt large values of $\\sigma$ the line profile $\\phi$ is small and Equation (\\ref{eq:diffusion_plugged_in}) becomes\n\\begin{eqnarray} \\label{eq:diffusion_at_large_sigma}\n\\frac{\\partial^2J}{\\partial\\sigma^2} \\approx \\frac{\\Delta^2\\kappa_n^2}{k^2} J,\n\\end{eqnarray}\nwhich has solutions \n\\begin{eqnarray}\nJ(n, \\sigma, \\omega)\\ {\\sim}\\ e^{\\pm \\kappa_n \\sigma \\Delta \/ k}.\n\\end{eqnarray}\nThis approximate solution implies a boundary condition at large $|\\sigma|$\n\\begin{eqnarray} \\label{eq:single_j_derivative}\n\\frac{\\partial J}{\\partial \\sigma} = \\mp \\frac{\\kappa_n\\Delta}{k} J,\n\\end{eqnarray}\nwhere a negative sign is taken for large $+\\sigma$ and a positive sign is taken for large $-\\sigma$ to choose the finite solution as $|\\sigma|\\to \\infty$. Numerical integrations are performed inward toward $\\sigma_s$ over several domains: from large $|\\sigma|$ to $\\sigma_s$, from large $|\\sigma|$ to 0, and from 0 to $\\sigma_s$, depending on whether $\\sigma_s$ is positive or negative. If $\\sigma_s=0$, just two integrations are performed inward from large $|\\sigma|$ to 0. Initial values for integration are obtained either by setting $J=1$ and $dJ\/d\\sigma$ from Equation (\\ref{eq:single_j_derivative}) at large $|\\sigma|$ or by matching $J$ and $dJ\/d\\sigma$ at 0. This gives $J$ and $J'$ on either side of $\\sigma_s$, where a prime indicates the derivative $\\partial\/\\partial \\sigma$. By enforcing the matching conditions, Equations (\\ref{eq:matching_condition_1}) and (\\ref{eq:matching_condition_2}), the eigenfunctions $J(n, \\sigma, \\omega)$ are obtained over the domain of photon frequencies $\\sigma$. Since the solutions are linear in the starting conditions, only two integrations with different starting values are necessary.\n\nWe now wish to reconstruct the specific mean intensity $J(r, \\sigma, t)$. While one might expect this could be expressed as a sum over eigenmodes, the analysis presented in Appendix \\ref{app:wkb} suggests this treatment is incomplete in the case where $x_s \\neq 0$ and the solution is asymmetric about the line center. This ansatz does, however, roughly agree with Monte Carlo results for $x_s=0$ based on numerical calculations of this result.\n\nLet us define the damping rate to be $\\gamma \\equiv i\\omega$, which is real and positive for damped solutions. At the eigenvalues $\\gamma=\\gamma_{nm}$, the response $J(n, \\sigma, \\omega)$ is resonant. We find that near these $\\gamma_{nm}$ poles an approximate expression for the resonant response of the eigenfunctions is\n\\begin{eqnarray} \\label{eq:jnsigmaomega_approx}\nJ(n,\\sigma,-i\\gamma) & \\simeq \\frac{ J_{nm}(\\sigma) }{\\gamma_{nm} - \\gamma} + C(\\gamma, \\sigma),\n\\end{eqnarray}\nwhere $C(\\gamma, \\sigma)$ varies slowly in $\\gamma$. If the $\\omega$-integral in Equation (\\ref{eq:jrsigmat_expansion}) could be closed at infinity and evaluated using the residue theorem, the result would be\n\\begin{eqnarray}\nJ(r,\\sigma,t) & \\simeq & j_0(\\kappa_n r) J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\end{eqnarray}\nSumming over all spatial modes $n$ and over all eigenmodes $m$ for a given $n$, we obtain\n\\begin{eqnarray} \\label{eq:Jrsigmat}\nJ(r,\\sigma,t) & = & \\sum_{n=1}^\\infty j_0(\\kappa_n r)  \\sum_{m=1}^{\\infty} J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\end{eqnarray}\nThis ansatz captures the contributions from $n \\times m$ simple poles. Taking a derivative with respect to $r$ and evaluating at the surface $r=R$, we use\n\\begin{eqnarray}\n\\frac{dj_0(\\kappa_n r)}{dr} \\bigg\\rvert_R & =& \\frac{d}{dr} \\left[ \\frac{\\sin(\\kappa_n r)}{\\kappa_n r} \\right]\\bigg\\rvert_R\n=  \\left( \\frac{\\cos(\\kappa_n R)}{R} - \\frac{\\sin(\\kappa_n R)}{\\kappa_n R^2} \\right)  =  \\frac{(-1)^n}{R}\n\\end{eqnarray}\nto obtain the flux, which is\n\\begin{eqnarray}\nF(R,\\sigma,t) & =& - \\frac{4\\pi}{3k\\phi} \\frac{dJ(R,\\sigma,t)}{dr} \n= - \\frac{4\\pi}{3k\\phi R}  \\sum_{nm} (-1)^n J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\end{eqnarray}\nMultiplying by $4\\pi R^2$ gives the energy per time per frequency emerging from the sphere to be\n\\begin{eqnarray}\n\\frac{dE}{dtd\\nu} & = & - \\frac{16\\pi^2 R}{3k\\phi}  \\sum_{nm} (-1)^n J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\label{eq:dEdtdnu}\n\\end{eqnarray}\nIntegrating over time yields a factor $1\/\\gamma_{nm}$, and by integrating over $d\\nu$ we find\n\\begin{eqnarray} \\label{eq:sum_rule}\nE & = &  \\sqrt{ \\frac{3}{2} } \\frac{16\\pi^2R\\Delta^2}{3k} \\sum_{nm} (-1)^{n+1} \\gamma_{nm}^{-1} \\int d\\sigma J_{nm}(\\sigma).\n\\end{eqnarray}\nThis non-trivial ``sum rule'' provides a check on the values of $\\gamma_{nm}$ and $J_{nm}(\\sigma)$. This expression can also be written as\n\\begin{eqnarray}\n1 & =& \\sum_{nm} P_{nm},\n\\label{eq:sumrule}\n\\end{eqnarray}\nwhere the contribution of each mode is\n\\begin{eqnarray} \\label{eq:pnmsoln}\nP_{nm} & \\equiv & \\sqrt{ \\frac{3}{2} } \\frac{16\\pi^2R\\Delta^2}{3kE}  (-1)^{n+1} \\gamma_{nm}^{-1} \\int d\\sigma J_{nm}(\\sigma).\n\\end{eqnarray}\nThese coefficients $P_{nm}$ are negative for odd values of $n$ and positive for even $n$. The size of each contribution scales roughly as $0.5\/(m-7\/8)^{2\/3}$, with a weak dependence on $n$. This indicates the need for a large number of $n$ and $m$ to converge, in that it takes roughly ten times as many $m$ modes for a given $n$ to reduce the size of $P_{nm}$ by a factor of ${\\sim}$5. The physical intuition for the convergence of these terms is that the $n$ spatial terms must provide sufficient spatial resolution to resolve the steep falloff in intensity at the surface of the sphere. Additionally, the function falls off steeply in frequency in the line wing, which requires more $m$ terms in the Fourier sum to resolve (also see the discussion of Figure \\ref{fig:steadystate} in Section \\ref{subsec:steadystatemc}). \n\n\\subsection{Numerical calculation}\n\nWe seek now to calculate the eigenmodes $J_{nm}(\\sigma)$ and eigenfrequencies $\\gamma_{nm}$ for a given spatial $n$. These will be labelled by an index $m=1, 2, ...$. To measure the size of the response to detect where resonances occur, we sum the absolute value of $J(n,\\sigma,-i\\gamma)$ over the array $\\sigma$. We call this response $f$, and use the index $j$ to represent the value of the response at discrete points $\\gamma_j$ over a range of $\\gamma$. In places where $f_j > f_{j-1}$ and $f_j>f_{j+1}$, we have bracketed a resonance that occurs in the interval $(\\gamma_{j-1},\\gamma_{j+1})$. To refine the value of the eigenfrequency before continuing the sweep in $\\gamma$, we evaluate $f_{j-1}$, $f_j$, and $f_{j+1}$ at the points $(\\gamma_{j-1},\\gamma_{j},\\gamma_{j+1})$. Assuming the form in Equation (\\ref{eq:jnsigmaomega_approx}), a guess at the correct eigenvalue $\\gamma_{nm}$ can be calculated by linear interpolation from\n\\begin{eqnarray}\n\\gamma_{\\rm guess} &=& \\frac{b\\gamma_{j-1} - \\gamma_{j+1}}{b - 1},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nb &=& \\left(\\frac{f_{j} - f_{j-1}}{f_{j} - f_{j+1}}\\right)\\left(\\frac{\\gamma_{j+1}-\\gamma_{j}}{\\gamma_{j-1}-\\gamma_{j}}\\right).\n\\end{eqnarray}\nThe error of the current guess is $|\\gamma_{\\rm guess} - \\gamma_{j}|$. This error is reduced iteratively by replacing initial points $(\\gamma_{j-1},\\gamma_{j},\\gamma_{j+1})$ with closer estimates while the size of the response grows as the resonance is approached. After iterating an eigenvalue $\\gamma_{nm}$ to convergence, we now find the corresponding eigenfunction $J_{nm}(\\sigma)$. We evaluate Equation \\ref{eq:jnsigmaomega_approx} at two points $\\gamma_1$ and $\\gamma_2$ near the resonance, subtracting them and solving for $J_{nm}(\\sigma)$ to find\n\\begin{eqnarray}\nJ_{nm}(\\sigma) & \\simeq & \\frac{ J(n,\\sigma,-i\\gamma_1)  - J(n,\\sigma,-i\\gamma_2) }{ 1\/(\\gamma_{nm}-\\gamma_1) - 1\/(\\gamma_{nm}-\\gamma_2)},\n\\end{eqnarray}\n\\noindent where  $C(\\gamma, \\sigma)$ has cancelled in the difference.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{Jsoln_n1_m100.pdf}\n    \\caption{Eigenfunctions $J_{nm}(x)$ for the lowest-order spatial eigenmode $n=1$, and $m=1, ..., 100$ with $x_s=0$ and $\\tau_0=10^7$. The scale of the $J_{nm}(\\sigma)$ are set by the factor $E\/(kR^3)$, which here is ${\\sim}10^{-37}$ in ergs\/(cm$^2$ Hz).}\n    \\label{fig:jsoln}\n\\end{figure}\n\nThe form of a single eigenmode $J_{nm}(\\sigma)$ is oscillatory out to some turning point, $\\sigma_{\\rm tp}$, at which point the function becomes evanescent. The location of the turning point can be found by ignoring the delta-function discontinuity at the source frequency $\\sigma_s$ in Equation (\\ref{eq:diffusion_plugged_in}) and examining the resulting homogeneous differential equation. We obtain\n\\begin{eqnarray} \\label{eq:wkb_differential_eqn}\n\\frac{d^2J}{d\\sigma^2} & = & \\left[ \\left( \\frac{\\kappa_n \\Delta }{k} \\right)^2 - \\frac{3\\phi \\gamma\\Delta^2}{ck}\\right] J,\n\\end{eqnarray}\nwhere the line profile is approximated as in Equation (\\ref{eq:app:line_profile_wing}). When the coefficient on the right hand side is positive, exponential growth or decaying evanescent solutions are found. This occurs in the line wings. When the coefficient on the right hand side is negative, oscillatory solutions are found (propagation), which occurs near the line core. The boundary between propagation and evanescence occurs at the turning point, given by\n\\begin{eqnarray} \\label{eq:sigma_tp}\n\\sigma_{\\rm tp} & = & \\sqrt{\\frac{2a}{\\pi}}\\left( \\frac{k \\gamma}{ \\kappa_n^2 c \\Delta} \\right)^{3\/2}\n\\end{eqnarray}\nThus, to ensure accuracy in each term of Equation (\\ref{eq:Jrsigmat}), the bounds of $\\sigma$ must be set sufficiently far outside of $\\sigma_{\\rm tp}$ such that the function is small at the edges. The scale of an $e$-folding in $J_{nm}(\\sigma)$ is $k\/(\\kappa_n \\Delta) = \\tau_0 \/ (\\sqrt{\\pi} n)$, so a grid of $\\sigma$ is chosen that spans a large enough number of $e$-foldings that no oscillatory behavior is present at the boundaries of the domain.\n\nThe eigenfunction's oscillatory forms have varying amplitudes which sum in Equation (\\ref{eq:Jrsigmat}) to create the final form of the mean intensity. The largest contribution at late times always comes from the ($n=1, m=1$) lowest-order eigenfunction. Figure \\ref{fig:jsoln} shows a set of eigenfunctions $J_{nm}(\\sigma)$ to illustrate their relative scales for different $m$ at a fixed spatial eigenmode $n$. The overall scale of the $J_{nm}(\\sigma)$ are set by the factor $E\/(kR^3)$ with $E$ arbitrarily set to 1. For H atoms with $T=10^4$ K and $\\tau_0=10^7$, an eigenfunction has typical size ${\\sim} E a \/ \\left(R^2 \\Delta \\right) = 10^{-37}$ in units of specific mean intensity times time. Additional terms add smaller-magnitude, faster-oscillating components that lead to higher accuracy upon summation with the lower-order terms in Equation (\\ref{eq:Jrsigmat}). The oscillations of various modes must cancel in the Fourier sum, so many modes $m$ and $n$ are required for convergence to the solution. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{gamma_nm.pdf}\n    \\caption{Dimensionless decay time vs. mode number. Overlapping solid lines are plotted for each $n$. The thickness of the lines decrease with $n$ in order to clearly show where they are in agreement. These numerically-obtained resonant frequencies are compared with the analytic expression in Equation (\\ref{eq:gamma_nm}), shown as a dashed line. The parameters for the calculation are $\\tau_0=10^7$ and $x_s=0$. The value of $n$ is given by the color bar on the right-hand side.}\n    \\label{fig:gamma_nm}\n\\end{figure}\nThe values of the $\\gamma_{nm}$ can be described approximately with Equation (\\ref{eq:gamma_nm}). Their values depend on $m$, $n$, and other physical parameters according to \n\\begin{eqnarray} \\label{eq:gamma_nm}\n\\gamma_{nm} = 2^{-1\/3} \\pi^{13\/6} n^{4\/3}\\left(m-\\frac{7}{8}\\right)^{2\/3}\\frac{c}{R}(a\\tau_0)^{-1\/3}\n\\end{eqnarray}\nas shown by setting the denominator of Equation \\ref{eq:a3\/a2} to zero in WKB approximation of Appendix \\ref{app:wkb}. The power law in $m$ is weak, requiring up to $m=1000$ to reduce the scale of $\\gamma_{nm}^{\\ \\ -1}$ by two orders of magnitude. When sweeping through to find resonances, Equation (\\ref{eq:gamma_nm}) is used to set the scale of the sweep points $\\gamma_j$ to ensure no $\\gamma_{nm}$ are missed. The close agreement with the analytic expression shown in Figure \\ref{fig:gamma_nm} indicate that the numerical solutions are accurate.\n\n\\subsection{Comparison with Steady State and Monte Carlo} \\label{subsec:steadystatemc}\n\nWe now calculate the wait time distribution for escape from the sphere. This is obtained by integrating Equation (\\ref{eq:dEdtdnu}) over all frequencies. We find\n\\begin{eqnarray}\nP(t)  & = & \\sqrt{\\frac{3}{2}} \\frac{16\\pi^2 R \\Delta^2 }{3kE}     \\sum_{nm} (-1)^{n+1}  e^{-\\gamma_{nm}t} \\int d\\sigma J_{nm}(\\sigma) \n\\nonumber \\\\ & = &  \\sum_{nm} P_{nm} \\gamma_{nm} e^{-\\gamma_{nm}t},\n\\label{eq:waittime}\n\\end{eqnarray}\nwhich is normalized to unity. For a sufficiently large number of spatial modes $n$ and frequency modes $m$, the result of this sum can agree with Monte Carlo escape time distributions when $x_s=0$. The late-time distribution is simply an exponential falloff. The rate constant of the falloff is the lowest-order eigenfrequency, $\\gamma_{11}$, and its scale is determined by the coefficient $P_{11}$ as in Equation (\\ref{eq:pnmsoln}). Thus, an approximate ``fitting function'' that captures both the peak of the escape time distribution and the exponential falloff is\n\\begin{eqnarray} \\label{eq:fitting_function}\nP(t) = \\exp{\\left[-\\left(\\frac{t_{\\rm diff}}{t}\\right)^2\\right]} \\times \\gamma_{11} P_{11} e^{-\\gamma_{11}t}.\n\\end{eqnarray}\nThe first term represents the early-time distribution, which then transitions to an exponential falloff past a point $ct_{\\rm diff}\/R = (a\\tau_0)^{1\/3}$, where $t_{\\rm diff}$ is the characteristic diffusion timescale.\n\nIn Figure \\ref{fig:tau_scaling}, the late-time decay timescale of the wait time distribution is shown as a function of $\\tau_0$. It is shown that the time constant of exponential decay in fitted Monte Carlo escape time distributions converges with $\\gamma_{11}^{-1}$ at sufficiently high $\\tau_0$, following a $t\\propto(a\\tau_0)^{1\/3}$ scaling. The coefficient of this scaling ($0.51$) is within a factor of 2 of the approximate ``light-trapping time'' defined in \\citet{2020MNRAS.497.3925L}, which predicts $ct\/R=0.901(a\\tau_0)^{1\/3}$. At lower $\\tau_0$, the effects of line core scattering are most important, leading to a larger discrepancy in the characteristic escape timescale. Here, the Monte Carlo accurately includes the photons which scatter in the core many times before escaping, while the semi-analytic solution does not capture this behavior as it uses only the Lorentzian piece of the line profile, and also does not use enough spatial modes to accurately model the frequency regime near line center. However, as $\\tau_0$ grows, the effect of core scattering becomes smaller and the approximations hold, agreeing better with the expected $(a\\tau_0)^{1\/3}$ scaling \\citep{1975ApJ...201..350A} for the rate constant at late times. The excess in the Monte Carlo data points due to core scattering decreases exponentially at higher $\\tau_0$, and though these points are not shown at $\\tau_0=10^8$ and $10^9$ due to computational expense, it is expected that the fractional error between the Monte Carlo and the $(a\\tau_0)^{1\/3}$ scaling would be less than 2\\% at $\\tau_0=10^9$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{tau_scaling.pdf}\n    \\caption{Late-time exponential decay timescale as a function of $\\tau_0$. The Monte Carlo points on this figure are obtained by fitting only the exponential piece of the escape time distribution to obtain the rate constant.}\n    \\label{fig:tau_scaling}\n\\end{figure}\n\nWe now evaluate the time-integrated spectrum (fluence) of the response to an impulse and compare it with the solution for the $H_0$ steady-state spectrum (Equation \\ref{eq:H0surf}). Integrating Equation (\\ref{eq:dEdtdnu}) over all times and dividing by the energy $E$, we find the fluence\n\\begin{eqnarray} \\label{eq:spectrum}\nP(x) & = &  \\frac{16\\pi^2 R \\Delta}{3k\\phi E}  \\sum_{nm} (-1)^{n+1} \\gamma_{nm}^{-1} J_{nm}(\\sigma).\n\\end{eqnarray}\nIntegrating over $\\nu$ then gives unity as required by the sum rule in Equation (\\ref{eq:sumrule}). \n\nIn Figure \\ref{fig:steadystate}, the fluence for $x_s=0$ and $\\tau_0=10^7$ is shown for a sum up to $n=20$ and $m=500$, labelled ``Time-integrated'', and is compared with two analytic solutions: the steady-state $H_0$ solution (Equation \\ref{eq:H0surf}), labelled ``Steady State'', and the result for summing a finite number of spatial modes in the steady-state eigenfunction expansion as in the first line of Equation (\\ref{eq:H0surf}), labelled ``Partial Sum''. Additional spatial modes $n$ increase the solutions' accuracy in the core of the line. If more spatial modes are included, the agreement with the steady-state spectrum extends further toward the line core. If additional frequency modes are included, faster-oscillating terms are incorporated into the Fourier sum over eigenmodes which create more perfect cancellations with the lower-order terms, reducing the ``ringing'' seen in the time-integrated spectrum. Extending the calculation deep into the line core by adding additional spatial modes could have an impact on the accuracy of the escape time distribution, but this would primarily affect the distribution at early times since the late time distribution is determined by the lowest order modes. This was the motivation for choosing a comparatively low number of spatial eigenmodes with respect to the number of frequency eigenmodes calculated.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{steadystate.pdf}\n    \\caption{Fluence $P(x)$ vs. $x$ (see Equation \\ref{eq:spectrum}). Fluence is the radiation flux integrated over time. Steady-state and time-integrated spectra for $n=1, ..., 20$ and $m=1, ..., 500$ are shown with $x_s=0$ and $\\tau_0=10^7$. Note that the x-axis begins near the edge of the line core, as we are only concerned with the solutions' accuracy near the line wing.}\n    \\label{fig:steadystate}\n\\end{figure}\n\n\nIn Figure \\ref{fig:escape_time}, the escape time distributions calculated from Equation (\\ref{eq:waittime}) are shown alongside Monte Carlo and the fitting function Equation (\\ref{eq:fitting_function}) for $\\tau_0=10^6, 10^7$ with $x_s=0$. The disagreement between the tail of the distribution and the Monte Carlo data is due to line core scattering which is not modeled by the eigenfunction solution, but improves for larger optical depth as seen in the figure. A large number of scatterings in the Doppler core affects the tail of the escape-time distribution, since photons with frequencies near line center will take longer to escape. Thus, the rate constant for the exponential falloff is overestimated slightly in the eigenfunction solution as compared with the Monte Carlo. The error in this rate constant is a function of $\\tau_0$ since the effect from the Doppler core is greatest when it extends into the peak of the spectrum.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{waittime.pdf}\n    \\caption{Wait-time distribution of escaping photons $P(t)$ vs. t  for $\\tau_0=10^6$ and $10^7$, including the fitting function from Equation (\\ref{eq:fitting_function}). A sum over 20 spatial eigenmodes and 500 frequency eigenmodes is labeled ``Eigenfunctions''. All calculations were performed with a monochromatic source of photons at line center ($x_s = 0$).}\n    \\label{fig:escape_time}\n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{Steady-State Source}\nA primary goal of this work is to present a solution for resonant scattering of photons near the line-center frequency $\\nu_0$ in a uniform sphere. We have generalized a spherically symmetric solution derived by \\citet{2006ApJ...649...14D} (called $H_0$ here) to allow a monochromatic source of photons with frequencies away from line center. We introduce a new term to this solution, $J_{\\rm bc}$, which allows the boundary condition $J=\\sqrt{3}H$ to be satisfied at the surface of the sphere. This is solved using a continuous Fourier expansion in frequency. The integrals are discretized and the Fourier coefficients solved for numerically. The resulting flux correction, $H_{\\rm bc}$, scales as $H_0(a\\tau_0)^{-1\/3}$. Thus, for large $a\\tau_0$, only a small correction to $H_0$ is needed, while larger errors are present in calculations performed at lower $a\\tau_0$. Since the Laplacian form for frequency redistribution in the differential equation is only correct for photons in the wing where the line profile is $\\phi \\approx a\/(\\pi x^2 \\Delta)$, our solutions do not accurately model the Doppler core of the \\lya line. Because the peak of the spectral energy distribution of escaping photons is $x_{\\rm peak} {\\sim} (a\\tau_0)^{1\/3}$, calculations performed at small $a\\tau_0$ are inaccurate due to the close proximity of the spectral peak and the Doppler core of the line.\n\nBy comparison with Monte Carlo simulations, we have shown that the enforcement of the correct frequency-dependent boundary conditions improves the accuracy of these analytic solutions for $a\\tau_0 \\gg 1$. Specifically, this solution shows improvement over previous solutions that utilized a $J=0$ surface boundary condition presented in \\citet{1973MNRAS.162...43H}, \\citet{1990ApJ...350..216N}, and \\citet{2006ApJ...649...14D}. Several papers have previously compared these analytic models to Monte Carlo and seen discrepancies on the order of this correction. For example, in the top-left panel of Figure 1 from \\citet{2006ApJ...649...14D}, the \\lya spectrum emergent from a sphere of uniform optical depth is shown for $\\tau_0=10^5, 10^6,$ and $10^7$ at a temperature of $T=10$ K, corresponding to $a=1.5 \\times 10^{-2}$. The dotted line showing their theoretically-derived spectrum ($H_0$) displays an excess at the peak of at least 5-10 percent as compared with the Monte Carlo for $\\tau_0=10^5$ and $10^6$. Another example is in \\citet{2015MNRAS.449.4336S}, where the peak excess in the \\lya spectrum is particularly noticeable for line center optical depths of $\\tau_0=10^6$ and $10^7$ in the top panel of their Figure 5, which used slab geometry and a gas temperature of $T=10^4$ K. Again, the error in their solution is of order 5-10 percent. Both of these solutions are too large at the spectral peaks and too small further out in the wing, and the error scales approximately as $(a\\tau_0)^{-1\/3}$. We show in our Figure \\ref{fig:sol_mc_tau} that the error present in $H_0$ is corrected by our treatment of the boundary condition at $\\tau_0 = 10^7$ for T$=10^4$ K, corresponding to $a=4.72 \\times 10^{-4}$. We note that our correction term $H_{\\rm bc}$ is positive in the line wing and negative at the peak of the spectrum, which matches with the discrepancies noted in the aforementioned solutions.\n\n\\subsection{Impulsive Source}\nThe time-dependent transfer equation is solved in order to characterize the distribution of photon escape times. A semi-analytic approach is used, utilizing an expansion in space, time, and photon frequency. This boundary value problem in frequency $\\sigma$ is solved to find the flux at the surface of the sphere as a function of $t$ and $\\nu$. This solution is expressed as a sum over spatial and frequency modes $n$ and $m$, respectively. Calculating additional spatial eigenmodes increases the accuracy nearer to line center, but convergence is slow due to each eigenmode's weak dependence on $n$. Additional frequency eigenmodes introduce fast-oscillating terms that improve the accuracy of the Fourier sum, as their contributions cancel with components of lower-order terms to better represent the true solution. Integrating the solution over time produces a fluence that is shown to broadly agree with the steady-state calculations in Section \\ref{sec:steadystate}, provided a sufficient number of terms in the sum and emission at the line-center frequency $\\nu_0$. Integrating the solution over frequency leads to a distribution of photon escape time, which can be compared directly with Monte Carlo simulations. The sum over eigenmodes produces an escape-time distribution that broadly captures the behavior shown by Monte Carlo data---a rise at early times, transitioning to exponential decay in the tail of the distribution. It is expected that the accuracy of the rate constant for the tail of the distribution is limited by the effect of the Doppler core, which can trap photons at high optical depths until they diffuse outward in frequency, weighting the distribution toward later times. This physics is not modeled by our solution for two reasons: 1) our calculations ignored the Gaussian component of the Voigt line profile, leaving the Lorentzian piece which is accurate only in the line wing, and 2) knowing the core is not modeled accurately, we do not include a large enough number of spatial eigenmodes in the sum to resolve it. However, an approximate fitting function dependent on parameters $a$ and $\\tau_0$ is found that adequately represents the escape time distribution of the Monte Carlo results within these constraints.\n\nOur characterization of the escape time distribution leads to a possible application of this work. Models of the interaction of stellar \\lya with the upper atmosphere of exoplanets and the associated transmission spectrum can be constructed with a treatment of resonant scattering in spherical geometry \\citep{2017ApJ...851..150H, 2021ApJ...907L..47Y}. The Monte Carlo method can be used for this problem, but is limited by its high computational demand for large $\\tau_0$ where there are many photon scatterings before escape. We seek to develop a method to accelerate the radiative transfer calculation. \n\nThere are several methods that are commonly used to accelerate Monte Carlo radiation transfer calculations, including core skipping methods \\citep{1968ApJ...153..783A,2002ApJ...567..922A} and hybrid diffusion methods \\citep{2018MNRAS.479.2065S}. Another approach with wide application is modified random walk methods, such as those discussed in \\citet{1984JCoPh..54..508F, 2009A&A...497..155M, 2010A&A...520A..70R}. In this approach, an outgoing photon is randomly sampled on the surface of the outgoing sphere by drawing its properties from distributions in outgoing frequencies, directions, and escape times, based on solutions to the diffusion equation. A method similar to this has been applied by \\citet{2006ApJ...645..792T} to Lyman $\\alpha$ transfer using the \\citet{1990ApJ...350..216N} solution, but this solution of course does not utilize the frequency-dependent boundary condition at the surface of the sphere. Furthermore, to perform a full radiation hydrodynamic simulation with Monte Carlo acceleration, it will be necessary to calculate radiation forces within each cell due to \\lya transfer. Similar calculations have been done in \\citet{1976ApJ...208..286W} in plane-parallel geometry. However, these solutions are limited to optical depths below $2.5 \\times 10^3$. For this work, it would be necessary to model line center optical depths of up to 1 million or more.\n\n\\section{Summary}\n\nWe have examined previous solutions to \\lya transfer including resonant scattering in the limit of large optical depth, noting that the separation of variables and treatment of the boundary condition in \\citet{1973MNRAS.162...43H}, \\citet{1990ApJ...350..216N}, \\citet{2006ApJ...649...14D} and others produces a discrepancy in the outgoing spectrum as compared with Monte Carlo. Here, we have derived the solution in spherical geometry with an appropriate treatment of the surface boundary condition. The key result is that the errors in the previously-cited works have been quantified via a correction term, $H_{\\rm bc}$, which explains an excess in flux at the spectral peak and a deficit in the line wing of the calculated spectrum of \\lya radiation as compared with Monte Carlo. The size of $H_{\\rm bc}\/H_0$ is of order unity when the spectral peaks are near the Doppler core, and diminishes at larger $\\tau_0$ following a $(a\\tau_0)^{-1\/3}$ scaling. \n\nThe time-dependent transfer equation for the impulsive source is solved numerically with an eigenfunction expansion. We demonstrate that it agrees with the steady-state spectrum for $x_s=0$ when integrated over time, though its rate of numerical convergence is slow and requires a sum over many modes to become accurate. The time-dependent solution is utilized to create wait-time distributions for photons escaping the sphere of optically-thick hydrogen gas. We compare the calculations from the time-dependent solution with Monte Carlo for a sample of $\\tau_0$, noting general agreement in the resulting escape time distributions. The solution derived in our work here may be used as the basis for a novel implementation of the modified random walk method, which would accelerate Monte Carlo \\lya transfer at large optical depths with potential applications in radiation hydrodynamic simulations of the atmospheres of exoplanets. \n\n\\acknowledgments\n\nThis research was funded by NASA ATP grant 80NSSC18K0696, ``Exoplanetary MHD Outflows Driven by EUV Heating, Lyman alpha Radiation Forces and Stellar Tides\". Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. We thank the referee for a detailed report that helped improve the presentation of our work.\n\\restartappendixnumbering\n\n\\software{\\texttt{numpy} \\citep{2020NumPy-Array}, \\texttt{scipy} \\citep{2020SciPy-NMeth}, Coblis - Color Blindness Simulator (\\href{https:\/\/www.color-blindness.com\/coblis-color-blindness-simulator\/}{color-blindness.com})}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introd}\n\n\nIn the last two decades, soft-glassy materials (SGM) have gained growing attention due to their applications in several industrial sectors. \nIn particular, emulsions and foams are employed to design novel soft mesoscale materials for chemical, food processing, manufacturing, and biomedical purposes \\cite{fernandez2016fluids,piazza2011soft,mezzenga2005understanding}.\nBesides the technological relevance, their major significant theoretical interest stems from their intriguing non-equilibrium effects, \nincluding long-time relaxation, yield-stress behavior, and highly non-Newtonian dynamics. \n\nIn this context, computational fluid dynamics (CFD) provides a valuable tool to improve the knowledge of the underlying physics of SGM.\nTo that purpose, a reliable SGM model alongside its software implementation is of apparent interest for the rational designing and shaping up of novel soft porous materials.\n\nIn this paper, we present and make available the CUDA Fortran code \\textbf{LBcuda}, specifically designed to simulate on GPUs bi-continuous systems with colloidal particles under a variety of different conditions. \\textbf{LBcuda} is a direct port of the LBsoft code \\cite{lbsoft}, an open-source software for simulations of soft glassy emulsions originally developed for CPU-architectures, which successfully combines the lattice-Boltzmann method (LBM) \\cite{succi2018lattice,kruger2017lattice,benzi1992lattice} with a Lagrangian solver to tackle the multi-scale coupling of fluids and particles \\cite{bernaschi2019mesoscopic}.\n\nNowadays, the straightforward parallelization of LBM makes the lattice-Boltzmann algorithm an excellent candidate for high-performance CFD, especially on GPU-based architectures, given the relative simplicity and locality of its underlying algorithm.\nAs a consequence, several LBM implementations have been developed for GPU architectures, both academic packages \nsuch as the GPU-enabled versions of WaLBerla \\cite{bauer2021walberla,holzer2021highly}, Palabos \\cite{latt2021palabos}, Ludwig \\cite{desplat2001ludwig}, MUPHY \\cite{bernaschi2009muphy}, and commercially \nlicensed software such as XFlow 2021 \\cite{holman2012solution}, \nto name a few. \n\nAs aforementioned, the model to describe colloidal particles is derived from the previous CPU-based LBSoft code, to which the reader is referred for further details \\cite{lbsoft}. Briefly, SGM modeling requires specific implementations of LBM and Lagrangian solvers to include the hydrodynamic interactions between solid particles and fluids, following several strategies reported in the literature \\cite{ladd2015lattice,ladd2001lattice,aidun1998direct,ladd1994numericala}. \nThis extension has opened the possibility to simulate complex colloidal systems, also referred to as Pickering emulsions \\cite{pickering1907cxcvi} which are of primary interest for the rational design of SGM \\cite{xie2017direct,liu2016multiphase, frijters2012effects,jansen2011bijels}. This intrinsically multi-scale approach can catch, for example, the dynamical transition from a bi-continuous interfacially jammed emulsion gel, also referred to as bijel (see Fig. \\ref{fig:bijel}), capturing the associated mechanical and spatial properties \\cite{sun2021pickering}.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{figure01.pdf}\n\\caption{A typical bijel configuration with colloidal particles (blue spheres) entrapping the interfaces between the two fluids (red and transparent green). }\n\\label{fig:bijel}\n\\end{center}\n\\end{figure}\n\n\nThe paper is structured as follows. In Section \\ref{sec:method} we report a very brief description of the underlying method, referring our previous paper for a deeper explanation of the details.\nIn Section \\ref{sec:impl} we describe the details of the data structures on the GPU, while in Section \\ref{sec:parallel} we explain the parallelization strategy and its impact on performance.\nIn Section \\ref{perfomance} we report a set of tests used to validate the implementation and we investigate the performance against the reference CPU version (LBsoft). Finally, conclusion and outlook on future development directions are discussed.\n\n\\section{Method}\n\\label{sec:method}\n\nIn this section, we briefly review the approach to the simulation of SGM implemented in LBcuda alongside the more significant algorithmic adaptations required by the GPU-based hardware. A more detailed illustration of the underlying algorithms can be found in Ref. \\cite{lbsoft}.  \n\nThe code combines two different levels of description: the first exploits a continuum approach for the dynamics of immiscible fluids, whereas the second manages individual rigid bodies representing colloidal particles or other suspended species. \nThe two levels exchange information at each step of the time integration scheme to describe the concurrent interaction among particles and surrounding fluids.\n\nIn the first level, the LBM exploits a fully discretized analog of the Boltzmann kinetic equation to model flows and hydrodynamic interactions in fluids.\n\nIn the LBM approach, the fundamental quantity is $f_{i}(\\vec{r};t)$, namely the probability of finding a \"fluid particle\" at the spatial point mesh $\\vec{r}$ and at time $t$ with velocity  $\\vec{c}_i$ selected from a \nfinite set of possible speeds. The LBcuda code implements the 3-dimension 19-speed cubic lattice scheme (D3Q19) with the discrete velocities $\\vec{c}_i$ with  $i \\in [0,...,18]$ connecting mesh points with spacing $\\Delta x$ (length lattice unit) to first and second mesh neighbours, located at distance $\\Delta x$ and $\\sqrt 2 \\Delta x$, respectively (in other words, D3Q19 neglects 8 out of 27 possible velocities: those having distance $\\sqrt 3\\Delta x$).\n\n\n\nDenoted $\\rho(\\vec{r};t)$ and $\\vec{u}(\\vec{r};t)$ respectively the fluid density and the fluid velocity, the lattice-Boltzmann equation is implemented in single-relaxation time (Bhatnagar-Gross-Krook equation) as follows:\n\\begin{equation}\\label{eq:bgk}\nf_{i}(\\vec{r}+\\vec{c}_{i};t+1)=(1-\\omega)f_{i}(\\vec{r};t)+\\omega f_{i}^{eq}(\\rho(\\vec{r};t),\\vec{u}(\\vec{r};t))\n\\end{equation}\nwhere $f^{eq}$ is the lattice local equilibrium, basically the local Maxwell-Boltzmann distribution (see Appendix A), and $\\omega$ is a frequency tuning the relaxation towards the local equilibrium on a timescale $\\tau=1\/\\omega$. The relaxation frequency $\\omega$ controls the kinematic viscosity of the fluids according to the relation:\n\\begin{equation}\n\\nu=c_{s}^{2}\\frac{\\Delta x^{2}}{\\Delta t} \\left(\\frac{1}{\\omega}-\\frac{1}{2}\\right),\n\\label{eq:viscosity}\n\\end{equation}\nwhere $\\Delta x$ and $\\Delta t$ are the physical length and time of the correspondent counterparts in lattice units. Note that the positivity of the kinematic viscosity requires the condition $0<\\omega<2$.\n\n\nIn order to model a two component systems we adopted, a color gradient (CG) algorithm, which enforces a diffuse interface between the two fluids \\cite{leclaire2017generalized}. \nIn short, in the update phase of the populations, the CG collision contains three sub-steps: a plain BGK collision, a perturbation operator, and a final recoloring step. It is worth stressing that the last two sub-steps act only near the interface between the two fluids. Further details are reported in Appendix.\n\nThe second level of description involves a Lagrangian solver for the particle evolution, where each particle (colloid) is represented by a closed surface ${\\mathcal S}$, taken, for simplicity, as a rigid sphere in the following. \n\nThe LBcuda code adopts the formulation given by Jansen and Harting \\cite{jansen2011bijels}, where only the exterior regions are filled with fluid, whereas the interior parts of the particles are solid nodes.\nThe solid--fluid interaction is managed via a simple generalization of the bounce-back rule including the correction due to the relative motion of the solid particle with respect to the surrounding fluid medium.\n\nHence, the particle position, speed $\\vec{v}_p$ and angular momentum\\index{angular!momentum} $\\vec{\\omega}_p$ are updated according to Newton's equations of motion:\n\n\\begin{equation}\n\\label{PARDYN}\n\\left\\{ \\begin{array}{lll}\n\\frac{d \\vec{r}_p}{d t} =  \\vec{v}_p,\\\\\nm_p \\frac{d \\vec{v}_p}{d t} =  \\vec{F}_p,\\\\\nI_p \\frac{d \\vec{\\omega}_p}{d t} = \\vec{T}_p,\n\\end{array}   \n\\right.\n\\end{equation}\nwhere $m_p$ and $I_p$ are the particle mass and moment of inertia, respectively.\n\nFollowing Ladd's seminal works \\cite{ladd1994numericala,ladd1994numericalb}, we advance in time eq. \\ref{PARDYN} with a leap-frog scheme, which is second order accurate in time.\nThis set of equations considers the full many-body hydrodynamic interactions since the forces and torques are computed with the actual flow, as dictated by the presence of all $N$ particles simultaneously.\n\n\n\\section{Implementation}\n\\label{sec:impl}\n\nThe code is implemented in CUDA Fortran, using modules to minimize code cluttering. The LBcuda code requires no external libraries besides the CUDA runtime and compiles using a simple Makefile. The code is written for the nvfortran compiler, with the GPU kernels confined in {\\em cuf} extension files, whereas the I\/O part and the main are coded in standard FORTRAN files.\n\nThe code is composed of 6 files:\n\\begin{itemize}\n\\item \\textbf{dimension.cuf}, which sets constants for the LB algorithm and the physical values of the simulation\n\\item \\textbf{kernels\\_fluid.cuf}, containing all GPU variables\n\\item \\textbf{kernels\\_fluid\\_CG.cuf}, containing the color-gradient GPU code \n\\item \\textbf{kernels\\_fluid\\_PART.cuf}, containing the particles GPU code\n\\item \\textbf{write\\_output.f90}, which outputs the VTK and VTI files for external visualization by  graphical programs (e.g, ParaView) \n\\item \\textbf{main.f90}, finally contains the driving code of all the subroutines.\n\\end{itemize}\n\nMost of the input for the simulation is defined by setting Fortran \\textbf{parameters} in dimension.cuf. In contrast, other runtime parameters, such as print frequency of VTK output files and average statistical quantities, can be set up without recompilation in a plain-text input file.\n\nThe data for each fluid component are organized in a five dimension matrix having x,y,z, then the population index, and finally two possible values for \\textit{switching} between old and new values during the collide-stream phases of the LB algorithm, also referred to as one-step two-grid algorithm \\cite{wittmann2013comparison}.\n\nAll data residing on the GPU are defined in kernels\\_fluid.cuf, whereas writing output files requires just a few memory passages from GPU to CPU at the printing frequency for fluid densities, flow field, and particle positions.\n\n\n\nIn order to solve the lattice-Boltzmann equation with particle dynamics, the algorithm proceeds executing the following sequence of subroutine calls:\n\\begin{itemize}\n    \\item Each thread of the GPU device computes fluid and particle quantities at a single spatial point, say located at the (i, j, k) node;\n    \\item In each node, the code proceeds according to three different cases:\n    \\begin{enumerate}\n    \\item If the node contains fluid far away from any particles, the thread will only advance the LB algorithm;\n    \\item If at that point a fluid touches a particle, the thread computes its part for the LB algorithm, then it computes its contribute for the force\/toque integral. Note that when there are touching particles, a point can contribute to more than one particle.\n    \\item If the point is inside a particle, no computation is performed and the following steps are skipped;\n    \\end{enumerate}\n    \\item Apply the collision step of Eq. \\ref{eq:bgk};\n    \\item Apply the halfway bounce-back rule at particle surface and the relative force terms on particles;\n    \\item Evolve position and angular velocity of particles (if present);\n    \\item Apply the stream step of Eq. \\ref{eq:bgk}.\n\\end{itemize}\nIt is worth stressing that the net force and torque exerted from the fluid on the particle center of mass is obtained by summing over all the particle surface nodes.\n\n\n\\section{Parallelization strategy: CUDA and MPI}\n\\label{sec:parallel}\n\nFor the LB part of the algorithm, the CUDA porting decomposes the global domain according to a 3D block distribution among the CUDA threads (see Fig. \\ref{Figdec}). The selection of the block distribution is fixed at compile-time, and it can be tuned to obtain the best performance given the global grid dimension and the compute capability of the GPU device. Usually, a 3D decomposition that emphasizes the x-axis dimension achieves the best performance since it exploits the high memory bandwidth due to the data continuity in the column-major order of the FORTRAN language. Each thread will be responsible for only one grid point of the fluid box in each CUDA block. In the LB part, the thread iterates over all the fluid populations in most kernels. On the other hand, the thread computes the contribution of the fluid grid point to the force and torque of the overlapping particles, defined as particles whose surface overlaps the fluid node owned by the thread.\n\n\nThe LBcuda code is designed to exploit multiple GPU devices.\nTo that purpose, the code resorts to MPI having one GPU card associated to each MPI task. The LB domain is divided into sub-domains of equal size, whereas the variables related to the particles are replicated in all the MPI processes. Thus, the LB solver proceeds locally on each GPU device, with the extra computational cost due to the communication of border information among the local sub-domains of the neighbor MPI tasks. \n\nEach MPI task computes the part (section) of the particles falling in its sub-domain in the particle solver. \nIn this framework, particles evolution is crucial for achieving a high computational throughput by avoiding excessive communication (or memory conflicts) among threads while integrating the particle quantities.\nThus, we have adopted a single particle list, which is stored on GPU devices. In particular, whenever the multi GPU is used, each MPI sub-domain has its list of owned particles on its GPU card. When a LB time step is completed, each thread in the sub-domain checks if it needs to compute the contribution of its fluid node to the computation of the surface integral for the force and torque that the fluid exerts on each particle surface node and vice versa.\nHence, a global MPI reduction is used to compute the corresponding total force and torque acting on the center of mass of each particle, so that particle positions, orientations, and velocities can be advanced in time on all the MPI processes. \nThe selection of the sub-domain particle list on each MPI process is made by a CUDA kernel, leveraging the parallel computing power of each GPU card that makes the required computing time almost negligible.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{figure02.pdf}\n\\end{center}\n\\caption{Sketch showing the domain decomposition strategy used for the particle data on the GPU device. On the left, each sub-domain (thread block) has a list of owned particles. On the right, the global GPU data vector stores all the particles in contiguous way over the sub-domains.\n}\n\\label{Figdec}\n\\end{figure}\n\nIt is worth highlighting that the overlap between particle and fluid node provides an unbalance in the work performed by the CUDA threads, notably wasting computational power at the surface particle node to treat the boundary conditions and the momentum exchange between the surrounding fluid and the particle. Nonetheless, we found that the overhead for the fluid solver is quite marginal, retaining an acceptable code scalability.\n\n\n\n\n\n\n\n\n\\section{Performance results}\n\\label{perfomance}\n\nIn order to analyze the computational performance of LBcuda, we consider two cubic boxes of side 128 and 256 lattice nodes\nwith periodic boundary along the three Cartesian axes. \nWe limited the size to a 256 cubic box, which is the largest grid that fits in the memory of a single GPU device.\nWe perform three different test cases. First, we examine the case of a single fluid in a cubic box with an initial density equal to one and zero velocity flow ($case \\; 1$).\nAs a second test ($case \\; 2$), a bi-component system is considered where all the fluid nodes are randomly filled with fluid mass density of the two fluids, red and blue component, to achieve the value of the order parameter, $\\phi=\\frac{\\rho_r-\\rho_b}{\\rho_r+\\rho_b}$, equal to 1 or -1 with zero velocity flow field with the subscripts r and b standing for \"red\" and \"blue\" fluids, respectively. The third test ($case \\; 3$) checks the entire LBcuda algorithm with\nthe two-component fluid combined with the particle solver \ndescribing the colloids in a rapid demixing emulsion.\nIn particular, we defined three sub-setups with different numbers of particles to assess the performance of the particle solver.\nHence, three values of the volume fraction occupied by particles are considered:\n0.1\\%, 1.0\\% and 10\\%, labeled $case \\; 3a$, $case \\; 3b$, and $case \\; 3c$, respectively.\n\nTo evaluate the performance of LBcuda, we compare the theoretical peak performance to the actual one achieved by our code. In particular, the roofline model \\cite{williams2009roofline} is used to rank the achievable computational performance in terms of Operational Intensity (OI), defined as the ratio between flops performed and data that need to be loaded\/stored from\/to memory. \nAt low OI (say, $ < 10 $), the performance is limited by the memory bandwidth, whereas for higher OI values, the limitation comes from the availability of floating-point units. \nIt is well known that LB is a bandwidth-limited numerical scheme, like most CFD models \\cite{towards}. The OI index for LB schemes is around 0.7 for double-precision (DP) simulations using a D3Q19 lattice. As a matter of fact, for a single fluid, since the number of floating-point operations per lattice site and time step is $F \\simeq 200 \\div 250 $ and the load\/store burden in bytes is $B= 19 \\times 2 \\times 8=304 $ (using double precision), the operational intensity is $F\/B \\sim 0.7$, whereas in single precision is $F\/B \\sim 1.4$ confirming that the code is bandwidth limited (see also Figure \\ref{fig:roof}).\n\nIn the following, we assess the efficiency of the LBcuda code by means of the Giga Lattice Updates Per Second (GLUPS) metrics. In particular, \nthe definition of GLUPS reads:\n\\begin{equation}\n\\label{eq:glups}\n\\text{GLUPS}=\\frac{L_x L_y L_z}{10^9 t_{\\text{s}}},\n\\label{eq:mlups}\n\\end{equation}\nwhere $L_x$, $L_y$, and $L_z$ are the domain sizes in the\n$x-$, $y-$, and $z-$ axis, and $t_{\\text{s}}$ is the run (wall-clock) time (in seconds) per single time step iteration.\n\n\n\\subsection{Single fluid}\n\nThe $case \\; 1$ with one fluid is tested using a plain BGK collision and a fused implementation (in which the collision and streaming step are performed simultaneously) of the LB time-advancing, the latter being the most popular approach for major LB codes. The results are reported for two different GPUs: a V100 and a GeForce RTX 2060S  in Table \\ref{tab:1}.\n\n\\FloatBarrier\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n        \\hline \n        \\textbf{Size} & \\textbf{GPU} & \\textbf{Time} & \\textbf{GLUPS} & \\textbf{Approach} \\\\\n        \\hline \n        $128^3$ & V100  &  1.09 ms\/iter & 1.923 & fused \\\\\n        $128^3$ & 2060S &  1.64 ms\/iter & 1.278 & fused \\\\\n        \\hline\n        $256^3$ & V100  &  8.34 ms\/iter & 2.011 & fused \\\\\n        $256^3$ & 2060S &  15.6 ms\/iter & 1.075 & fused \\\\\n        \\hline \n        $128^3$ & V100  &  2.09 ms\/iter & 1.003 & plain \\\\\n        $128^3$ & 2060S &  3.14 ms\/iter & 0.667 & plain \\\\\n        \\hline\n        $256^3$ & V100  &  15.34 ms\/iter & 1.093 & plain \\\\\n        $256^3$ & 2060S &  30.1 ms\/iter & 0.557 & plain \\\\\n        \\hline \n    \\end{tabular}\n    \\caption{Timings alongside with GLUPS of $128^3$ and $256^3$ cubic boxes for the $case \\; 1$ using both the optimized fused and plain approach single fluid on a Tesla V100 and a GeForce RTX 2060S in single precision.}\n    \\label{tab:1}\n\\end{table}\n\\FloatBarrier\n\nThe key differences between the two GPUs are: the V100 has 5132 Cuda cores offering a peak performance of 14TFlop\/s and a memory bandwidth of 900GB\/s, whereas the GeForce RTX 2060S has 2176 cores for a peak of 6.4TFlop\/s and a memory bandwidth of 448GB\/s. On the other hand, we observe that the obtained 2.011 GLUPS for a cubic box of side 256 is comparable with the state-of-the-art represented, for instance, by the highly optimized code by G. Falcucci et al. in Ref. \\cite{falcucci2021extreme} which reaches 3.406 GLUPS on a single V100 with the fused implementation on the same cubic box size.\n\nAlthough the fused approach reduces of a factor two the number of memory accesses, it is worth highlighting that the particle solver requires the mandatory use of the plain approach, showing a decrease of the performance of about a factor two for $case \\; 1$.\nIndeed, the particle boundaries require using the plain Lattice-Boltzmann algorithm, where the collision is first computed for all the fluid lattice nodes. Then, the boundary conditions are applied (internal walls or particles), and finally the streaming of the populations is carried out (see Section \\ref{sec:impl}).\n\n\n\\begin{figure} \\begin{center}\n\\includegraphics[width=1.0\\linewidth]{figure03.pdf}\n\\end{center}\n\\caption{Roofline model for single fluid using a GeForce RTX 2060S, as measured by NVIDIA NSight in the plain LB approach and in single precision. Bandwidth and Float point computation limits were obtained performing {\\em memory-stream} and High Performance Computing Linpack benchmark. Note that the LBcuda code lies on the left part of the plot showing that it is bandwidth limited.}\n\\label{fig:roof}\n\\end{figure}\n\n\nFrom an in-depth analysis using the NVidia dedicated tool (NSight), we observe that on the GeForce RTX 2060S the main kernel (the LB time-stepping before the streaming substep)128x1x1 achieves better performance  achieves almost $\\sim 520 GFlop\/s$ with an arithmetic intensity of 2.0 in single precision (see Figure \\ref{fig:roof}). This result shows that we are far from an intensity $\\sim 15.0$ which should give the peak performance, and in the $case \\; 1$ we attain about 60\\% of memory bandwidth utilization due to its non-optimal use following the plain approach.\n \n\n\\subsection{Two fluid test}\n\nThe $case \\; 2$ is related to the simulation of a two-component system by the color-gradient model (see Section \\ref{sec:method}). We remark that the time integration is implemented using the plain approach with a standard collide-stream 2-pass algorithm. Table \\ref{tab:cg} highlights the measured performance for the $128^3$ and $256^3$ cubic box. In particular, we observe an increase of about a factor 3 with respect to the previous $case \\; 1$, which is mainly due to the larger number of operations, more than doubled, in the color gradient collision operator containing three steps (see Section \\ref{sec:method}) instead of the single step of the plain BGK single fluid case. \n\n\\FloatBarrier\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n        \\hline \n        \\textbf{Size} & \\textbf{GPU} & \\textbf{CUDA Block} & \\textbf{Time} & \\textbf{GLUPS} \\\\\n        \\hline \n        $128^3$ & V100  & 8x4x4 &  4.5 ms\/iter & 0.466 \\\\\n        $128^3$ & 2060S & 8x4x4 &  11 ms\/iter & 0.190 \\\\\n        \\hline\n        $256^3$ & V100  & 8x4x4 &  34.1 ms\/iter & 0.492 \\\\\n        $256^3$ & 2060S & 8x4x4 &  82.3 ms\/iter & 0.204 \\\\\n        \\hline \n        \\hline \n        $128^3$ & V100  & 128x1x1 &  3.96 ms\/iter & 0.554 \\\\\n        $128^3$ & 2060S & 128x1x1 &  8.96 ms\/iter & 0.245 \\\\\n        \\hline\n        $256^3$ & V100  & 128x1x1 &  27.9 ms\/iter & 0.601 \\\\\n        $256^3$ & 2060S & 128x1x1 &  64.5 ms\/iter & 0.261 \\\\\n        \\hline \n    \\end{tabular}\n    \\caption{Timings alongside with GLUPS for $128^3$ and $256^3$, 2 fluids with CG using a Tesla V100 and a GeForce RTX 2060S with the plain approach in single precision with two different decompositions of threads in CUDA block. Note that the CUDA block configuration 128x1x1 achieves better performance exploiting the contiguous data over $x$ in the population arrays.}\n    \\label{tab:cg}\n\\end{table}\n\\FloatBarrier\n\n\n\n\\subsection{Particles}\n\nThe entire LBcuda algorithm is evaluated with the two-component colour gradient method (see Section \\ref{sec:method}) combined\nwith the particle solver to model the colloids in a rapid demixing emulsion. To assess\nthe performance of the particle solver, we prepared three simulation setups with different numbers of particles with radius equal to 5.5 lattice units in a cubic box of side 256 lattice points.\nThe three cases, in the following labelled $case \\; 3a$, $3b$, and $3c$, differ in the ratio between the volume occupied by the particles compared to the box volume equal to $\\varphi=$ 0.1\\%, 1.0\\% and 10\\%, respectively, in order to study the impact of the particle evolution on the simulation time.\nThe performance impact of having particles goes from negligible in $case \\; 3a$ to be comparable with the LB computation in $ case \\; 3c$ in which the time for each iteration almost doubles, as shown in table \\ref{tab:partic}.\n\n\\FloatBarrier\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n        \\hline \n        \\textit{\\textbf{case}} & \\textbf{GPU}  & $\\varphi$ & \\textbf{Time} & \\textbf{GLUPS} \\\\\n        \\hline \n        $3a$ & V100  &  0.1 \\% & 34.6 ms\/iter &  0.484 \\\\\n        $3a$ & 2060S &   0.1 \\% & 84 ms\/iter &  0.199  \\\\\n        \\hline \n        $3b$ & V100  &   1.0 \\% & 37.6 ms\/iter &  0.446 \\\\\n        $3b$ & 2060S &   1.0 \\% & 88 ms\/iter &  0.190 \\\\\n        \\hline \n        $3c$ & V100  &   10 \\% & 56 ms\/iter &  0.299 \\\\\n        $3c$ & 2060S &   10 \\% & 113 ms\/iter &  0.148 \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{Timings alongside with GLUPS for the three cases with particle volume fraction, $\\varphi$, equal to 0.1\\%, 1.0\\%, and 10\\%, respectively, in a cubic box of $256^3$ lattice nodes in single precision.}\n    \\label{tab:partic}\n\\end{table}\n\\FloatBarrier\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\linewidth]{figure04.pdf}\n\\caption{Renderings for $256^3$ simulation without (plot  d) and with particles (plot a,b,c). From top-left to bottom-right) a) Initial condition for particle simulation.  b) Density field after 50k iterations with 10\\% volume fraction.  c) Density field after 100k iterations (10\\% vol. fraction).  d) Density field after 100k iterations without particles starting from a mixed bi-component fluid system (similar to plot a).}\n\\label{fig:stopInterface}\n\\end{figure}\n\nIt is worth highlighting that LBcuda code implements a double precision accumulator because of floating point accuracy problems related to the momentum transfer from the fluid to each particle. In particular, the particle force and torque computation suffer from floating accuracy problems due to strong cancellation between addends of alternating sign over the nodes of the two-fluids interface.\n\nThe benchmark results can be also compared to the corresponding box size without particles reported in Table \\ref{tab:cg}.\nFor the test in $case \\; 3$ the particle radius is equal to 5.5 lattice units, and\nthe particle positions are randomly distributed in the box. \nThe initial particle velocity is zero in all runs.\nThe particle wettability is tuned to set an angle equal to $90^{\\circ}$ \nwith respect to the axis $\\vec{x}^{\\star}$ in the local reference frame \nof each particle. The impenetrability among particles is avoided by an hertzian repulsive contact force computed by means of neighbor's lists (see Section \\ref{sec:parallel}).\nThe lubrication force is also considered by adding an extra force term whenever two particles are located at a mutual distance lower than 2\/3 lattice unit, as reported in previous simulations \\cite{jansen2011bijels}. \nThe particle mass was estimated as the weight corresponding to a particle \nmade of silica \\cite{herzig2007bicontinuous}. \n\n\n\n\nAll runs were simulated on both the CPU and GPU architectures using the previous LBSoft code and the corresponding GPU ported version, LBcuda. In all the $cases \\; 3$ we observe only a small deviation in the position always lower than $10^{-4}$ in lattice units, mainly due to the aforementioned floating point accuracy problem. Indeed, the order of the addends over the particle surface nodes is completely random on the GPU device. \nIn the $case \\; 3c$ we observe the arrest of the phase demixing process with particles located at the fluid-fluid interface entrapping the demixing process into a metastable state, the bi-continuous jammed gel state \\cite{stratford2005colloidal}. In Figure \\ref{fig:stopInterface} we show two of these numerical experiments, in which a random mixture of two fluids evolves in a very different way in the presence of a high number of particles ($case \\; 3c$) and without particles ($case \\; 2$). Indeed, the particles stop the complete spinodal decomposition of the two fluids corresponding to the condition of minimal energy and minimal interface between them. Whenever a high volume fraction of particles is in the simulation box, the separation surface becomes way more \"corrugated\" showing the formation of the bi-continuous jammed gel state.\nFigure \\ref{fig:stopInterface} shows the initial condition, the fluids after 50k (with particles), and final configurations of both simulations after 100k iterations ($case \\; 3c$ and $case \\; 2$).\n\n\n\n\n\\subsection{Multi GPUs Performance}\n\nThe LBcuda code resorts to the Message Passing Interface (MPI) library to exchange data among GPU devices running in parallel.\nThe performance obtained using multiple GPUs show a good scaling behavior as reported in Tables \\ref{tab:multiGPU} and  \\ref{tab:multiGPU2}. The benchmark in Table \\ref{tab:multiGPU} was carried out on Marconi100 at CINECA, a cluster of V100 cards, each with 16 GB of global memory using a different number of GPU devices, for three cubic boxes: $256^3$, $512^3$, and $1024^3$. The cluster is made of nodes, each endowed with four GPU cards so that the MPI communication does not incur network latency unless the job is using more than 4 GPU cards. The benchmark in Table \\ref{tab:multiGPU2} ran on a cluster made of NVIDIA DGX A100. Each NVIDIA DGX A100 is equipped with 8 A100 NVIDIA GPU with 80 GB of RAM interconnected intra-node through the NVswitch and 8 NVIDIA Infiniband HDR (one NIC for each GPU) for multi-node scaling.\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|c|}\n        \\hline \n        \\textbf{Grid} & \\textbf{1} & \\textbf{2} & \\textbf{4}  & \\textbf{8}  & \\textbf{16} & \\textbf{32} \\\\\n        \\hline \n     \n        $256^3 V100\\ 16GB$ & 0.60 & 1.17 & 2.24 & 2.93 & 2.71 & 3.15 \\\\\n        \\hline \n        $512^3 V100\\ 16GB$ & x & x & 2.67 & 4.57 & 6.23 & 9.26 \\\\\n        \\hline\n        $1024^3 V100\\ 16GB$ &  x     &   x    &  x    &  x    &   x    & 12.48 \\\\\n        \\hline\n        \\end{tabular}\n    \\caption{Performance for $256^3$, $512^3$, $1024^3$, measured in GLUPS running on multiple NVIDIA Volta V100 GPUs (each card equipped with 16 GB of RAM). Note that the symbol (x) denotes a grid size too big to fit in the local GPU memory. For the $256^3$, performance degrade dramatically when using more than 8 V100 GPUs and more than 64 A100 because each local domain becomes too small (256x256x8) to offset the cost of scheduling the GPU task.}\n    \\label{tab:multiGPU}\n\\end{table}\n\n\n\\begin{table}[h!]\n    \\centering\n      \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n        \\hline\n        \\textbf{Grid} & \\textbf{8} & \\textbf{16} & \\textbf{32}  & \\textbf{64} & \\textbf{128} &  \\textbf{256} & \\textbf{512}   \\\\\n        \\hline\n        $256^3 A100\\ 80GB$ & 6.17 & 9.42 & 12.38 & 15.38 &  & & \\\\\n        \\hline \n        $512^3 A100\\ 80GB$ & 7.63 & 14.13 & 23.13 & 36.05  & & &\\\\\n        \\hline\n        $1024^3 A100\\ 80GB$ & 11.65 & 20.57 & 28.59 & 49.11  & 76.91 & & \\\\\n        \\hline\n        $2048^3 A100\\ 80GB$ & x & x & x & 55.22  & 102.3 & 160.9 & 204.5 \\\\\n        \\hline\n                \n    \\end{tabular}\n    \\caption{Performance for $256^3$, $512^3$, $1024^3$, $2048^3$ measured in GLUPS running on multiple NVIDIA Volta A100 GPUs (each card equipped with 80 GByte of RAM). Note that the symbol (x) denotes a grid size too big to fit in the local GPU memory.}\n    \\label{tab:multiGPU2}\n\\end{table}\n\nAll benchmarks reported in Tables \\ref{tab:multiGPU} and  \\ref{tab:multiGPU2} were carried out on a bi-component fluid system without particles ($case \\; 2$). \nThe data for the cubic box sizes $256^3$, $512^3$ are also plotted in Figure \\ref{fig:GPUMlups} with an evident decrease in the scalability whenever the code starts to run on more than one node (more than 4 GPUs).\nHence, the three tests, $case \\; 3a$, $3b$, and $3c$, with different values in the particle volume fraction, $\\varphi$, were performed to probe the efficiency of the particles solver as a function of the particles number in the system. All benchmarks were carried out on eight GPU cards: both V100 cards with a 16 GB RAM and A100 cards with a 40 GB RAM. In Table \\ref{tab:partmultiGPU}, we observe a clear communication penalty as the particles number increases that is mainly due to the replicated data strategy used for the particle solver parallelization. Indeed, the replicated data parallel approach replicates the physical quantities of all particles across the MPI processes, performing local updating and global MPI sum reductions in order to advance the system in time, which decreases the performance as the quantity of particles data increases.\nNonetheless, the analysis of the performance as a function of the number of particles shows that the code is able to reach about 3.80 GLUPS in a system with 10\\% in the particle volume fraction.\n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.5\\linewidth]{figure05.pdf}\n\\caption{Top panel: Measured GLUPS for different numbers of V100 cards with a 16 GB RAM, running two cubic boxes of side 256 and 512, respectively. Bottom panel: Measured GLUPS for different numbers of A100 cards with a 80 GB RAM, running two cubic boxes of side 1024 and 2048, respectively.}\n\\label{fig:GPUMlups}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n        \\hline \n        \\textbf{GPU} & \\textbf{No particles} & \\bm{$\\varphi=0.1\\%$} & \\bm{$\\varphi=1.0\\%$}& \\bm{$\\varphi=10\\%$} \\\\\n        \\hline \n         8 V100@16 & 30 ms (4.47) & 40 ms (3.35) & 51 ms (2.63) & 95 ms (1.41) \\\\\n        \\hline\n         8 A100@40 & 18 ms (7.55) & 19 ms (7.06) & 22 ms (6.10) & 35 ms (3.83) \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{Time per single iteration alongside with GLUPS in parenthesis  for $512^3$ without and with particles (at different particle volume fraction $\\varphi$)  on 2 different machines: using eight V100 with 16 GB of RAM on 2 nodes (connected by InfiniBand) and using eight A100 with 40 GB in a single node (without the latency time due to the InfiniBand communication).}\n    \\label{tab:partmultiGPU}\n\\end{table}\n\n\n\n\n\\subsection{Comparing LBcuda with LBsoft}\n\nFor the sake of completeness, we probe the gain provided by the CUDA port reported in the present article. Thus, the $case \\; 2$ bi-component system was initialized with the same values in density and flow field in both LBcuda and LBsoft code. The cubic box size is equal to $512^3$ lattice points. Although it is difficult to compare two completely different computing architectures, we measure the wall-clock time obtained on a GPU cluster made of V100 cards with 16 GB RAM and a CPU cluster containing two Intel Cascade Lake 8260 CPUs at 2.40 GHz with 48 cores and 384 GB of RAM per node.\nWe note that the code produces the same results in single precision unless a slight difference of $10^{-4}$ order of magnitude in the particle positions due to floating point accuracy problems (mainly due to different order of summation). The wall-clock time for iteration results in 12 ms on 32 V100 cards of LBcuda versus 88 ms on 528 cores of LBSoft, confirming the clear advantage in running the CUDA ersion on a GPU HPC cluster.\n\n\n\\section{Conclusion}\n\n\nWe have presented LBcuda, a CUDA port of LBsoft, an open-source software aimed at simulating specifically colloidal systems. \nLBcuda is written in CUDA Fortran and\npermits to simulate large system sizes running on multiple GPU devices by exploiting an efficient parallel domain decomposition implementation.\n\nIn particular, the code shows good scaling behavior of the fluid solver achieving the performance peak of 200 GLUPS on 512 NVIDIA A100 cards with a grid of eight billion lattice points.\nOn the other hand, the particle solver combined with the LB approach shows a very satisfactory performance in terms of scalability in both system size and number of processing cores, especially using the Nvidia Ampere A100 cards.\n\nIn this work, the main structure of LBcuda has been outlined along with the key steps of its implementation. \nFurthermore, several cases have been introduced to test the code over typical problems that the LBcuda code can deal with. \nIn particular, the simulations with particles demonstrate the capabilities of the present code to reproduce the complex dynamics of bi-jel systems in a rapid de-mixing emulsion.\n\nThe LBcuda code is open source and completely accessible at the public repository GitHub, which is in line with the spirit of open-source software, mainly to promote the contribution of independent developers.\n\n\n\\section{Acknowledgments}\nThe research leading to these results has received\nfunding from the European Research Council under the European\nUnion's Horizon 2020 Framework Programme (No. FP\/2014-\n2020)\/ERC Grant Agreement No. 739964 ``COPMAT'' and from MIUR under the project \"3D-Phys\" (No. PRIN 2017PHRM8X). The CINECA is acknowledged for the support granted by the ISCRA project ``porting LBSOft in CUda on multi-node GPUs (LBSOCU)''.\n\n\n\n\\section*{Appendix}\n\nFor the details of the color gradient (GC) model of the Lattice Boltzmann method employed in the bi-component systems \\cite{leclaire2017generalized}, we recall some notions.\n\nIn the color gradient LB for two-component flows, two\nsets of distribution functions are defined to track the evolution of the two fluid components, which occurs via a streaming-collision algorithm:\n\n\\begin{equation} \\label{CGLBE}\nf_{i}^{k} \\left(\\vec{x}+\\vec{c}_{i}\\Delta t,\\,t+\\Delta t\\right) =f_{i}^{k}\\left(\\vec{x},\\,t\\right)+\\Omega_{i}^{k}( f_{i}^{k}\\left(\\vec{x},\\,t\\right)),\n\\end{equation}\n\nwhere $f_{i}^{k}$ is the discrete distribution function, representing\nthe probability of finding a particle of the $k^{th}$ component at position $\\vec{x}$ and time\n$t$ with discrete velocity $\\vec{c}_{i}$ . \n\nIn the last Eq. $i$ is the index running over the lattice discrete directions $i = 0,...,b$, where $b=18$ for a three dimensional 19 speed lattice (D3Q19) implemented in LBcuda.\nThe lattice time step $\\Delta t$ has been taken as $1$ (in lattice units) for convenience.\nThe density $\\rho^{k}$ of the $k^{th}$ component is given  by the zeroth moment of the distribution functions:\n\\begin{equation}\n\\rho^{k}\\left(\\vec{x},\\,t\\right) = \\sum_i f_{i}^{k}\\left(\\vec{x},\\,t\\right),\n\\end{equation}\nwhile the total fluid density is assessed as $\\rho=\\sum_k \\rho^k$, and the total momentum of the mixture is given as the sum of the linear momentum of the two components:\n\\begin{equation}\n\\rho \\vec{u} = \\sum_k  \\sum_i f_{i}^{k}\\left(\\vec{x},\\,t\\right) \\vec{c}_{i}.\n\\end{equation}\n\nThe collision operator in the CG model is made of three parts: \n\n\\begin{equation}\n\\Omega_{i}^{k} = \\left(\\Omega_{i}^{k}\\right)^{(3)}\\left[\\left(\\Omega_{i}^{k}\\right)^{(1)}+\\left(\\Omega_{i}^{k}\\right)^{(2)}\\right].\n\\end{equation}\n\nIn the above, $\\left(\\Omega_{i}^{k}\\right)^{(1)}$ stands for the standard collisional relaxation which reads:\n\\begin{equation}\n\\left(\\Omega_{i}^{k}\\right)^{(1)}=\\omega(f_i^{k,eq} - f_i^k),\n\\label{1coll}\n\\end{equation}\nwhere $\\omega=2\/(6\\bar{\\nu} -1)$ is the effective relaxation parameter being $\\bar{\\nu}$ the mean viscosity of the bi-component system computed as\n$\\frac{1}{\\bar{\\nu}}=\\frac{\\rho_1}{(\\rho_1+\\rho_2)}\\frac{1}{\\nu_1} + \\frac{\\rho_2}{(\\rho_1+\\rho_2)}\\frac{1}{\\nu_2}$ ($\\nu_1$ and $\\nu_2$ are the kinematic viscosities of the two pure components in the bulk). \nThe equilibrium distribution function of the $k^{th}$ component $f_i^{k,eq}$ is given by a low-Mach, second-order, expansion\nof a local Maxwellian, namely:\n\\begin{equation}\nf_i^{k,eq}=w_i \\rho^k (1 + \\frac{ \\vec{c_i} \\cdot \\vec{u}}{c_s^2} +\\frac{(\\vec{c_i} \\cdot \\vec{u})^2}{2c_s^4} - \\frac{\\vec{u} \\cdot \\vec{u}}{2 c_s^2}),\n\\label{eq:LEQ}\n\\end{equation}\nwhere $c_s=1\/\\sqrt{3}$ is the sound speed of the model.\nThe symbol $\\left(\\Omega_{i}^{k}\\right)^{(2)}$ denotes the perturbation step, which \ncontributes to the build up of an interfacial tension. Finally, $\\left(\\Omega_{i}^{k}\\right)^{(3)}$ is the recoloring step, which promotes the segregation  between species, so as to minimize their mutual diffusion.\n\nIn order to reproduce the correct form of the stress tensor, the perturbation operator can be constructed by exploiting the concept of the continuum surface force.\nFirstly, the perturbation operator must satisfy the following  conservation constraints:\n\n\\begin{eqnarray} \\label{consconstr}\n\\sum_i \\left(\\Omega_{i}^{k}\\right)^{(2)}=0 \\\\\n\\sum_k \\sum_i \\left(\\Omega_{i}^{k}\\right)^{(2)} \\vec{c}_i=0\n\\end{eqnarray}\n\nBy performing a Chapman-Enskog expansion, it can be shown that the hydrodynamic limit of Eq.\\ref{CGLBE} is represented by a\nset of equations for the conservation of mass and linear momentum:\n\n\\begin{eqnarray} \\label{NSE}\n\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot {\\rho \\vec{u}}=0 \\\\\n\\frac{\\partial \\rho \\vec{u}}{\\partial t} + \\nabla \\cdot {\\rho \\vec{u}\\vec{u}}=-\\nabla p + \\nabla \\cdot [\\rho \\nu (\\nabla \\vec{u} + \\nabla \\vec{u}^T)] + \\nabla \\cdot \\bm{\\Sigma}\n\\end{eqnarray}\n\nwhere $p=\\sum_k p_k$ is the pressure and $\\nu=c_s^2(\\tau-1\/2)$ is the kinematic viscosity of the mixture, being $\\tau$ the single relaxation time.\n\n\nThe stress tensor in the momentum equation is given by:\n\n\\begin{equation}\n\\bm{\\Sigma}=-\\tau \\sum_i \\sum_k \\left( \\Omega_{i}^{k} \\right)^{(2)} \\vec{c}_i \\vec{c}_i\n\\end{equation}\n\n\n\nSince the perturbation operator is responsible for generating interfacial tension, the following \nrelation must hold:\n\n\\begin{equation} \n\\nabla \\cdot \\bm{\\Sigma} = \\vec{F}\n\\label{SeqF}\n\\end{equation}\n\nDenoting $\\Theta =(\\rho^1-\\rho^2)\/(\\rho^1+\\rho^2)$ \nthe phase field, by choosing the second operator as:\n\\begin{equation} \n\\left(\\Omega_{i}^{k}\\right)^{(2)}= \\frac{A_k}{2} |\\nabla \\Theta|\\left[w_i \\frac{(\\vec{c}_i \\cdot \\nabla \\Theta)^2}{|\\nabla \\Theta|^2} -B_i \\right], \n\\label{2coll}\n\\end{equation}\nsubstituting it into Eqs \\ref{consconstr} and \\ref{SeqF} and by imposing that the set $B_i$ must satisfy the following isotropy constraints\n\n\\begin{eqnarray}\n\\sum_i B_i= \\frac{1}{3} \\; ; \\sum_i B_i \\vec{c}_i=0 \\;  ; \\sum_i B_i \\vec{c}_i \\vec{c}_i= \\frac{1}{3} \\mathbf{I},\n\\end{eqnarray}\n\nwe obtain an equation for the surface tension of the model:\n\\begin{equation}\\label{sigmaA}\n\\sigma=\\frac{2}{9}(A_1+A_2)\\frac{1}{\\omega}=\\frac{4}{9}A\\frac{1}{\\omega}.\n\\end{equation}\nThe above relation shows a direct link between the surface tension and the parameter $A$ with $A_1=A_2$.\nIn actual practice, after choosing the viscosity of the two components and the surface tension of the model, at each time step, one locally computes the $A$  coefficient by using the formula reported in eq. (\\ref{sigmaA}).\n\n\nAs pointed out above, the perturbation operator generates an interfacial tension in compliance with the capillary-stress tensor of the Navier-Stokes equations for a multicomponent fluid system.\n\nNonetheless, the perturbation operator alone does not guarantee the immiscibility of different fluid components.\nFor this reason, a further step is needed (i.e. the recoloring step) to minimize the mutual diffusion between components.\n\nFollowing the work of Latva-Kokko and\nRothman, the recoloring operator for the two sets of distributions takes the following form:\n\n\\begin{eqnarray}\n\\left(\\Omega_{i}^{1}\\right)^{(3)} =\\frac{\\rho^1}{\\rho} f_i^* + \\beta \\frac{\\rho^1\\rho^2}{\\rho^2} \\cos{\\phi_i} f_i^{eq,0} \\\\\n\\left(\\Omega_{i}^{2}\\right)^{(3)} =\\frac{\\rho^2}{\\rho} f_i^* - \\beta \\frac{\\rho^1\\rho^2}{\\rho^2} \\cos{\\phi_i} f_i^{eq,0},\n\\label{3coll}\n\\end{eqnarray}\n\nwhere $f_i^*=\\sum_k f_i^{k,*}$ denotes the set of post-perturbation distributions, $\\rho=\\rho^1 + \\rho^2$,  $\\cos{\\phi_i}$ is the angle between the phase field gradient and the $i^{th}$ lattice vector and $f_i^{eq,0}=f_i(\\rho,\\vec{u}=0)^{eq}=\\sum_k f_i^k(\\rho,\\vec{u}=0)^{eq}$ is the total zero-velocity equilibrium distribution function. In Eq. \\ref{3coll}, the coefficient $\\beta$ is a free parameter which tunes the interface width, thus playing the role of an inverse diffusion length scale. The coefficients used in Eqs \\ref{eq:LEQ} and \\ref{2coll} are reported in Table.\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{ c c c c }\n        \\hline \n        \\hline \n        D3Q19 & $\\{ i:\\left| c \\right|^2 = 0 \\}$ & $\\{ i:\\left| c \\right|^2 = 1 \\}$ & $\\{ i:\\left| c \\right|^2 = 2 \\}$ \\\\\n        \\hline \n        \\hline \n         $w_i$ & 1\/3 &  1\/18 & 1\/36  \\\\\n        \n         $B_i$ & -2\/9 & 1\/54 & 1\/27  \\\\\n        \n        \\hline\n    \\end{tabular}\n    \\caption*{D3Q19 lattice velocity and weights \\cite{leclaire2017generalized}.}\n    \\label{tab:coef}\n\\end{table}\n\nFor the details of the particle time evolution, a full explanation is in Ref. \\citep{lbsoft}. For the sake of completeness, the main key steps are outlined in the following.\n\n\nThe velocity at a boundary node of $p$-th particle is given by:\n\\begin{equation}\n\\label{UPI}\n\\vec{u}_{b} = \\vec{v}_p + \\vec{r}_b \\times \\vec{\\omega}_p,\n\\end{equation}\nwhere, $r_{b} = \\vec{r}_s + \\frac {1}{2} \\vec{c}_i$ is the \nlocation of the moving wall along the $i$-th link connecting\nthe solid node $\\vec{r}_s$ to the \nfluid node $\\vec{r}_i = \\vec{r}_s +\\vec{c}_i$.\nAll coordinates are relative to the center of the $p$-th particle, located at position $\\vec{r}_p$ and moving with translation and angular velocities $\\vec{v}_p$ and $\\vec{\\omega}_p$, respectively.\n\nThe timestep is made unit for simplicity. \nThis velocity sets the bias between colliding pairs:\n\\begin{eqnarray}\n\\label{LADDFLBE}\nf_i \\left( \\vec{r}+\\vec{c}_i, t+1 \\right) =\nf_{\\bar i}  \\left( \\vec{r}+\\vec{c}_i, t' \\right) + 2 \\rho w_i u_{bi}, \\\\\nf_{\\bar i} \\left( \\vec{r}, t+1 \\right) =  f_i \\left( \\vec{r}, t' \\right) \n- 2 \\rho w_i u_{bi}\n\\end{eqnarray}\nwhere $t'$ denotes post-collisional states and we have set\n$$\nu_{bi}= \\frac{\\vec{u}_b \\cdot \\vec{c}_i}{c_s^2}\n$$\nNote that these rules reduce to the usual bounce-back conditions for a solid at rest, $\\vec{v}_p=\\vec{\\omega}_p=0$.\n\nThese collision rules produce a net momentum transfer between the fluid and the solid site:\n\\begin{equation}\n\\label{FORCEPP}\n\\vec{F}_i(\\vec{r}_b,t+\\frac{1}{2}) = 2 \\vec{c}_i \n[f_i (\\vec{r},t') - f_{\\bar i}(\\vec{r}_i,t') - 2 \\rho w_i u_{bi}]\n\\end{equation}\n\nThe net force acting upon particle $p$ is obtained by summing over all boundary sites $\\vec{r}_b$ and associated interacting links, namely:\n\\begin{equation}\n\\label{FORCEPARTEQ}\n\\vec{F}_p = \\sum_{{\\vec{r}_b,i} \\in \\Sigma_p} \\vec{F}_{i}(\\vec{r}_b).\n\\end{equation}\nSimilarly, the total torque\\index{torque} $\\vec{T}_p$ is computed as\n\\begin{equation}\n\\label{TORQUEPARTEQ}\n\\vec{T}_p = \\sum_{{\\vec{r}_b,i} \\in \\Sigma_p} \\vec{F}_{i}(\\vec{r}_b)\n\\times \\vec{r}_b,\n\\end{equation}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe main goal of this article is to study the uniqueness property of the following mean field equations with singularities:\n\\begin{equation}\\label{m-equ}\n\\Delta_g v+\\rho\\bigg(\\frac{he^v}{\\int_M h e^v{\\rm d}\\mu}-\\frac{1}{vol_g(M)}\\bigg)=\\sum_{j=1}^N 4\\pi \\alpha_j (\\delta_{q_j}-\\frac{1}{vol_g(M)}) \\quad {\\rm in} \\ \\; M,\n\\end{equation}\nwhere $(M,g)$ be a Riemann surface with the metric $g$, $\\Delta_g$ is the Laplace-Beltrami operator ($-\\Delta_g\\ge 0$), $h$ is a positive smooth function on $M$, $q_1,\\cdots,q_N$ are distinct points on $M$, $\\rho>0,\\alpha_j>-1$ are constants, $\\delta_{q_j}$ is the Dirac measure at $q_j\\in M$. Equation (\\ref{m-equ}) is one of the most extensively studied elliptic PDE in the past few decades, partly due to its immense and profound connections with many branches of mathematics and Physics. In conformal geometry, (\\ref{m-equ}) represents a metric on M with conic singularity (see \\cite{fang-lai,troy,wei-zhang-pacific}). Also it is derived from the mean\nfield limit of point vortices in the Euler flow \\cite{caglioti-1,caglioti-2} and serves as a model equation\nin the Chern-Simons-Higgs theory \\cite{spruck-yang,taran-1,y-yang} and in the electroweak theory \\cite{ambjorn}, etc. The literature for the study of various form of (\\ref{m-equ}) is just too numerous to be listed in any reasonable way.\n\n\nRecently it was found by Lin-Yan \\cite{lin-yan-uniq} that the uniqueness property is particularly important for equations with concentration phenomenon. In their work \\cite{lin-yan-uniq} they proved the first uniqueness property for bubbling solutions of Chern-Simon-Higgs equation and computed the exact number of solutions in certain special cases. In an important work \\cite{bart-4} Bartolucci, et. al, extended Lin-Yan's result for mean field equation (\\ref{m-equ}) if the blowup points are not singular sources. Our goal in this article is to further extend the uniqueness property to the case that some singular sources coincide with blowup points.\n\n\\smallskip\n\nTo write the main equation in an equivalent form, we invoke the standard Green's function$G(x,p)$:\n\\begin{equation}\\label{gf}\n\\left\\{\\begin{array}{ll}\n-\\Delta_g G(x,p)=\\delta_p-1\\quad {\\rm in}\\ \\; M\n\\\\\n\\int_{M}G(x,p){\\rm d}\\mu=0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere the volume of $M$ is assumed to be $1$ for convenience. Then it is well known that in a neighborhood of $p$, $G(x,p)$ can be written as\n$$G(x,p)=-\\frac 1{2\\pi }\\log dist(x,p)+R(x,p)$$\nwhere $dist(x,p)$ is the geodesic distance from $p$ to $x$ for $x$ close to $p$.\n\nUsing $G(x,p)$ we write (\\ref{m-equ}) as\n\\begin{equation}\\label{r-equ}\n\t\\Delta_g w+\\rho\\bigg(\\frac{He^w}{\\int_M H e^w{\\rm d}\\mu}-1\\bigg)=0 \\quad {\\rm in}\\ \\; M,\n\\end{equation}\nwhere\n\\begin{equation}\\label{r-sol}\n\tw(x)=v(x)+4\\pi \\sum_{j=1}^N \\alpha_j G(x,q_j),\n\\end{equation}\nand\n\\begin{equation}\\label{H1}\n\tH(x)=h(x)\\prod_{j=1}^N e^{-4\\pi\\alpha_j G(x,q_j)}.\n\\end{equation}\nNote that in a local coordinate near $q_j$,\n\\begin{equation}\\label{H2}\nH(x)=h_j(x)|x-q_j|^{2\\alpha_j},\\quad |x-q_j|\\ll 1,\\quad 1\\leq j\\leq N,\n\\end{equation}\nfor some $h_j(x)>0$.\n\nWe say that $\\{v_k\\}$ is a sequence of bubbling solutions of (\\ref{m-equ}) if the corresponding $w_k$ defined by (\\ref{r-sol}) tends to infinity as $k$ goes to infinity. The places that $w_k$ tends to infinity are called blowup points of $v_k$ or $w_k$. In this article we use $p_1,...,p_m$ to denote blowup points. Let $q_1,...,q_N$ be the location of singular sources. If none of $p_1,...p_m$ is a singular source, Bartolucci, et. al have obtained the uniqueness of the blow up solution in \\cite{bart-4}. Thus in this article we consider two cases: either all blowup points are singular sources or part of blowup points coincide with singular sources. In more precise terms let\n\\begin{equation}\\label{pq}\n\\left\\{\\begin{array}{ll}\np_j=q_j\\quad &{\\rm if} \\ 1\\leq j \\leq \\tau,\n\\\\\np_j \\notin \\{q_1,\\cdots,q_N\\}\\quad &{\\rm if} \\ \\tau+1\\leq j\\leq m,\n\\end{array}\n\\right.\n\\end{equation}\nfor some $1\\le \\tau\\le m$. Thus if $\\tau=m$ all blowup points are singular sources, if $1\\le \\tau<m$, some blowup points are singular sources and some are not. Let $4\\pi\\alpha_j$ be the strength of the singular source at $p_j$, so we have $\\alpha_j=0$ if $j>\\tau$. Since the largest $\\alpha_j$ matters the most we require the first $t$ of them to have this strength:\n\\begin{equation}\n\\alpha_1=\\cdots =\\alpha_t>\\alpha_l, \\quad l\\geq t+1, \\quad \\mbox{where } 1\\le t\\le \\tau.\n\\end{equation}\n\nIt is well known that equation (\\ref{r-equ}) is the Euler-Lagrange equation of the variational form:\n$$I_{\\rho}(w)=\\frac 12 \\int_M |\\nabla w|^2+\\rho\\int_Mw-\\rho \\log \\int_M He^w, $$\nfor $w\\in H^1(M)$. Since adding a constant to any solution of (\\ref{r-equ}) certainly gives to another solution, the space of solutions for (\\ref{r-equ}) is the set of all $H^1(M)$ function with average equal to $0$. The discussion on the variational structure of (\\ref{r-equ}) can be found in \\cite{machiodi-1}.\n\n\\smallskip\n\nTo state the main results we use the following notations:\n\\begin{align}\n&G_j^*(x)=8\\pi (1+\\alpha_j)R(x,p_j)+8\\pi \\sum_{l\\neq j}^{1,\\cdots,m}(1+\\alpha_l)G(x,p_l),  \\label{G_j*}\n\\\\\n&L(\\mathbf{p})=\\sum_{j=1}^t \\big[\\Delta \\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big] (h_j(p_j))^{\\frac{1}{1+\\alpha_1}}e^{\\frac{G_j^*(p_j)}{1+\\alpha_1}},   \\label{L}\n\\\\\n&D(\\mathbf{p})=\n\\begin{pmatrix}\n\\nabla(\\log h_1+G_1^*)(p_1) \\\\\n\\cdots \\\\\n\\nabla(\\log h_t+G_t^*)(p_t)\n\\end{pmatrix}\n,  \\label{D}\n\\end{align}\nwhere $h_j$ is defined in (\\ref{H2}), and\n\\begin{equation*}\n  \\rho_*=8\\pi\\sum_{j=1}^m(1+\\alpha_j),\\quad N^*=4\\pi\\sum_{j=1}^N\\alpha_j\n\\end{equation*}\n\nOur first result is when all blowup points are singular sources:\n\\begin{thm}\\label{main-theorem}\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of bubbling solutions of (\\ref{m-equ}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $\\alpha_j\\in\\mathbb{R}^+\\setminus\\mathbb{N}\\,(1\\leq j\\leq m)$. If $L(\\mathbf{p})\\neq0$ and $D(\\mathbf{p})=0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\n\nNote that we use $\\mathbb N$ to denote the set of positive integers. The assumption that $\\alpha_j\\in \\mathbb{R}^+\\setminus\\mathbb{N}$ implies that all blowup points are singular sources. It is also very essential to require $\\alpha_j$ to be non-integer, since quantized singular sources ( if the strength is $4\\pi N$) exhibit non-simple blowup phenomenon \\cite{kuo-lin}\\cite{wei-zhang-19}\nthat has to be studied in a separate work in the future.\n\nThe assumption of $D({\\bf p})$ is also very interesting. It is well known that if $p$ is not a singular source, the vanishing rate of $D({\\bf p})$ is very fast for a regular blowup point ( \\cite{gluck},\\cite{chen-lin-sharp}).\n\\medskip\n\nOur second main result is about the uniqueness of bubbling solutions when some blowup points are non-quantized singular sources and some are regular points. So in this case we require $1\\le \\tau<m$\nand for $(x_{\\tau+1},\\cdots,x_m)\\in M\\times\\cdots \\times M$, we define\n\\begin{equation}\\label{f*}\n  f^*(x_{\\tau+1},\\cdots,x_m)=\\sum_{j=\\tau+1}^m\\big[\\log h(x_j)+4\\pi R(x_j,x_j)\\big]+4\\pi \\sum_{l\\neq j}^{\\tau +1,\\cdots,m}G(x_l,x_j).\n\\end{equation}\nIt is well known that $(p_{\\tau+1},\\cdots,p_m)$ is a critical point of $f^*$.\n\n\n\\begin{thm}\\label{main-theorem-2}\n\t\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of bubbling solutions of (\\ref{m-equ})  with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $0\\leq\\alpha_j<1(1\\leq j\\leq m)$. Suppose $1\\le \\tau<m$, $L(\\mathbf{p})\\neq0$, $D(\\mathbf{p})=0$ and $\\det \\big(D^2f^*(p_{\\tau+1},\\cdots,p_m)\\big)\\neq 0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\nThe notation $D^2f^*$ in Theorem \\ref{main-theorem-2} stands for the Hessian tensor field on $M$. Theorems \\ref{main-theorem} and Theorem \\ref{main-theorem-2} are clearly extensions of the main theorem in \\cite{bart-4}, where the uniqueness of bubbling solutions around regular blowup points is established. Here in our work, the assumptions of $L(\\mathbf{p})$ and $D(\\mathbf{p})$\nare only placed on singular sources with the strongest strength.\n\nIn addition to the importance of application, the proof of the main theorems requires extremely delicate local analysis, just like the argument in \\cite{bart-4}. Our argument relies heavily on the result of the second author in \\cite{zhang2}, Chen-Lin's refined estimates in \\cite{chen-lin,chen-lin-deg-2} and the argument used by Lin-Yan \\cite{lin-yan-uniq} and Bartolucci-et-al \\cite{bart-4}. Even though the outline of our paper is similar to those used in \\cite{lin-yan-uniq,bart-4} we have to establish accurate estimates for certain terms in an iterative manner.\n\nThe proof of Theorem \\ref{main-theorem} and Theorem \\ref{main-theorem-2} can also be applied to solve the following locally defined Dirichlet boundary problem:\nLet $\\Omega$ be an open and bounded domain in $\\mathbb{R}^2$ with regular boundary $\\partial\\Omega\\in C^2$, $v$ be a solution of\n\\begin{equation}\\label{equ-flat}\n\\left\\{\\begin{array}{lll}\n\\Delta v+\\rho \\frac{he^v}{\\int_{\\Omega} h e^v{\\rm d}x}=\\sum_{j=1}^N 4\\pi \\alpha_j \\delta_{q_j}  &{\\rm in} \\;\\ \\Omega,\n\\\\\nv=0  &{\\rm on} \\;\\ \\partial\\Omega,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $h>0$ is a $C^1$ funation in $\\Omega$, $q_1,\\cdots,q_N$ are distinct points in $\\Omega$, $\\rho>0$, $\\alpha_j>0$ are constants.\n\nLet $\\{v_k\\}$ be a sequence of solutions to (\\ref{equ-flat}) with $\\rho=\\rho_k$. We say\n\\begin{equation}\\label{blowup-flat}\nv_k \\  {\\rm blows} \\  {\\rm up} \\  {\\rm at} \\  p_j\\in\\Omega,\\quad 1\\leq j\\leq m,\n\\end{equation}\nif $\\rho \\frac{he^v}{\\int_{\\Omega} h e^v{\\rm d}x}\\rightharpoonup8\\pi\\sum_{j=1}^N(1+\\alpha_j)\\delta_{p_j}$ in $\\Omega$ in the sense of measure, where $\\alpha_j=0$ if $p_j\\notin\\{q_1\\cdots,q_N\\}$. Similar to notations for the first part, we assume there exist $1\\leq t\\leq\\tau\\leq m$ such that $\\alpha_1=\\cdots\\alpha_t>\\alpha_i$, $i\\ge t+1$ and $\\alpha_{\\tau+1}=\\cdots\\alpha_m$.\n\nLet $G_{\\Omega}$ be the Green's function defined by\n\\begin{equation*}\n\\left\\{\\begin{array}{lll}\n-\\Delta G_{\\Omega}(x,p)=\\delta_{p}  &{\\rm in} \\;\\ \\Omega,\n\\\\\nG_{\\Omega}(x,p)=0  &{\\rm on} \\;\\ \\partial\\Omega,\n\\end{array}\n\\right.\n\\end{equation*}\nand $R_{\\Omega}(x,p)=G_{\\Omega}(x,p)+\\frac{1}{2\\pi}\\log |x-p|$ be the regular part of $G_{\\Omega}(x,p)$. In order to state the uniqueness results of (\\ref{equ-flat}) we denote $N^*=4\\pi\\sum_{j=1}^m\\alpha_j$ and\n\\begin{align*}\n&G_{j,\\Omega}^*(x)=8\\pi (1+\\alpha_j)R_{\\Omega}(x,p_j)+8\\pi \\sum_{l\\neq j}^{1,\\cdots,m}(1+\\alpha_l)G_{\\Omega}(x,p_l),\n\\\\\n&L_{\\Omega}(\\mathbf{p})=\\sum_{j=1}^t \\big[\\Delta \\log h(p_j)-N^*\\big] (h_j(p_j))^{\\frac{1}{1+\\alpha_1}}e^{\\frac{G_j^*(p_j)}{1+\\alpha_1}},\n\\\\\n&D_{\\Omega}(\\mathbf{p})=\n\\begin{pmatrix}\n\\nabla(\\log h_1+G_1^*)(p_1) \\\\\n\\cdots \\\\\n\\nabla(\\log h_t+G_t^*)(p_t)\n\\end{pmatrix}\n.\n\\end{align*}\n\nThen we have the following result similar to Theorem \\ref{main-theorem}.\n\n\\begin{thm}\\label{main-theorem-3}\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of solutions of (\\ref{equ-flat}) (\\ref{blowup-flat}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $\\alpha_j\\in\\mathbb{R}^+\\setminus\\mathbb{N}\\,(1\\leq j\\leq m)$. If $L_{\\Omega}(\\mathbf{p})\\neq0$ and $D_{\\Omega}(\\mathbf{p})=0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\n\nIf the set of blowup points is a mixture of non-quantized singular sources and regular points, we also have a uniqueness result. Let\n\\begin{equation*}\nf_{\\Omega}^*(x_{\\tau+1},\\cdots,x_m)=\\sum_{j=\\tau+1}^m\\big[\\log h(x_j)+4\\pi R(x_j,x_j)\\big]+4\\pi \\sum_{l\\neq j}^{\\tau +1,\\cdots,m}G(x_l,x_j),\n\\end{equation*}\nand $D^2f_{\\Omega}^*$ be the Hessian tensor field on $M$. In this case, $(p_{\\tau+1},\\cdots,p_m)$ is a critical point of $f_{\\Omega}^*$. Then, we obtain the following result.\n\n\\begin{thm}\\label{main-theorem-4}\n\t\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of solutions of (\\ref{equ-flat}) (\\ref{blowup-flat}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $0\\leq\\alpha_j<1(1\\leq j\\leq m )$. If $L_{\\Omega}(\\mathbf{p})\\neq0$, $D_{\\Omega}(\\mathbf{p})=0$ and $\\det \\big(D^2f_{\\Omega}^*(p_{\\tau+1},\\cdots,p_m)\\big)\\neq 0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\n\nWhen we were in the final stage of writing this article, we found that Bartolucci, et, al \\cite{bart-4-2} posted an article on arxiv.org about the same topic. Their theorem is a special case of our results and both works were carried out independently.\n\n\\smallskip\n\nThe organization of this paper is as follows. Section \\ref{preliminary} is dedicated to notations and preliminary sharp estimates for bubbling solutions of equation (\\ref{m-equ}).  In section \\ref{difference} we consider the differences between two bubbling sequences and establish many estimates near each blowup point and away from all blowup points. In section \\ref{anal-pohozaev} we derive some Pohozaev-type identities and evaluate each term carefully. These Pohozaev identities play a key role in the proof of the main theorems. Finally the proof of Theorem \\ref{main-theorem} is placed in section \\ref{pf-uni-1} and that of Theorem \\ref{main-theorem-2} can be found in section \\ref{pf-uni-2}. At the end of section \\ref{pf-uni-2}, we list the brief sketch of the proof of Theorems \\ref{main-theorem-3} and \\ref{main-theorem-4} based on well known facts \\cite{ma-wei}.\n\n\n\n\\section{Preliminary Estimates}\\label{preliminary}\n\nSince the proof of the main theorems requires very delicate analysis, in this section we list some established estimates in \\cite{chen-lin-sharp,chen-lin,zhang1,zhang2}.\n\n\n\n\nLet $w_k$ be a sequence of solutions of (\\ref{r-equ}) with $\\rho =\\rho_k$. Suppose that $w_k$ blows up at $m$ points $\\{p_1 \\cdots,p_m\\}$ as we have stated in section one. To describe the bubbling profile of $w_k$ near $p_j$, we set\n\\begin{equation}\\label{n-sol}\n\tu_k=w_k-\\log\\bigg(\\int_M He^{w_k}{\\rm d}\\mu \\bigg)\n\\end{equation}\n and write the equation for $u_k$ as\n\\begin{equation}\\label{n-equ}\n\\Delta_g u_k+\\rho_k(He^{u_k}-1)=0\\quad {\\rm in} \\ \\; M.\n\\end{equation}\nIt is easy to observe from the definition of $u_k$ that $$\\int_{M}He^{u_k}{\\rm d}\\mu=1.$$\n\n\nFrom previous works of Liouville equations ( for example \\cite{chen-lin-sharp} ),\n\\begin{equation}\\label{local-cov}\n u_k-\\bar{u}_k \\ \\to \\sum_{j=1}^m 8\\pi(1+\\alpha_j)G(x,p_j) \\quad {\\rm in} \\ \\; {\\rm C}_{loc}^2(M\\backslash \\{p_1,\\cdots,p_m\\})\n\\end{equation}\nwhere $\\bar{u}_k$ is the average of $u_k$ on $M$:\n$$\\bar{u}_k=\\int_{M}u_k{\\rm d}\\mu.$$\n\nFor the convenience later we fix $r_0>0$ small  and $M_j\\subset M, 1\\leq j\\leq m$ such that\n\\begin{equation}\\label{Mj}\nM=\\bigcup_{j=1}^m \\overline{M}_j;\\quad M_j\\cap M_l=\\varnothing,\\  {\\rm if}\\ j\\neq l;\\quad B(p_j,3r_0)\\subset M_j, \\ j=1,\\cdots,m.\n\\end{equation}\nAccording to this definition $M_1=M$, if $m=1$.\n\n\nThen we use  $\\lambda_{k,j}$ to denote\n\\begin{equation}\\label{lambda_kj}\n\\lambda_{k,j}=\\left\\{\n\\begin{array}{lcl}\nu_k(p_j) && {\\rm if}\\ \\,\\alpha_j\\neq 0, \\\\\nu_k(p_{k,j}) \\mathrel{\\mathop:}=\\max_{B(p_j,r_0)}u_k  && {\\rm if}\\ \\,\\alpha_j= 0.\n\\end{array}\n\\right.\n\\end{equation}\nand let $U_{k,j}$ be a global solution of\n\\begin{equation}\\label{U_kj-equ}\n\\Delta U_{k,j}+\\rho_kh_j(p_{k,j})|x-p_{k,j}|^{2\\alpha_j}e^{U_{k,j}}=0 \\quad  {\\rm in} \\ \\; \\mathbb{R}^2\n\\end{equation}\nwith the expression ($U_{k,j}$ is called a standard bubble):\n\\begin{equation}\\label{U_kj}\nU_{k,j}(x)=\\lambda_{k,j}-2\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}}|x-p_{k,j}|^{2(1+\\alpha_j)}\\Big).\n\\end{equation}\n\nIt is well-known \\cite{li-cmp,Bartolucci-Chen-Lin-T} that $u_k$ can be approximated by the standard bubbles $U_{k,j}$ near $p_j$ with $O(1)$ error:\n\\begin{equation}\\label{standard-bubble}\n  \\big|u_k(x)-U_{k,j}(x)\\big| \\leq C, \\quad x\\in B(p_j,r_0).\n\\end{equation}\n\n As a consequence,\n\\begin{equation}\\label{lambda-ij}\n|\\lambda_{k,i}-\\lambda_{k,j}|\\leq C, \\quad 1\\leq i,j \\leq m.\n\\end{equation}\nfor some $C$ independent of $k$.\nFurthermore, it is established in \\cite{bart-taran-mass} that $\\rho_*=\\lim_{k\\to +\\infty}\\rho_k$.\n\n\\medskip\n\n\nLater, sharper estimates were obtained in \\cite{zhang2,chen-lin} for $1\\leq j \\leq \\tau$ and in \\cite{chen-lin-sharp,zhang1,gluck} for $\\tau+1\\leq j\\leq m$. In order to apply those estimates, we might consider the equation in terms of the flat metric and introduce the following notations.\n\n\\medskip\n\nIn $B(p_j,r_0)$, the flat metric is ${\\rm d}s^2=e^{\\phi_j}\\big(({\\rm d}x_1)^2+({\\rm d}x_2)^2\\big)$ with $\\phi_j$ satisfying\n\\begin{equation}\\label{phi-equ}\n\\left\\{\\begin{array}{lcl}\n\\Delta \\phi_j+2Ke^{\\phi_j}=0, &&  {\\rm in} \\ \\; B(p_j,r_0),\n\\\\\n\\phi_j(0)=|\\nabla \\phi_j(0)|=0, && \\quad\n\\end{array}\n\\right.\n\\end{equation}\nwhere $0$ is the coordinate of $p_j$, $\\Delta =\\sum _{i=1}^{2}\\frac{\\partial^2}{\\partial x_i^2}$. In this local coordinate, equation (\\ref{n-equ}) is equivalent to\n\\begin{equation}\\label{flat-equ}\n\\Delta u_k+\\rho_ke^{\\phi_j}(He^{u_k}-1)=0\\quad {\\rm in} \\ \\; B(p_j,r_0).\n\\end{equation}\nIf we denote $\\tilde{h}_j=h_je^{\\phi_j}$, then (\\ref{flat-equ}) can be written as follows:\n\\begin{equation}\\label{flat-equ-2}\n\\Delta u_k+\\rho_k\\tilde{h}_j|x-p_j|^{2\\alpha_j}e^{u_k}-\\rho_ke^{\\phi_j}=0\\quad {\\rm in} \\ \\; B(p_j,r_0).\n\\end{equation}\n\nTo state the more refined asymptotic analysis we introduce the following notations:\n\\begin{equation}\\label{loc-mass}\n\\rho_{k,j}=\\int_{B(p_{k,j},r_0)}\\rho_kHe^{u_k}{\\rm d}\\mu,\\quad 1\\leq j\\leq m,\n\\end{equation}\n\\begin{equation}\\label{sigma1}\n\\sigma_k(x)=u_k(x)-\\bar{u}_k-\\sum_{j=1}^m\\rho_{k,j} G(x,p_{k,j}), \\quad x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j},\\frac{r_0}{2}),\n\\end{equation}\n\\begin{equation}\\label{G_kj}\nG_{k,j}(x)=\\rho_{k,j}R(x,p_{k,j})+\\sum_{l\\neq j}^{1,\\cdots,m}\\rho_{k,l}G(x,p_{k,l}),\\quad x\\in B(p_{k,j},r_0),\n\\end{equation}\nwhere $R(x,p_{k,j})$ is the regular part of $G(x,p_{k,j})$. Finally\nfor $x\\in B(p_{k,j},r_0)$, set\n\\begin{equation*}\n\\tilde{u}_{k,j}(x)=u_k(x)-\\big(G_{k,j}(x)-G_{k,j}(p_{k,j})\\big),\n\\end{equation*}\n\\begin{equation}\\label{eta1}\n\\eta_{k,j}(x)=\\tilde{u}_{k,j}(x)-U_{k,j}(x).\n\\end{equation}\n\n\\subsection{Sharper estimates}\n\\quad\n\\medskip\n\nIf $\\alpha_j\\in \\mathbb{R}^+\\setminus\\mathbb{N}$, in order to obtain the refined estimates of the bubbling solutions, the second author considered the harmonic function $\\psi_{k,j}$ in \\cite{zhang2}, which satisfies\n\\begin{equation}\\label{psi-equ}\n\\left\\{\\begin{array}{lcl}\n\\Delta \\psi_{k,j}=0 && {\\rm in} \\ \\; B(p_{k,j},r_0),\n\\\\\n\\psi_{k,j} =\\tilde{u}_{k,j}-\\frac{1}{2\\pi r_0}\\int_{\\partial B(p_{k,j},r_0)}\\tilde{u}_{k,j} {\\rm d}s && {\\rm on} \\ \\; \\partial B(p_{k,j},r_0).\n\\end{array}\n\\right.\n\\end{equation}\n\nWith the help of $\\psi_{k,j}$, Zhang and Chen-Lin proved the following sharp estimate in \\cite{zhang2}.\n\\begin{thmA}\\label{Theorem zhang}\\cite{zhang2,chen-lin}\n\t\n\tFor $x\\in B(p_{k,j},r_0)$, it holds that\n\t\\begin{equation}\\label{zhang}\n\t\\begin{split}\n\t\\eta_{k,j}(x)=&\\psi_{k,j}(x)-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{\\langle a,x-p_{k,j}\\rangle}{1+\\frac{\\rho_k h_j(p_{(k,j)})}{8{(1+\\alpha_j)^2}}e^{\\lambda_{k,j}}|x-p_{k,j}|^{2(1+\\alpha_j)}}\\\\\n\t& +d_j\\log \\big(2+e^{\\frac{\\lambda_{k,j}}{2(1+\\alpha_j)}}|x-p_{k,j}|\\big)e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_j}}+O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_j}}),\n\t\\end{split}\n\t\\end{equation}\n\twhere $a=\\nabla(\\log h_j+G_{k,j})(p_{k,j}) \\in \\mathbb{R}^2$ and\n\t\\begin{equation*}\n\t\td_j=\\frac{\\pi}{(1+\\alpha_j)\\sin \\frac{\\pi}{1+\\alpha_j}}\\Big(\\frac{8(1+\\alpha_j)^2}{\\rho_kh_j(p_{k,j})}\\Big)^{\\frac{1}{1+\\alpha_j}}\\big[\\Delta\\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big].\n\t\\end{equation*}\n\\end{thmA}\n\nIn \\cite{chen-lin},the following estimates for $\\psi_{k,j}$ and $\\sigma_k$ are established:\n\\begin{thmA}\\label{Theorem chen-lin1}\\cite{chen-lin}\n\t\\begin{equation}\\label{psi-est}\n\t|\\psi_{k,j}(x)|=O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}),\\quad x\\in B(p_{k,j},r_0).\n\t\\end{equation}\n\t\\begin{equation}\\label{sigma2}\n\t|\\sigma_k(x)|+|\\nabla\\sigma_k(x)|=O(e^{-\\frac{\\lambda_{k,1}}{1+\\alpha_1}}),\\quad x\\in M\\backslash \\big(\\bigcup_{j=1}^mB(p_{k,j},\\frac{r_0}{2})\\big).\n\t\\end{equation}\n\\end{thmA}\n\nThen, by Theorem \\ref{Theorem zhang} and Theorem \\ref{Theorem chen-lin1}, we have\n\\begin{equation}\\label{eta2}\n|\\eta_{k,j}(x)|=O(e^{-\\frac{\\lambda_{k,j}}{2(1+\\alpha_j)}}+e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}),\\quad x\\in B(p_{k,j},r_0),\\quad 1\\leq j\\leq \\tau.\n\\end{equation}\nFor the case $\\tau+1\\leq j \\leq m $, the estimate for $\\eta_{k,j}$, established in \\cite{chen-lin-sharp}\\cite{zhang1}\\cite{gluck}, is\n\\begin{equation}\\label{eta3}\n|\\eta_{k,j}(x)|=O(\\lambda_{k,j}e^{-\\lambda_{k,j}}),\\quad x\\in B(p_{k,j},r_0),\\quad \\tau+1\\leq j\\leq m.\n\\end{equation}\n\nMoreover, according to the proof of Theorem 3.5 in \\cite{chen-lin}, the following estimate holds:\n\\begin{equation}\\label{uk-ave-1}\n\\bar{u}_k+\\lambda_{k,j}+2\\log\\dfrac{\\rho_kh_j(p_{k,j})}{8(1+\\alpha_j)^2}+G_{k,j}(p_{k,j})+\\frac{d_j}{2(1+\\alpha_j)}\\lambda_{k,j} e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}=O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}).\n\\end{equation}\nAs a consequence, we have\n\\begin{equation}\\label{uk-ave-2}\n\\lambda_{k,j}-\\lambda_{k,1}=2\\log\\dfrac{(1+\\alpha_j)^2h_1(p_{k,1})}{(1+\\alpha_1)^2 h_j(p_{k,j})}+G_{k,1}(p_{k,1})-G_{k,j}(p_{k,j})+O(e^{-\\frac{\\lambda_{k,1}}{2(1+\\alpha_1)}}).\n\\end{equation}\n\nFor the difference between $\\rho_k$ and $\\rho_*$, $\\rho_k$ and $8\\pi(1+\\alpha_j)$, the following estimates also have been proved in \\cite{chen-lin,chen-lin-sharp}.\n\\begin{thmA}\\label{Theorem chen-lin2}\\cite{chen-lin,chen-lin-sharp}\t\n\t\\begin{align}\n\t&\\rho_{k,j}-8\\pi(1+\\alpha_j)=2\\pi d_je^{-\\frac{\\lambda_{k,j}}{1+\\alpha_j}}+O\\big(e^{-\\frac{1+\\gamma}{1+\\alpha_1}\\lambda_{k,1}}\\big), && 1\\leq j\\leq \\tau, \\label{rho-kj-1}\n\t\\\\\n\t&\\rho_{k,j}-8\\pi=O\\big(\\lambda_{k,j}e^{-\\lambda_{k,j}}\\big), &&\\tau+1\\leq j\\leq m,  \\label{rho-kj-2}\n\t\\\\\n\t&\\rho_k-\\rho_*=L^*e^{-\\frac{\\lambda_{k,1}}{1+\\alpha_1}}+O\\big(e^{-\\frac{1+\\gamma}{1+\\alpha_1}\\lambda_{k,1}}\\big),&&  \\label{rho-k}\n\t\\end{align}\n\twith fixed $\\gamma\\in (0,\\min({\\alpha_1,\\frac{1}{2}}))$ small and\n\t$$L^*=\\dfrac{2\\pi^2}{(1+\\alpha_1)\\sin\\frac{\\pi}{1+\\alpha_1}}e^{-\\frac{G_1^*(p_{1})}{1+\\alpha_1}}\\Big(\\frac{8(1+\\alpha_1)^2}{\\rho_* h_1(p_1)^2}\\Big)^{\\frac{1}{1+\\alpha_1}}L(\\mathbf{p}).$$\n\\end{thmA}\n\n\\smallskip\n\nIf $\\tau<m$, as in \\cite{chen-lin-sharp} and \\cite{bart-4}, non-degeneracy condition $\\det\\big(D^2f^*(p_{\\tau+1},\\cdot,p_m)\\big)\\neq0$ leads to\n\\begin{equation}\\label{p_kj-location}\n|p_{k,j}-p_j|=O(\\lambda_{k,j}e^{-\\lambda_{k,j}}),\\quad \\tau+1\\leq j\\leq m.\n\\end{equation}\nFuthermore, in \\cite{chen-lin}, the authors showed that\n\\begin{equation}\\label{first-deriv-est}\n\\nabla(\\log h+G_j^*)(p_{k,j})=O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}),\\quad \\tau+1\\leq j\\leq m.\n\\end{equation}\n\n\n\\subsection{The kernel of the linearized equations}\n\\quad\n\\medskip\n\nIn the proof of the uniqueness, we need some facts about the linearized equation after the appropriate rescale.\n\n\\medskip\n\nFor $\\alpha\\in \\mathbb{R}^+\\setminus\\mathbb{N}$, Chen-Lin proved the following lemma in \\cite{chen-lin}.\n\\begin{lemA}\\label{linear-lem-1}\n\tSuppose $ \\alpha>0 $ is not an integer, $ \\varphi $ is a $C^2$-function that satisfies\n\t\\begin{equation*}\n\t\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta \\varphi+|x|^{2\\alpha}e^{U_\\alpha}\\varphi=0\\quad & {\\rm in} \\ \\; \\mathbb{R}^2,\n\t\t\\\\\n\t\t|\\varphi| \\leq (1+|x|)^{\\kappa} \\quad & {\\rm in} \\ \\;\\mathbb{R}^2,\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation*}\n\twhere $ U_{\\alpha}(x)=\\log\\frac{8(1+\\alpha)^2}{(1+|x|^{2(1+\\alpha)})^2} $ and $\\kappa\\in(0,1)$. Then there exists some constant $b_0$ such that\n\t\\begin{equation*}\n\t\\varphi(x)= b_0\\frac{1-|x|^{2(1+\\alpha)}}{1+|x|^{2(1+\\alpha)}}.\n\t\\end{equation*}\n\\end{lemA}\n\n\nFor $\\alpha=0$, Chen-Lin proved the following lemma in \\cite{chen-lin-sharp}.\n\\begin{lemA}\\label{linear-lem-2}\n\tLet  $ \\varphi $ be a $ C^2 $-function of\n\t\\begin{equation*}\n\t\\left\\{\\begin{array}{ll}\n\t\\Delta \\varphi+e^U\\varphi=0\\quad & {\\rm in} \\ \\;\\mathbb{R}^2,\n\t\\\\\n\t|\\varphi| \\leq c\\big(1+|x|\\big)^{\\kappa} \\quad & {\\rm in} \\ \\;\\mathbb{R}^2,\n\t\\end{array}\n\t\\right.\n\t\\end{equation*}\n\twhere $ U(x)=\\log\\frac{8}{(1+|x|^2)^2} $ and $ \\kappa \\in[0,1) $. Then there exist constants $b_0$, $b_1$, $b_2$ such that\n\t\\begin{equation*}\n\t\t \\varphi= b_0\\varphi_0+b_1\\varphi_1+b_2\\varphi_2,\n\t\\end{equation*}\n\twhere\n\t\\begin{equation*}\n\t\\varphi_0(x)= \\frac{1-|x|^2}{1+|x|^2},\\quad \\varphi_1(x)= \\frac{x_1}{1+|x|^2},\\quad \\varphi_2(x)= \\frac{x_2}{1+|x|^2}.\n\t\\end{equation*}\n\t\n\\end{lemA}\n\n\\section{The difference between $ u_k^{(1)} $ and $ u_k^{(2)} $}\\label{difference}\n\nThe way we prove the main theorems is by contradiction. So we assume that $ u_k^{(1)} $ and $ u_k^{(2)} $\nare two different sequences of solutions to (\\ref{r-equ}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(2)}$, and common blowup points located at $p_1,\\cdots,p_m$. For $i=1,2$, we use\nthe following notations\n$$\\lambda_{k,j}^{(i)}, u_{k,j}^{(i)}, v_{k,j}^{(i)}, \\rho_{k,j}^{(i)}, \\bar{u}_k^{(i)}, U_{k,j}^{(i)}, G_{k,j}^{{(i)}}, \\psi_{k,j}^{(i)}, \\eta_{k,j}^{(i)}, \\epsilon_{k,j}^{(i)},  \\sigma_k^{(i)}, p_{j}^{(i)}, $$\nwith obvious interpretations in the context.\nFinally the following three functions are defined by the difference of $u_1^k$ and $u_2^k$:\n\\begin{align}\n&\t\\varsigma_k(x)=\\dfrac{u_k^{(1)}(x)-u_k^{(2)}(x)}{\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}}, \\label{varsigma}\n\\\\\n&f_k(x)=\\rho_k H(x)\\frac{e^{u_k^{(1)}(x)}-e^{u_k^{(2)}(x)}}{\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}}, \\label{f}\n\\\\\n&c_k(x)=\\dfrac{e^{u_k^{(1)}(x)}-e^{u_k^{(2)}(x)}}{u_k^{(1)}(x)-u_k^{(2)}(x)}.  \\label{c}\n\\end{align}\nClearly $\\varsigma_k$ satisfies\n\\begin{equation}\\label{sigma-equ}\n\\Delta_g\\varsigma_k(x)+f_k(x)=\\Delta_g\\varsigma_k(x)+\\rho_kH(x)c_k(x)\\varsigma_k(x)=0,\\quad x\\in M.\n\\end{equation}\n\nAs the first step of our proof, we give an initial estimate of $ \\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}$ using $L({\\bf p})\\neq 0$:\n\\begin{lem}\\label{est1}\nUnder the assumption of $L(\\mathbf{p})\\neq 0$, we have\n\\begin{equation}\\label{u-est1}\n\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}=O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}).\n\\end{equation}\n\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\t\n\\textbf{Step 1.}\nFor $x\\in B(p_{k,j},r_0), 1\\leq j\\leq m$, by (\\ref{eta1}) (\\ref{eta2}) (\\ref{loc-mass}) (\\ref{G_kj}) and Theorem \\ref{Theorem chen-lin2}, we have\n\\begin{align*}\n&u_k^{(1)}(x)-u_k^{(2)}(x)\\\\\n=&\\,U_{k,j}^{(1)}(x)-U_{k,j}^{(2)}(x)+\\eta_{k,j}^{(1)}(x)-\\eta_{k,j}^{(2)}(x)+G_{k,j}^{(1)}(x)-G_{k,j}^{(2)}(x)\\\\\n&\\,+G_{k,j}^{(1)}(p_{k,j}^{(1)})-G_{k,j}^{(2)}(p_{k,j}^{(2)})\\\\\n=&\\,\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)}-2\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}\\big|x-p_{k,j}^{(1)}\\big|^{2(1+\\alpha_j)}\\Big)\\\\\n&\\,+2\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(2)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(2)}}\\big|x-p_{k,j}^{(2)}\\big|^{2(1+\\alpha_j)}\\Big)+O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{2(1+\\alpha_1)}}\\Big).\n\\end{align*}\nTheorem \\ref{Theorem chen-lin2} and $L(\\mathbf{p})\\neq 0$ give rise to\n\\begin{equation*}\ne^{-\\frac{1}{1+\\alpha_1}(\\lambda_{k,1}^{(1)}-\\lambda_{k,1}^{(2)})}=1+O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big),\n\\end{equation*}\nwhich immediately implies\n\\begin{equation}\\label{lam-est-1}\n\\lambda_{k,1}^{(1)}-\\lambda_{k,1}^{(2)}=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big).\n\\end{equation}\nThen by (\\ref{lam-est-1}) and (\\ref{uk-ave-2}), what holds for one point is also true at other blowup points:\n\\begin{equation}\\label{lam-est-2}\n\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)}=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big),\\quad 1\\leq j\\leq m.\n\\end{equation}\n\nOn the other hand, using (\\ref{p_kj-location}) in direct computation, we have,\n\\begin{align*}\n&\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}\\big|x-p_{k,j}^{(1)}\\big|^{2(1+\\alpha_j)}\\Big)-\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(2)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(2)}}\\big|x-p_{k,j}^{(2)}\\big|^{2(1+\\alpha_j)}\\Big)\\\\\n&=O(\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)})\n\\end{align*}\nThus $u_k^{(1)}$ and $u_k^{(2)}$ are close in the interior of the ball $B(p_{k,j}^{(1)},r_0)$:\n\\begin{equation}\\label{est-1}\n\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(B(p_{k,j}^{(1)},r_0))}=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big)=O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}).\n\\end{equation}\n\n\n\n\\textbf{Step 2.}\nFor $x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0)$, we first use the Green's representation formula to write $u_k^{(1)}-u_k^{(2)}-\\big(\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}\\big)$ in three parts:\n\n\\begin{equation}\n\\begin{split}\n & u_k^{(1)}-u_k^{(2)}-\\big(\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}\\big) \\notag\\\\\n=& \\int_{M} G(y,x)\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n=& \\sum_{j=1}^{m}\\int_{B(p_{k,j}^{(1)},\\frac{r_0}{2})} \\big(G(y,x)-G(p_{k,j}^{(1)},x)\\big)\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n&+ \\sum_{j=1}^{m}G(p_{k,j}^{(1)},x)\\int_{B(p_{k,j}^{(1)},\\frac{r_0}{2})}\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y)  \\\\\n&+ \\int_{M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},\\frac{r_0}{2})} G(y,x)\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n=&\\mathrel{\\mathop:}I_1+I_2+I_3.\n\\end{split}\n\\end{equation}\nBefore we evaluate each one of them we recall a few facts: First\n\\begin{equation*}\n\tp_{k,j}^{(1)}-p_{k,j}^{(2)}=\\left\\{\\begin{array}{ll}0, \\quad \\mbox{for}\\  1\\le j\\le \\tau, \\\\\n\tO(\\sum_{i=1}^2\\lambda_{k,j}^{(i)}e^{-\\lambda_{k,j}^{(i)}}) \\quad \\mbox{if}\\  j>\\tau \\ \\mbox{ (see (\\ref{p_kj-location}))}.\n\t\\end{array}\n\t\\right.\n\\end{equation*}\nNext for $x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0)$, $ y\\in B(p_{k,j}^{(1)},\\frac{r_0}{2}) $,\n\\begin{equation*}\n\tG(y,x)-G(p_{k,j}^{(1)},x)=\\langle\\partial_yG(y,x)\\big|_{y-p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle+O(|y-p_{k,j}^{(1)}|^2)\n\\end{equation*}\nThen using symmetry, scaling, and the closeness between $u_k^{(i)}$ with standard bubbles, we have\n\n\\begin{align*}\n&I_1\n=\\sum_{j=1}^{m}\\sum_{i=1}^2\\int_{B(p_{k,j}^{(i)},\\frac{r_0}{2})}\\frac{\\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(i)}},y-p_{k,j}^{(i)}\\rangle\\rho_k\\tilde{h}_j(y)|y-p_{k,j}^{(i)}|^{2\\alpha_j}}{\\big(1+\\frac{\\rho_kh_j(p_{k,j}^{(i)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(i)}}|y-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\big)^2}\\\\\n&\\times \\Big(1+O(|y-p_{k,j}^{(i)}|)+O(e^{-\\frac{\\lambda_{k,j}^{(i)}}{2(1+\\alpha_j)}})+O(e^{-\\frac{\\lambda_{k,j}^{(i)}}{1+\\alpha_1}})\\Big){\\rm d}y+\n O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big),\\\\\n &=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big).\n \\end{align*}\n\nThe closeness between $\\rho_{k,j}^{(1)}$ and $\\rho_{k,j}^{(2)}$ leads to the smallness of $I_2$ (see\n (\\ref{loc-mass}) (\\ref{rho-kj-1}) and (\\ref{rho-kj-2})):\n\\begin{equation}\\label{I2}\nI_2=\\sum_{j=1}^{m}G(p_{k,j}^{(1)},x)(\\rho_{k,j}^{(1)}-\\rho_{k,j}^{(2)})=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big).\n\\end{equation}\nFor $I_3$, the magnitude of $u_k^{(i)}$ outside the bubbling area determines the smallness of $I_3$:\n$$\nI_3=\\rho_k\\int_{M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},\\frac{r_0}{2})} G(y,x)H(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y)=O\\Big(\\sum_{i=1}^{2}e^{-\\lambda_{k,1}^{(i)}}\\Big).\n$$\nTherefore\n\\begin{equation}\\label{step2-2}\n u_k^{(1)}-u_k^{(2)}-\\big(\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}\\big)=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big)\\quad \\mbox{in}\\quad\n M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0).\n\\end{equation}\nTo eliminate the averages in (\\ref{step2-2}) we take advantage of (\\ref{uk-ave-1}) and (\\ref{lam-est-1}):\n\\begin{equation}\\label{step2-3}\n\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}=-(\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)})+O\\Big(\\sum_{i=1}^{2}\\lambda_{k,j}^{(i)} e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big)=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big),\n\\end{equation}\nUsing (\\ref{step2-3}) in (\\ref{step2-2}) we arrive at\n\\begin{equation}\\label{step2-4}\nu_k^{(1)}(x)-u_k^{(2)}(x)=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big)=O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}).\n\\end{equation}\nfor all $x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0)$.\n Lemma \\ref{est1} is established.\n\n\\end{proof}\n\n\nAs an immediate application, Lemma \\ref{est1} gives ( see (\\ref{c}) )\n\\begin{equation}\\label{c-est}\n\tc_k(x)=e^{u_k^{(1)}(x)}\\big(1+O(\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)})\\big)=e^{u_k^{(1)}(x)}\\big(1+O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}})\\big).\n\\end{equation}\n\nTo simply the notations, we set\n\\begin{equation}\n\t\\epsilon_{k,j}=\\bigg(\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}\\bigg)^{-\\frac{1}{2(1+\\alpha_j)}}e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}.\n\\end{equation}\nand\n\\begin{equation}\\label{varsigma-kj}\n\\varsigma_{k,j}(z)=\\varsigma_k(\\epsilon_{k,j}z+p_{k,j}^{(1)}),\\quad |z|<\\frac{r_0}{\\epsilon_{k,j}},\\quad 1\\leq j\\leq m,\n\\end{equation}\nwhich satisfies\n\\begin{equation}\\label{sigma-t-equ}\n\\Delta\\varsigma_{k,j}+\\frac{8(1+\\alpha_j)^2}{\\rho_kh_j(p_{k,j}^{(1)})}\\rho_k\\tilde{h}_j(\\epsilon_{k,j}z+p_{k,j}^{(1)})e^{-\\lambda_{k,j}^{(1)}}|z|^{2\\alpha_j}c_k(\\epsilon_{k,j}z+p_{k,j}^{(1)})\\varsigma_{k,j}=0.\n\\end{equation}\nfor $ |z|<r_0\\epsilon_{k,j}^{-1}$.\n\n\\medskip\n\nThe following lemma determines the limit of $\\varsigma_{k,j}$ in both situations:\n\n\\begin{lem}\\label{lem-limit-1}\n{\\rm( The limit of} $\\varsigma_{k,j}$ {\\rm )}\n\n{\\rm (i)} For $1\\leq j \\leq \\tau$,\n\\begin{equation*}\n\\varsigma_{k,j}\\rightarrow b_{j,0}\\varphi_{j,0}\\quad {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2).\n\\end{equation*}\nwhere $\\varphi_{j,0}$ is\n\\begin{equation*}\n\\varphi_{j,0}(z)=\\frac{1-|z|^{2(1+\\alpha_j)}}{1+|z|^{2(1+\\alpha_j)}}.\n\\end{equation*}\n\n\n{\\rm (ii)} If $\\tau<m$ and $\\tau+1\\leq j \\leq m$, there exist constants $ b_{j,0} $, $ b_{j,1} $ and $ b_{j,2} $ such that\n\\begin{equation*}\n\\varsigma_{k,j}\\rightarrow  b_{j,0}\\varphi_{j,0}+b_{j,1}\\varphi_{j,1}+b_{j,2}\\varphi_{j,2}\\quad {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2),\n\\end{equation*}\nwhere $\\varphi_{j,i}$ are\n\\begin{equation*}\n\\varphi_{j,0}(z)=\\frac{1-|z|^2}{1+|z|^2}, \\quad\n\\varphi_{j,1}(z)=\\frac{z_1}{1+|z|^2}, \\quad\n\\varphi_{j,2}(z)=\\frac{z_2}{1+|z|^2}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\n(i) For $1\\leq j \\leq \\tau$, it is easy to use (\\ref{c-est}) (\\ref{eta1}) (\\ref{eta2}) and (\\ref{eta3}) to obtain\n\\begin{align*}\n&\\frac{8(1+\\alpha_j)^2}{\\rho_kh_j(p_{k,j}^{(1)})}\\rho_k\\tilde{h}_j(\\epsilon_{k,j}z+p_{k,j}^{(1)})e^{-\\lambda_{k,j}^{(1)}}c_k(\\epsilon_{k,j}z+p_{k,j}^{(1)})  \\\\\n=& \\frac{8(1+\\alpha_j)^2}{\\rho_kh_j(p_{k,j}^{(1)})}\\rho_k\\tilde{h}_j(\\epsilon_{k,j}z+p_{k,j}^{(1)})e^{-\\lambda_{k,j}^{(1)}}e^{U_{k,j}^{(1)}+G_{k,j}^{(1)}(\\epsilon_{k,j}z+p_{k,j}^{(1)})-G_{k,j}^{(1)}(p_{k,j}^{(1)}) }\\big(1+o(1)\\big)\\\\\n=& \\frac{8(1+\\alpha_j)^2}{\\big(1+|z|^{2(1+\\alpha_j)}\\big)^2}\\big(1+O(\\epsilon_{k,j}|z|)+o(1)\\big)\n\\to \\frac{8(1+\\alpha_j)^2}{\\big(1+|z|^{2(1+\\alpha_j)}\\big)^2} \\quad {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2).\n\\end{align*}\nTherefore, $\\varsigma_{k,j}\\rightarrow\\varsigma_j$ in $C_{loc}(\\mathbb{R}^2)$ and $\\varsigma_j$ satisfies\n\\begin{equation}\\label{limit-equ1}\n\\Delta\\varsigma_j(z)+\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{\\big(1+|z|^{2(1+\\alpha_j)}\\big)^2}\\varsigma_j(z)=0 \\quad {\\rm in} \\ \\; \\mathbb{R}^2.\n\\end{equation}\n\n\nSince it is obvious to have $|\\varsigma_j|\\leq 1$ from $|\\varsigma_{k,j}|\\leq 1$, we apply Lemma \\ref{linear-lem-1} to have $\\varsigma_j=b_{j,0}\\varphi_{j,0}$ for some constant $b_{j,0}$ and\n\\begin{equation*}\n\\varphi_{j,0}(z)=\\frac{1-|z|^{2(1+\\alpha_j)}}{1+|z|^{2(1+\\alpha_j)}}.\n\\end{equation*}\n That is $\\varsigma_{k,j}\\rightarrow b_{j,0}\\varphi_{j,0}$ in $ C_{loc}(\\mathbb{R}^2)$.\n\n\\medskip\n\n(ii) For $\\tau+1\\leq j \\leq m$,  by (\\ref{c-est}) (\\ref{eta1}) and (\\ref{eta3}), we have $\\varsigma_{k,j}\\rightarrow\\varsigma_j$ in $C_{loc}(\\mathbb{R}^2)$, where\n\\begin{equation*}\n\\left\\{\\begin{array}{lcl}\n\\Delta\\varsigma_j(z)+\\frac{8}{(1+|z|^{2})^2}\\varsigma_j(z)=0 && {\\rm in}\\ \\; \\mathbb{R}^2,\n\\\\\n|\\varsigma_j|\\leq 1  &&{\\rm in}\\ \\; \\mathbb{R}^2.\n\\end{array}\n\\right.\n\\end{equation*}\nIn this case we use Lemma \\ref{linear-lem-2} to conclude that\n\\begin{equation*}\n\\varsigma_j(z)=b_{j,0}\\varphi_{j,0}(z)+b_{j,1}\\varphi_{j,1}(z)+b_{j,2}\\varphi_{j,2}(z),\n\\end{equation*}\nfor some constants $ b_{j,0} $, $ b_{j,1} $ and $ b_{j,2} $. Lemma \\ref{lem-limit-1} is established.\n\n\\end{proof}\n\nOur next goal is to prove that all $b_{j,0}$ are the same, and equal to the limit of $\\varsigma_k$ away from the bubbling area. Our approach is similar to the corresponding parts in\n\\cite{lin-yan-uniq} for the Chern-Simons-Higgs equation and in \\cite{bart-4} for regular mean field equations.\n\n\\begin{lem}\\label{lem-limit-2}\n\t\nThere exists a constant $b_0$ such that\n\\begin{equation*}\n\\varsigma_k\\rightarrow -b_0\\quad {\\rm in}\\ \\; C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\}).\n\\end{equation*}\t\nMoreover, $b_{j,0}=b_0$ for all $1 \\leq j \\leq m$.\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\nStarting from the equation for $\\varsigma_k$:\n\\begin{equation*}\n\\Delta_g\\varsigma_k(x)+\\rho_k H(x)c_k(x)\\varsigma_k(x)=0\\quad in\\ \\; M,\n\\end{equation*}\nwe observe from\n (\\ref{c-est}) (\\ref{sigma1}) and (\\ref{sigma2}) that $c_k\\rightarrow 0$ in $C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\})$. Since $\\|\\varsigma_k\\|_{L^{\\infty}(M)}\\leq 1$,  $\\varsigma_k\\rightarrow \\varsigma_0$ in $C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\})$, where $\\varsigma_0$ satisfies\n\\begin{equation}\\label{limit-equ2}\n\\Delta_g\\varsigma_0=0\\quad {\\rm in}\\ \\; M\\backslash\\{p_1,\\cdots,p_m\\}.\n\\end{equation}\nThe bound for $\\varsigma$: $\\|\\varsigma_0\\|_{L^{\\infty}(M)}\\leq 1$, which comes from  $\\|\\varsigma_k\\|_{L^{\\infty}(M)}\\leq 1$, yields the smoothness of $\\varsigma_0$ on the whole manifold. Thus $\\varsigma_0\\equiv -b_0$ in $M$ for some constant $b_0$. In particular,\n\\begin{equation}\\label{limit-equ3}\n\\varsigma_k\\rightarrow -b_0\\quad {\\rm in}\\ \\; C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\}).\n\\end{equation}\n\nFor $1\\leq j \\leq m$, let\n\\begin{equation*}\n\t\\varphi_{k,j}(x)=\\frac{1-\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}}{1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}},\\quad x\\in B(p_{k,j}^{(1)},r_0).\n\\end{equation*}\t\nbe a sequence of solutions of\n\\begin{equation*}\n-\\Delta\\varphi_{k,j}(x)=\\rho_kh_j(p_{k,j}^{(1)})|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{U_{k,j}^{(1)}}\\varphi_{k,j}(x),\\quad x\\in B(p_{k,j}^{(1)},r_0).\n\\end{equation*}\t\nRecall that\n\\begin{equation*}\n-\\Delta\\varsigma_k(x)=\\rho_k \\tilde{h}_j(x)|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\frac{e^{u_k^{(1)}(x)}-e^{u_k^{(2)}(x)}}{u_k^{(1)}(x)-u_k^{(2)}(x)}\\varsigma_k(x),\\quad x\\in B(p_{k,j}^{(1)},r_0).\n\\end{equation*}\t\nUsing (\\ref{p_kj-location}) (\\ref{eta1}) (\\ref{eta2}), (\\ref{eta3}) and integration by parts, we find, for $ d\\in (0,r_0) $, that\n\\begin{align*}\n&\\int_{\\partial B(p_{k,j}^{(1)},d)}\\big(\\varphi_{k,j}\\frac{\\partial\\varsigma_k}{\\partial\\nu}-\\varsigma_k\\frac{\\partial\\varphi_{k,j}}{\\partial\\nu}\\big){\\rm d}\\sigma =\\int_{B(p_{k,j}^{(1)},d)}\\big(\\varphi_{k,j}\\Delta\\varsigma_k-\\varsigma_k\\Delta\\varphi_{k,j}\\big){\\rm d}x \\\\\n=& \\int_{B(p_{k,j}^{(1)},d)}\\rho_k\\varsigma_k\\varphi_{k,j}\\Big(-\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\frac{e^{u_k^{(1)}}-e^{u_k^{(2)}}}{u_k^{(1)}-u_k^{(2)}}+h_j(p_{k,j}^{(1)})|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{U_{k,j}^{(1)}}\\Big){\\rm d}x \\\\\n=& \\int_{B(p_{k,j}^{(1)},d)}\\rho_k\\varsigma_k\\varphi_{k,j}|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\Big(-\\tilde{h}_je^{u_k^{(1)}}\\big(1+O(|u_k^{(1)}-u_k^{(2)}|)\\big)+h_j(p_{k,j}^{(1)})e^{U_{k,j}^{(1)}}\\Big){\\rm d}x  \\\\\n=& \\int_{B(p_{k,j}^{(1)},d)}\\rho_k\\varsigma_k\\varphi_{k,j}|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\Big(-\\tilde{h}_je^{U_{k,j}^{(1)}+G_{k,j}^{(1)}-G_{k,j}^{(1)}(p_{k,j}^{(1)})+\\eta_{k,j}^{(1)}}\\big(1+O(|u_k^{(1)}-u_k^{(2)}|)\\big)  \\\\\n&\\qquad \\quad+h_j(p_{k,j}^{(1)})e^{U_{k,j}^{(1)}}\\Big){\\rm d}x .\n\\end{align*}\n\nBy scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$, (\\ref{eta2}), (\\ref{eta3}) and the estimate of\n\t$\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}$, it is not hard to obtain\n\\begin{equation}\\label{div-est}\n\t\\int_{\\partial B(p_{k,j}^{(1)},d)}\\big(\\varphi_{k,j}\\frac{\\partial\\varsigma_k}{\\partial\\nu}-\\varsigma_k\\frac{\\partial\\varphi_{k,j}}{\\partial\\nu}\\big){\\rm d}\\sigma=O\\big(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}\\big).\n\\end{equation}\n\nLet $\\varsigma_{k,j}(r)$ be the spherical average of $\\varsigma_k$:\n$$\\varsigma_{k,j}^*(r)=\\frac 1{2\\pi}\\int_{0}^{2\\pi}\\varsigma_k(r\\cos \\theta,r\\sin \\theta){\\rm d}\\theta, $$ where $r=|x-p_{k,j}^{(1)}|$. Then (\\ref{div-est}) yields\n\\begin{equation*}\n(\\varsigma_{k,j}^*)'(r)\\varphi_{k,j}(r)-\\varsigma_{k,j}^*(r)\\varphi_{k,j}'(r)=\\frac{1}{r}O\\big(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}\\big),\\quad r\\in(R\\epsilon_{k,j},r_0).\n\\end{equation*}\nFor any $r \\in (R\\epsilon_{k,j},r_0)$, we also notice that\n\\begin{align*}\n&\\varphi_{k,j}(r)=-1+\\frac{1}{r^{2(1+\\alpha_j)}}O(e^{-\\lambda_{k,j}^{(1)}}),\\\\\n&\\varphi_{k,j}'(r)=\\frac{1}{r^{2\\alpha_j+3}}O(e^{-\\lambda_{k,j}^{(1)}}).\n\\end{align*}\nThen we conclude\n\\begin{equation}\\label{ratial-d-est}\n(\\varsigma_{k,j}^*)'(r)=\\frac{1}{r}O\\big(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}\\big)+\\frac{1}{r^{2\\alpha_j+3}}O(e^{-\\lambda_{k,j}^{(1)}}),\\quad r\\in(R\\epsilon_{k,j},r_0).\n\\end{equation}\nIntegrating (\\ref{ratial-d-est}) from $R\\epsilon_{k,j}$ to $r$, we get for all $r\\in(R\\epsilon_{k,j},r_0)$\n\\begin{align}\\label{ratial-est}\n\\begin{split}\n\\varsigma_{k,j}^*(r)\n=& \\varsigma_{k,j}^*(R\\epsilon_{k,j})+O\\big(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,j}^{(1)}}\\big)\\big(\\log r+\\log R+\\lambda_{k,j}^{(1)}\\big)  \\\\\n&+O\\big(e^{-\\lambda_{k,j}^{(1)}}\\big) \\big(r^{-2(1+\\alpha_j)}+e^{\\lambda_{k,j}^{(1)}}R^{-2(1+\\alpha_j)}\\big) \\\\\n=& \\varsigma_{k,j}^*(R\\epsilon_{k,j})+o(1)\\log R+O(R^{-2(1+\\alpha_j)}) .\n\\end{split}\n\\end{align}\n\nThe first term of (\\ref{ratial-est}) is almost a constant ( Lemma \\ref{lem-limit-1} ):\n\\begin{equation*}\n\\varsigma_{k,j}^*(R\\epsilon_{k,j})=- b_{j,0}+o(1)+o_R(1),\n\\end{equation*}\nwhere $\\lim_{R\\to +\\infty}o_R(1)=0$. Then it is easy to see from  (\\ref{ratial-est}) and (\\ref{limit-equ3}) that $b_{j,0}=b_0$ for all $1\\le j\\le m$.\n\\end{proof}\n\nNext we introduce a few quantities to be used later. For $1 \\leq j\\leq m$, let\n\\begin{align}\\label{phi-kj}\n\\begin{split}\n\\phi_{k,j}(x)=&\\frac{\\rho_k}{\\sum_{l=1}^m(1+\\alpha_l)}\\bigg\\{(1+\\alpha_j)\\Big(R(x,p_{k,j}^{(1)})-R(p_{k,j}^{(1)},p_{k,j}^{(1)})\\Big) \\\\\n&+\\sum_{l\\neq j}^{1,\\cdots,m}(1+\\alpha_l)\\Big(G(x,p_{k,l}^{(1)})-G(p_{k,j}^{(1)},p_{k,l}^{(1)})\\Big)\\bigg\\},\n\\end{split}\t\n\\end{align}\n\\begin{equation}\\label{G-k-tilde}\n\\tilde{G}_k(x)=8\\pi\\sum_{l=1}^{m}(1+\\alpha_l)G(x,p_{k,l}^{(1)}),\n\\end{equation}\nIt is easy to see that in $M\\setminus \\{p_{k,1}^{(1)},\\cdots,p_{k,m}^{(1)}\\}$,\n\\begin{equation}\\label{G-phi-equ}\n\\nabla\\big(\\tilde{G}_k(x)-\\phi_{k,j}(x)\\big)=-4(1+\\alpha_j)\\frac{x-p_{k,j}^{(1)}}{|x-p_{k,j}^{(1)}|^2},\\quad \\Delta\\big(\\tilde{G}_k(x)-\\phi_{k,j}(x)\\big)=0.\n\\end{equation}\nSet\n\\begin{equation}\\label{v-kj}\nv_{k,j}^{(i)}(x)=u_k^{(i)}(x)-\\phi_{k,j}(x),\\quad  i=1,2,\n\\end{equation}\nand\n\\begin{equation}\\label{A-kj}\nA_{k,j}=\\int_{M_j} f_k \\, {\\rm d}\\mu,\n\\end{equation}\nthen we estimate $\\nabla v_{k,j}^{(i)}$ away from the bubbling area:\n\n\\begin{lem}\\label{lem-Dv-kj}\n\t\n\tFor any $\\theta\\in(0,r_0)$ small enough and $x\\in B(p_{k,j}^{(1)},2r_0)\\setminus B(p_{k,j}^{(1)},\\theta)$, the gradient of $v_{k,j}^{(i)}$ is very close to that of a harmonic function:\n\t\\begin{align}\\label{Dv-kj-est}\n\t\\nabla v_{k,j}^{(i)}(x)=-4(1+\\alpha_j)\\frac{x-p_{k,j}^{(1)}}{|x-p_{k,j}^{(1)}|^2}+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{align}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tBy (\\ref{sigma1}) (\\ref{sigma2}) and Theorem \\ref{Theorem chen-lin2}, we have\n\t\\begin{align*}\n\tO(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}})=\\nabla \\sigma_k^{(i)}=\\nabla (v_{k,j}^{(i)}+\\phi_{k,j}-\\tilde{G}_k)+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}), \\quad x\\in B(p_{k,j}^{(1)},2r_0)\\setminus B(p_{k,j}^{(1)},\\theta)\n\t\\end{align*}\n\t for $i=1,2$, $1\\leq j\\leq m$. Consequently using (\\ref{phi-kj})$\\sim$(\\ref{G-phi-equ}) we have\n\t\\begin{align*}\n\t\\nabla v_{k,j}^{(i)}(x)=&\\nabla \\big(\\tilde{G}_k(x)-\\phi_{k,j}(x)\\big)+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}})\\\\\n\t=&-4(1+\\alpha_j)\\frac{x-p_{k,j}^{(1)}}{|x-p_{k,j}^{(1)}|^2}+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{align*}\n\t\n\\end{proof}\nNext we estimate $\\varsigma_k$ and its derivatives away from blowup points.\n\\begin{lem}\\label{lem-C1-est}\n\t\n\tGiven $\\theta\\in(0,r_0)$ small enough, we have\n\t\\begin{align}\\label{GRF-est-1}\n\t\\begin{split}\n\t\\varsigma_k-\\bar{\\varsigma}_k=\\sum_{j=1}^m A_{k,j}G(p_{k,j}^{(1)},x)+o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2(1+\\alpha_1)}}) \\quad {\\rm in}\\ \\; M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta).\n\t\\end{split}\n\t\\end{align}\n\t\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\tFrom the Green's representation formula for $u_k^{(i)}$ and the definition of $\\varsigma_k$, we have the following expression of $\\varsigma_k$:\n$$\\varsigma_k(x)-\\bar \\varsigma_k=\\int_MG(y,x)f_k(y)d\\mu(y). $$\nThen for $ x\\in M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)$ we evaluate the integral in three parts:\n\t\\begin{align}\\label{J1+J2+J3}\n\t\\begin{split}\n\t\\varsigma_k(x)-\\bar{\\varsigma}_k=&\\sum_{j=1}^m A_{k,j}G(p_{k,j}^{(1)},x)+\\sum_{j=1}^\\tau\\int_{M_j}\\big(G(y,x)-G(p_{k,j}^{(1)},x)\\big)f_k(y){\\rm d}\\mu(y) \\\\\n\t&+\\sum_{j=\\tau+1}^m\\int_{M_j}\\big(G(y,x)-G(p_{k,j}^{(1)},x)\\big)f_k(y){\\rm d}\\mu(y) \\\\\n\t=&\\mathrel{\\mathop:}J_1+J_2+J_3\n\t\\end{split}\n\t\\end{align}\n\tNote that $J_3=\\emptyset$ if  $\\tau=m$.\n\tThen it follows from the definition of $\\eta_{k,j}^{(1)}$, (\\ref{eta2}) and (\\ref{eta3}) that\n\t\\begin{align*}\n\t&\\int_{M_j}\\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle f_k(y){\\rm d}\\mu(y)  \\\\\n\t=&\\int_{B(p_{k,j}^{(1)},r_0)} \\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle f_k(y)e^{\\phi_j(y)}{\\rm d}y+O(e^{-\\lambda_{k,j}^{(1)}}) \\\\\n\t=&\\int_{B(p_{k,j}^{(1)},r_0)} \\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle \\rho_kh_j(p_{k,j}^{(1)})|y-p_{k,j}^{(1)}|^{2\\alpha_j}e^{U_{k,j}^{(1)}} \\varsigma_k(y)  \\\\\n\t&\\times\\big(1+O(|y-p_{k,j}^{(1)}|+\\epsilon_{k,j}+\\epsilon_{k,1}^2)\\big)(1+o(1)){\\rm d}y+O(e^{-\\lambda_{k,j}^{(1)}}).\n\t\\end{align*}\n\t\n\tFor $1\\leq j\\leq \\tau$, using $y=\\epsilon_{k,j}z+p_{k,j}^{(1)}$ in the evaluation of the identity above, we have\n\t\\begin{align*}\n\t&\\int_{B(p_{k,j}^{(1)},r_0)} \\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle f_k(y)e^{\\phi_j}{\\rm d}y  \\\\\n\t=&\\epsilon_{k,j}\\int_{|z|<\\frac{r_0}{\\epsilon_{k,j}}}\\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},z\\rangle\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\\Big(b_0\\frac{1-|z|^{2(1-\\alpha_j)}}{1+|z|^{2(1+\\alpha_j)}}+o(1)\\Big)  \\\\\n\t&\\times\\big(1+O(\\epsilon_{k,j}|z|+\\epsilon_{k,j}+\\epsilon_{k,1}^2)+o(1)\\big){\\rm d}z.\t \\\\\n\t=&o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}),\n\t\\end{align*}\n\n\t\n\tIf $\\tau <m$, let us recall that $\\alpha_j=0$ for $\\tau+1\\leq j\\leq m$. Similarly, by the standard scaling, Lemma \\ref{lem-limit-1} and symmetry, we have\n\t\\begin{align*}\n\t&\\int_{B(p_{k,j}^{(1)},r_0)} \\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle  f_k(y)e^{\\phi_j}{\\rm d}y  \\\\\n\t=&\\epsilon_{k,j}\\int_{|z|<\\frac{r_0}{\\epsilon_{k,j}}}\\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(1)}},z\\rangle\\frac{8}{(1+|z|^2)^{2}}\\varsigma_{k,j}(z){\\rm d}z+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}})\t \\\\\n\t=&e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j +o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}),\n\t\\end{align*}\n\twhere\n\t\\begin{equation}\\label{Bj}\n\t\tB_j=4\\sqrt{\\frac{8}{\\rho_kh_j(p_{k,j}^{(1)})}}\\displaystyle\\int_{\\mathbb{R}^2}\\frac{|z|^2}{(1+|z|^2)^3}{\\rm d}z .\n\t\\end{equation}\n\tFor the second order terms in the expansion of $G$, we have\n\t\\begin{align*}\n\t&\\int_{M_j}|y-p_{k,j}^{(1)}|^2f_k{\\rm d}\\mu(y) \\\\\n\t=&\\int_{B(p_{k,j}^{(1)},r_0)}|y-p_{k,j}^{(1)}|^2f_ke^{\\phi_j}{\\rm d}y+O(e^{-\\lambda_{k,j}^{(1)}})   \\\\\n\t=&O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}})\\int_{|z|<\\frac{r_0}{\\epsilon_{k,j}}}\\frac{|z|^{2(1+\\alpha_j)}}{(1+|z|^{2(1+\\alpha_j)})^2}(1+o(1)){\\rm d}z+O(e^{-\\lambda_{k,j}^{(1)}})  \\\\\n\t=&\\left\\{\n\t\\begin{array}{lcl}\n\tO(e^{-\\frac{1}{1+\\alpha_j}\\lambda_{k,j}^{(1)}}), && 1\\leq j\\leq \\tau, \\\\\n\tO(\\lambda_{k,j}^{(1)}e^{-\\lambda_{k,j}^{(1)}}),  && \\tau+1\\leq j\\leq m.\n\t\\end{array}\n\t\\right.\n\t\\end{align*}\n\t\n\tConsequently for $J_2$ and $J_3$ we have\n\t\\begin{equation}\\label{J2}\n\t\tJ_2=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2(1+\\alpha_1)}}),\n\t\\end{equation}\n\t\\begin{equation}\\label{J3}\n\t\tJ_3=\\sum_{j=\\tau+1}^me^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j +o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\t\\end{equation}\n\t\n  Observing (\\ref{J1+J2+J3}) (\\ref{J2}) and (\\ref{J3}), we conclude that (\\ref{GRF-est-1}) holds. Then by standard estimates we also have\n  \\begin{align}\\label{Dsigma_k-1}\n  \\nabla\\varsigma_k(x)=\\sum_{j=\\tau+1}^m A_{k,j}\\nabla_xG(p_{k,j}^{(1)},x)+o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2(1+\\alpha_1)}}), \\quad x\\in M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta).\n  \\end{align}\n\t\n\\end{proof}\n\n\\section{Estimates associated with Pohozaev identities}\\label{anal-pohozaev}\n\n In this section, we establish some sharp estimates for certain terms crucial for evaluation of Pohozaev identities.\n\n\\smallskip\n\nThe first important quantity is $A_{k,j}$, defined in (\\ref{GRF-est-1}) and the study of which is through the following Pohozaev identity:\n\n\\begin{lem}\\label{Pohozaev identity-1}\n\n\tFor $1 \\leq j \\leq m$ and any $r\\in(0,r_0)$, it holds that\n\t\\begin{align}\\label{PI-1}\n\t\\begin{split}\n\t& \\frac{1}{2}\\int_{\\partial B(p_{k,j}^{(1)},r)}r\\langle \\nabla v_{k,j}^{(1)}+\\nabla v_{k,j}^{(2)},\\nabla \\varsigma_k\\rangle{\\rm d}\\sigma \\\\\n\t&  -\\int_{\\partial B(p_{k,j}^{(1)},r)}r\\langle\\nu,\\nabla v_{k,j}^{(1)}+\\nabla v_{k,j}^{(2)}\\rangle  \\langle\\nu,\\nabla\\varsigma_k\\rangle {\\rm d}\\sigma \\\\\n\t=& \\int_{\\partial B(p_{k,j}^{(1)},r)}\\frac{r\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\big(e^{v_{k,j}^{(1)}+\\phi_{k,j}}-e^{v_{k,j}^{(2)}+\\phi_{k,j}}\\big)}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)} } {\\rm d}\\sigma \\\\\n\t&  - 2(1+\\alpha_j)\\int_{B(p_{k,j}^{(1)},r)} \\frac{\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\big(e^{v_{k,j}^{(1)}+\\phi_{k,j}}-e^{v_{k,j}^{(2)}+\\phi_{k,j}}\\big)}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}} {\\rm d}x\\\\\n\t&  -\\int_{B(p_{k,j}^{(1)},r)} \\frac{\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\big(e^{v_{k,j}^{(1)}+\\phi_{k,j}}-e^{v_{k,j}^{(2)}+\\phi_{k,j}}\\big)}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}} \\langle \\nabla\\big(\\log \\tilde{h}_j+\\phi_{k,j}\\big),x-p_{k,j}^{(1)}\\rangle {\\rm d}x.\n\t\\end{split}\n\t\\end{align}\n\t\n\\end{lem}\nThis Pohozaev identity has been used in \\cite{bart-4} and \\cite{lin-yan-uniq}, we include the proof for the convenience of the readers.\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tFirst we observe that for any two smooth functions $u$ and $v$,\n\t\\begin{align}\\label{divergence}\n\t\\begin{split}\n\t&\\Delta u\\{\\nabla v\\cdotp (x-p_{k,j}^{(1)})\\}+\\Delta u\\{\\nabla v\\cdotp (x-p_{k,j}^{(1)})\\} \\\\\n\t=\\,&{\\rm div}\\Big\\{\\nabla u\\big[\\nabla v\\cdotp(x-p_{k,j}^{(1)})\\big]+\\nabla v\\big[\\nabla u\\cdotp(x-p_{k,j}^{(1)})\\big] - \\nabla u\\cdotp\\nabla v (x-p_{k,j}^{(1)})\\Big\\}.\n\t\\end{split}\n\t\\end{align}\n\t\nReplacing $u$, $v$ by $v_{k,j}^{(1)}- v_{k,j}^{(2)}$ and $v_{k,j}^{(1)}+ v_{k,j}^{(2)}$ respectively in (\\ref{divergence}), we have\n\t\\allowdisplaybreaks\n\t\\begin{align}\\label{div-formula-1}\n\t\\begin{split}\n\t&\\Delta (v_{k,j}^{(1)}- v_{k,j}^{(2)})\\big\\{\\nabla (v_{k,j}^{(1)}+ v_{k,j}^{(2)})\\cdotp (x-p_{k,j}^{(1)})\\big\\} +\\Delta (v_{k,j}^{(1)}+ v_{k,j}^{(2)})\\big\\{\\nabla (v_{k,j}^{(1)}- v_{k,j}^{(2)})\\cdotp (x-p_{k,j}^{(1)})\\big\\} \\\\\n\t=&{\\rm div}\\Big\\{\\nabla (v_{k,j}^{(1)}- v_{k,j}^{(2)})\\big[\\nabla (v_{k,j}^{(1)}+ v_{k,j}^{(2)})\\cdotp(x-p_{k,j}^{(1)})\\big] \\\\\n\t+&\\nabla (v_{k,j}^{(1)}+ v_{k,j}^{(2)})\\big[\\nabla (v_{k,j}^{(1)}- v_{k,j}^{(2)})\\cdotp(x-p_{k,j}^{(1)})\\big]-\\nabla (v_{k,j}^{(1)}- v_{k,j}^{(2)})\\cdotp\\nabla (v_{k,j}^{(1)}+ v_{k,j}^{(2)})(x-p_{k,j}^{(1)})\\Big\\}.\n\t\\end{split}\n\t\\end{align}\n\tBy the definition of $v_{k,j}^{(i)}$, we see that, for $x\\in B(p_{k,j}^{(1)},r_0)$,\n\t\\begin{align}\\label{v-kj-equ}\n\t\\Delta(v_{k,j}^{(1)}\\pm v_{k,j}^{(2)})+\\rho_ke^{\\phi_{k,j}}\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}(e^{u_k^{(1)}}\\pm e^{u_k^{(2)}})=0.\n\t\\end{align}\n\tUsing (\\ref{v-kj-equ}) and\n $$g_i=e^{v_{k,j}^{(i)}+\\phi_{k,j}+\\log \\tilde{h}_j+2\\alpha_j\\log|x-p_{k,j}^{(1)}|}$$\n the right hand side (RHS) of (\\ref{div-formula-1}) can be written as:\n\t\\begin{align*}\n\t&{\\rm (RHS)}\\ {\\rm of}\\ (\\ref{div-formula-1})\\\\\n\t=&-\\rho_k(g_1-g_2)\\big\\{\\nabla (v_{k,j}^{(1)}+ v_{k,j}^{(2)})\\cdotp (x-p_{k,j}^{(1)})\\big\\} -\\rho_k(g_1+g_2)\\big\\{\\nabla (v_{k,j}^{(1)}- v_{k,j}^{(2)})\\cdotp (x-p_{k,j}^{(1)})\\big\\}\\\\\n\t=&-2\\rho_kg_1\\big\\{\\nabla v_{k,j}^{(1)}\\cdotp (x-p_{k,j}^{(1)})\\big\\}+2\\rho_kg_2\\big\\{\\nabla v_{k,j}^{(2)}\\cdotp (x-p_{k,j}^{(1)})\\big\\}\\\\\n\t=&-{\\rm div}\\big\\{2\\rho_k(g_1+g_2) (x-p_{k,j}^{(1)})\\big\\}+2\\rho_k(g_1-g_2)\\big\\{\\nabla (\\log \\tilde{h}_j+\\phi_{k,j})\\cdotp (x-p_{k,j}^{(1)})\\big\\}\\\\\n\t&+2\\rho_k(g_1-g_2)\\big\\{2\\alpha_j\\nabla \\log |x-p_{k,j}^{(1)}|\\cdotp (x-p_{k,j}^{(1)})\\big\\}+4\\rho_k(g_1-g_2)\\\\\n\t=&-{\\rm div}\\big\\{2\\rho_k(g_1+g_2) (x-p_{k,j}^{(1)})\\big\\}+4(1+\\alpha_j)\\rho_k(g_1-g_2)\\\\\n\t&+2\\rho_k(g_1-g_2)\\big\\{\\nabla (\\log \\tilde{h}_j+\\phi_{k,j})\\cdotp (x-p_{k,j}^{(1)})\\big\\}.\n\t\\end{align*}\n\t Since $\\varsigma_k=\\frac{v_{k,j}^{(1)} -v_{k,j}^{(2)}}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}}$ and $\\nu=\\frac{x-p_{k,j}^{(1)}}{r}$, we have\n\t\\begin{align}\\label{div-formula-2}\n\t\\begin{split}\n\t&\\int_{\\partial B(p_{k,j}^{(1)},r)}\\frac{{\\rm (RHS)}\\ {\\rm of}\\  (\\ref{div-formula-1})}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}}{\\rm d}\\sigma\\\\\n\t=&-2\\int_{\\partial B(p_{k,j}^{(1)},r)}\\frac{r\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\big(e^{v_{k,j}^{(1)}+\\phi_{k,j}}+e^{v_{k,j}^{(2)}+\\phi_{k,j}}\\big)}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)} } {\\rm d}\\sigma\\\\\n\t&  + 4(1+\\alpha_j)\\int_{B(p_{k,j}^{(1)},r)} \\frac{\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\big(e^{v_{k,j}^{(1)}+\\phi_{k,j}}-e^{v_{k,j}^{(2)}+\\phi_{k,j}}\\big)}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}} {\\rm d}x\\\\\n\t&  +2\\int_{B(p_{k,j}^{(1)},r)} \\frac{\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}\\big(e^{v_{k,j}^{(1)}+\\phi_{k,j}}-e^{v_{k,j}^{(2)}+\\phi_{k,j}}\\big)}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}} \\langle \\nabla\\big(\\log \\tilde{h}_j+\\phi_{k,j}\\big),x-p_{k,j}^{(1)}\\rangle {\\rm d}x.\n\t\\end{split}\n\t\\end{align}\n\tOn the other hand,\n\t\\begin{align}\\label{div-formula-3}\n\t\\begin{split}\n\t&\\int_{\\partial B(p_{k,j}^{(1)},r)}\\frac{{\\rm (LHS)}\\ {\\rm of}\\  (\\ref{div-formula-1})}{\\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)} \\parallel_{L^{\\infty}(M)}}{\\rm d}\\sigma\\\\\n\t=&-\\int_{\\partial B(p_{k,j}^{(1)},r)}r\\langle \\nabla v_{k,j}^{(1)}+\\nabla v_{k,j}^{(2)},\\nabla \\varsigma_k\\rangle{\\rm d}\\sigma \\\\\n\t&+2\\int_{\\partial B(p_{k,j}^{(1)},r)}r\\langle\\nu,\\nabla v_{k,j}^{(1)}+\\nabla v_{k,j}^{(2)}\\rangle \\langle\\nu,\\nabla \\varsigma_k\\rangle {\\rm d}\\sigma\n\t\\end{split}\n\t\\end{align}\n\tThen (\\ref{PI-1}) follows from (\\ref{div-formula-1}), (\\ref{div-formula-2}) and (\\ref{div-formula-3}).\n\t\n\\end{proof}\n\n\\begin{rem}\n\tIt is easy to see Pohozaev-type identity (\\ref{PI-1}) also holds for $\\alpha_j>-1$.\n\\end{rem}\n\n\n\\begin{lem}\\label{lem-PI1-left}\n\nFor all $1\\leq j\\leq m$,\n\t\\begin{equation}\\label{PI-1-l}\n\t{\\rm (LHS)}\\ {\\rm of}\\  (\\ref{PI-1})=-4(1+\\alpha_j)A_{k,j}+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\sum_{l=1}^{m}|A_{k,l}|)+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tFrom (\\ref{Dv-kj-est}) and (\\ref{Dsigma_k-1}), we find that\n\t\\begin{align*}\n\t&{\\rm (LHS)}\\ {\\rm of}\\  (\\ref{PI-1})\\\\\n\t=&\\,4(1+\\alpha_j)\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle\\nu,D\\varsigma_k\\rangle{\\rm d}\\sigma+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\parallel D\\varsigma_k\\parallel_{L^{\\infty}(\\partial B(p_{k,j}^{(1)},r))})\\\\\n\t=&\\,4(1+\\alpha_j)\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle\\nu,D\\varsigma_k\\rangle{\\rm d}\\sigma+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\sum_{l=1}^{m}|A_{k,l}|)+o(e^{-\\frac{3}{2(1+\\alpha_1)}\\lambda_{k,j}^{(1)}}).\n\t\\end{align*}\n\tFor  $x\\in\\partial B(p_{k,j}^{(1)},r)$ we use the Green's representation formula to estimate $\\varsigma_k(x)$:\n\t\\begin{align*}\n\t&\\varsigma_k(x)-\\bar{\\varsigma}_k=\\int_{M}G(y,x)f_k(y){\\rm d}\\mu(y)\\\\ =&\\sum_{l=1}^mA_{k,l}G(p_{k,l}^{(1)},x)+\\sum_{l=1}^m\\sum_{h=1}^2B_{k,l,h}\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}+\\frac{1}{2}\\sum_{l=1}^m\\sum_{h,i=1}^2C_{k,l,h,i}\\partial_{y_hy_i}^2G(y,x)\\big|_{y=p_{k,l}^{(1)}}\\\\\n\t&+O(1)\\sum_{l=1}^m\\int_{M_l}|y-p_{k,j}^{(1)}|^3f_k{\\rm d}\\mu(y)  \\quad {\\rm in}\\,\\ C^1\\big(B(p_{k,j}^{(1)},2r_0)\\setminus B(p_{k,j}^{(1)},\\frac{r}{2})\\big),\n\t\\end{align*}\n\twhere\n\t\\begin{align*}\n\t&\tB_{k,l,h}=\\int_{M_l}(y-p_{k,l}^{(1)})_hf_k(y){\\rm d}\\mu(y),\n\t\\\\\n\t& C_{k,l,h,i}=\\int_{M_l}(y-p_{k,l}^{(1)})_h(y-p_{k,l}^{(1)})_if_k(y){\\rm d}\\mu(y).\n\t\\end{align*}\n\t\n\tIt is easy to see that the last term is rather minor:\n\t\\begin{align*}\n\t&\\sum_{l=1}^m\\int_{M_l}|y-p_{k,j}^{(1)}|^3f_k(y){\\rm d}\\mu(y)   \\\\ =&\\sum_{l=1}^m\\int_{B(p_{k,l}^{(1)},r)}\\frac{e^{\\lambda_{k,l}^{(1)}}|y-p_{k,l}^{(1)}|^{2\\alpha_l+3}}{\\big(1+e^{\\lambda_{k,l}^{(1)}}|y-p_{k,l}^{(1)}|^{2(1+\\alpha_l)}\\big)^2}{\\rm d}y +O(e^{-\\lambda_{k,1}^{(1)}})  \\\\\n\t=&\\sum_{l=1}^mO(e^{-\\frac{3}{2(1+\\alpha_l)}\\lambda_{k,l}^{(1)}})\\int_{|z|<\\frac{r}{\\epsilon_{k,l}}}\\frac{|z|^{2\\alpha_l+3}}{(1+|z|^{2(1+\\alpha_l)})^2}{\\rm d}z+O(e^{-\\lambda_{k,1}^{(1)}})  \\\\\n\t=&o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}).\n\t\\end{align*}\nSetting\n\t\\begin{align*}\n\t\\bar{G}_k(x)=&\\bar{\\varsigma}_k(x)+\\sum_{j=1}^m A_{k,j}G(p_{k,j}^{(1)},x)+\\sum_{l=1}^m\\sum_{h=1}^2B_{k,l,h}\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}\\\\\n\t&+\\frac{1}{2}\\sum_{l=1}^m\\sum_{h,i=1}^2C_{k,l,h,i}\\partial_{y_hy_i}^2G(y,x)\\big|_{y=p_{k,l}^{(1)}},\n\t\\end{align*}\nwe now have\n\t\\begin{equation*}\n\t\t\\nabla\\varsigma_k(x)-\\nabla\\bar{G}_k(x)=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation*}\n\tThus\n\t\\begin{align}\\label{pi-l-1}\n\t\\begin{split}\n\t&{\\rm (LHS)}\\ {\\rm of}\\  (\\ref{PI-1})\\\\\n\t=&\\,4(1+\\alpha_j)\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle\\nu,\\nabla\\bar{G}_k\\rangle{\\rm d}\\sigma+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\sum_{l=1}^{m}|A_{k,l}|)+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{split}\n\t\\end{align}\n\tNow we take the global cancellation property into consideration: for any fixed $\\theta\\in(0,r)$,\n\t\\begin{equation}\\label{Gbar-equ}\n\t\\Delta \\bar{G}_k=\\sum_{l=1}^{m}A_{k,l}=\\int_{M}f_k\\,{\\rm d}\\mu=0,\\quad {\\rm in}\\ \\;B(p_{k,j}^{(1)},2r_0)\\setminus B(p_{k,j}^{(1)},\\theta).\n\t\\end{equation}\n\tUsing (\\ref{G-phi-equ}) (\\ref{Gbar-equ}) and (\\ref{divergence}), we have\n\t\\begin{align*}\n\t0=&\\int_{B_r\\setminus B_{\\theta}} \\Big\\{\\Delta\\bar{G}_k\\big\\{\\nabla(\\tilde{G}_k-\\phi_{k,j})\\cdotp (x-p_{k,j}^{(1)})\\big\\}+\\Delta(\\tilde{G}_k-\\phi_{k,j})\\big\\{\\nabla \\bar{G}_k\\cdotp (x-p_{k,j}^{(1)})\\big\\}\\Big\\}{\\rm d}x\\\\\n\t=&-4\\pi(1+\\alpha_j)\\int_{\\partial(B_r\\setminus B_{\\theta})}\\frac{\\partial\\bar{G}_k}{\\partial\\nu} \\,{\\rm d}\\sigma,\n\t\\end{align*}\n\twhere $B(p_{k,j}^{(1)},r)$ and $B(p_{k,j}^{(1)},\\theta)$ are replaced by $B_r$, $B_{\\theta}$  respectively for simplicity. Therefore\n\t\\begin{equation}\\label{pi-l-2}\n\t\\int_{\\partial B_r}\\frac{\\partial\\bar{G}_k}{\\partial\\nu} {\\rm d}\\sigma=\\int_{\\partial B_{\\theta}}\\frac{\\partial\\bar{G}_k}{\\partial\\nu} {\\rm d}\\sigma.\n\t\\end{equation}\n\tFurther direct computation yields\n\t\\begin{align}\\label{pi-l-3}\n\t\\begin{split}\n\t&\\int_{\\partial B_{\\theta}}\\langle\\nu,\\sum_{l=1}^{m}A_{k,l}\\nabla_xG(p_{k,l}^{(1)},x)\\rangle{\\rm d}\\sigma\\\\\n\t=& -A_{k,j}\\int_{\\partial B_{\\theta}}\\langle\\nu,\\nabla_xG(p_{k,l}^{(1)},x)\\rangle{\\rm d}\\sigma+o_{\\theta}(1)\\\\\n\t=& -A_{k,j}\\int_{\\partial B_{\\theta}}\\langle\\nu,\\nabla_x\\frac{1}{2\\pi}\\log |x-p_{k,l}^{(1)}|\\rangle{\\rm d}\\sigma+o_{\\theta}(1)  \\\\\n\t=&-A_{k,j}+o_{\\theta}(1),\n\t\\end{split}\n\t\\end{align}\n\twhere $\\lim_{\\theta\\to 0}o_{\\theta}(1)=0$, and we have used the fact that all the terms related to $l\\neq j$ are minor. Let us observe that\n\t\\begin{equation}\\label{pi-l-4}\n\t\\begin{split}\n\t&\\int_{\\partial B(0,\\theta)}\\langle\\nu,\\nabla_x\\partial_{y_h}\\log|z|\\rangle{\\rm d}\\sigma=-\\int_{\\partial B(0,\\theta)}\\sum_{i=1}^2\\frac{z_i}{|z|}\\frac{\\delta_{ih}|z|^2-2z_iz_h}{z^4}{\\rm d}\\sigma=0, \\\\\n\t&\\int_{\\partial B(0,\\theta)}\\langle\\nu,\\nabla_x\\frac{\\partial^2}{\\partial y_h^2}\\log|z|\\rangle{\\rm d}\\sigma=-\\int_{\\partial B(0,\\theta)}\\big(\\frac{2}{|z|^3}-\\frac{4z_h^2}{|z|^5}\\big){\\rm d}\\sigma=0, \\\\\n\t&\\int_{\\partial B(0,\\theta)}\\langle\\nu,\\nabla_x\\frac{\\partial^2}{\\partial y_hy_i}\\log|z|\\rangle{\\rm d}\\sigma=-\\int_{\\partial B(0,\\theta)}\\big(\\frac{4z_hz_i}{|z|^5}-\\frac{8z_hz_i}{|z|^5}\\big){\\rm d}\\sigma=0,\n\t\\end{split}\n\t\\end{equation}\n\t\n\tObviously, from (\\ref{pi-l-2})$\\sim$(\\ref{pi-l-4}) we can see that\n\t\\begin{equation*}\n\t\t\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle \\nu,\\nabla\\bar{G}_k\\rangle{\\rm d}\\sigma=-A_{k,j}+o_{\\theta}(1).\n\t\\end{equation*}\n\twhich together with (\\ref{pi-l-1}), concludes the proof of Lemma \\ref{lem-PI1-left}.\n\t\n\\end{proof}\n\n\\begin{lem}\\label{lem-PI1-right}\n\t\\begin{equation}\\label{PI-1-r-1}\n\t\\begin{split}\n\t&{\\rm (RHS)}\\ {\\rm of}\\  (\\ref{PI-1})\\\\\n\t=&-2(1+\\alpha_j)A_{k,j}-\\frac{4\\pi^2\\big[\\Delta \\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big]}{\\big(\\rho_kh_j(p_{k,j}^{(1)})\\big)^{\\frac{1}{1+\\alpha_1}}(1+\\alpha_1)\\sin \\frac{\\pi}{1+\\alpha_1}}b_0e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\\\\n\t&+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\qquad 1\\leq j\\leq t.\n\t\\end{split}\n\t\\end{equation}\n\t\\begin{align}\n\t&{\\rm (RHS)}\\ {\\rm of}\\  (\\ref{PI-1})=-2(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_1)}}),\\quad t+1\\leq j\\leq \\tau. \\label{PI-1-r-2} \\\\\n\t&{\\rm (RHS)}\\ {\\rm of}\\  (\\ref{PI-1})=-2(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad \\tau+1\\leq j\\leq m.    \\label{PI-1-r-3}\n\t\\end{align}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tWe use $K_1,K_2,K_3$ to denote the three terms on the right hand of (\\ref{PI-1}). The first two terms are quite easy to estimate:\n\t\\begin{equation}\\label{K1}\n\tK_1=\\int_{\\partial B(p_{k,j}^{(1)},r)}rf_k\\,{\\rm d}\\sigma=\\int_{\\partial B(p_{k,j}^{(1)},r)}\\rho_k\\tilde{h}_jr^{2\\alpha_j+1}e^{u_k^{(1)}}(\\varsigma_k+o(1)){\\rm d}\\sigma=O(e^{-\\lambda_{k,j}^{(1)}}),\n\t\\end{equation}\n\t\\begin{equation}\\label{K2}\n\tK_2=-2(1+\\alpha_j)A_{k,j}.\n\t\\end{equation}\n\tMore work is needed for\n\t\\begin{equation*}\n\tK_3=-\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log \\tilde{h}_j+\\phi_{k,j}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x.\n\t\\end{equation*}\n\tFirst we use $\\nabla\\phi_j(p_{k,j}^{(1)})=0$ to write $\\nabla (\\log \\tilde h_j+\\phi_{k,j})(x)$ as\n\t\\begin{equation}\\label{expansion}\n\t\\begin{split}\n\t&\\nabla(\\log \\tilde{h}_j+\\phi_{k,j})(x)\\\\\n\t=&\\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)})+\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle+O(|x-p_{k,j}^{(1)}|^2).\n\t\\end{split}\n\t\\end{equation}\n\tThen we evaluate $K_3$ in three cases:\n\n\t$\\mathbf{Case\\ 1:}$ $1\\leq j\\leq t$ $(\\alpha_j=\\alpha_1)$. The assumption $D(\\mathbf{p})=0$ and (\\ref{phi-kj}) (\\ref{rho-k}) imply\n\t\\begin{equation*}\n\t\t\\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)})=O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation*}\n\tThus, after the scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$, the first order term can be estimated as follows:\n\t\\begin{equation}\\label{K3-first-1}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(e^{-(\\frac{1}{2(1+\\alpha_j)}+\\frac{1}{1+\\alpha_1})\\lambda_{k,j}^{(1)}}).\n\t\\end{equation}\n\tFor the second order term that contains $D^2(\\log\\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})$, we have\n\t\\begin{align*}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x  \\\\\n\t=&\\int_{B(p_{k,j}^{(1)},r)}\\frac{\\rho_kh_j(p_{k,j}^{(1)})e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2\\alpha_j}}{\\Big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\Big)^2}\\big(1+O(|x-p_{k,j}^{(1)}|+\\epsilon_{k,j}+\\epsilon_{k,1}^2)\\big)  \\\\\n\t&\\times\\varsigma_k(x)(1+o(1))\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x   \\\\\n\t=&\\epsilon_{k,j}^2\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})z,z\\rangle\\varsigma_{k,j}(z){\\rm d}z+o(\\epsilon_{k,j}^2).\n\t\\end{align*}\n\tThe expression above can be greatly simplified by this beautiful identity:\n\t\\begin{equation*}\n\t\\int_{0}^{\\infty}\\frac{8(1+\\alpha_j)^2s^{2\\alpha_j+3}}{(1+s^{2(1+\\alpha_j)})^2}\\frac{1-s^{2(1+\\alpha_j)}}{1+s^{2(1+\\alpha_j)}}{\\rm d}s=-\\frac{4\\pi}{(1+\\alpha_j)\\sin\\frac{\\pi}{1+\\alpha_j}}.\n\t\\end{equation*}\n\tConsequently, Lemma \\ref{lem-limit-1} and the two identities above lead to\n\t\\begin{equation}\\label{K3-second-1}\n\t\\begin{split}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x  \\\\\n\t=&\\epsilon_{k,j}^2\\pi\\Delta(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})b_0\\int_{0}^{\\infty}\\frac{8(1+\\alpha_j)^2s^{2\\alpha_j+3}}{(1+s^{2(1+\\alpha_j)})^2}\\frac{1-s^{2(1+\\alpha_j)}}{1+s^{2(1+\\alpha_j)}}{\\rm d}s+o(\\epsilon_{k,j}^2)  \\\\\n\t=&-\\frac{4\\pi^2\\big[\\Delta \\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big]}{\\big(\\rho_kh_j(p_{k,j}^{(1)})\\big)^{\\frac{1}{1+\\alpha_1}}(1+\\alpha_1)\\sin \\frac{\\pi}{1+\\alpha_1}}b_0e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{split}\n\t\\end{equation}\n\tAlso elementary estimate gives\n\t\\begin{equation}\\label{K3-third-1}\n\t\\begin{split}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle O(|x-p_{k,j}^{(1)}|^2),x-p_{k,j}^{(1)}\\rangle{\\rm d}x  \\\\\n\t=&O(1)e^{-\\frac{3}{2(1+\\alpha_j)}\\lambda_{k,j}^{(1)}}\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{|z|^{2\\alpha_j+3}}{(1+|z|^{2(1+\\alpha_j)})^2}{\\rm d}z \\\\\n\t=& O(1)\\big(e^{-\\frac{3}{2(1+\\alpha_j)}\\lambda_{k,j}^{(1)}}+e^{-\\lambda_{k,j}^{(1)}}\\big) =\to(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}}).\t\n\t\\end{split}\n\t\\end{equation}\n\tTherefore, we complete the proof of (\\ref{PI-1-r-1}) by using (\\ref{K1}) (\\ref{K2}) and (\\ref{K3-first-1})$\\sim$(\\ref{K3-third-1}). Note that the leading term in the second order term is ignored at this stage, since the requirement of error in the current step is still crude.\n\t\n\\smallskip\n\t$\\mathbf{Case\\ 2:}$ $t+1\\leq j\\leq \\tau$ $(0<\\alpha_j<\\alpha_1)$. For the first term it is easy to see that\n    \\begin{equation}\\label{K3-first-2}\n    \\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}).\n    \\end{equation}\n    For the second order term we have\n    \\begin{equation}\\label{K3-second-2}\n    \\begin{split}\n    &\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x\\\\\n    =&O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}})=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\n    \\end{split}\n    \\end{equation}\n    where we used the scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$ and $\\alpha_j<\\alpha_1$. Similar to (\\ref{K3-third-1}), we know\n    \\begin{equation}\\label{K3-third-2}\n    \\begin{split}\n    \\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle O(|x-p_{k,j}^{(1)}|^2),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}}).\t\n    \\end{split}\n    \\end{equation}\n    Therefore (\\ref{PI-1-r-2}) follows from (\\ref{K1}), (\\ref{K2}) and (\\ref{K3-first-2})$\\sim$(\\ref{K3-third-2}).\n\n\\smallskip\n\t$\\mathbf{Case\\ 3:}$ $\\tau+1\\leq j\\leq m$ $(\\alpha_j=0)$. In view of (\\ref{first-deriv-est}), we get\n\t\\begin{equation*}\n\t\t\\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)})=\\nabla(\\log h+G_j^*)(p_j)+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}})=O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation*}\n\tThe first order term is rather small:\n\t\\begin{equation}\\label{K3-first-3}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(e^{-(\\frac{1}{2}+\\frac{1}{1+\\alpha_1})\\lambda_{k,j}^{(1)}}).\n\t\\end{equation}\n\tFor the second order term we have\n\t\\begin{align*}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x  \\\\\n\t=&\\epsilon_{k,j}^2\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{8}{(1+|z|^{2})^2}\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})z,z\\rangle \\varsigma_{k,j}(z){\\rm d}z+O(e^{-\\lambda_{k,j}^{(1)}})  \\\\\n\t=&\\epsilon_{k,j}^2\\pi\\Delta(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})b_0\\int_{0}^{\\frac{r}{\\epsilon_{k,j}}}\\frac{8s^3}{(1+s^{2})^2}\\frac{1-s^{2}}{1+s^{2}}{\\rm d}s+O(e^{-\\lambda_{k,j}^{(1)}}),\n\t\\end{align*}\n\twhere we have used Lemma \\ref{lem-limit-1} and symmetry. It is easy to see\n\t\\begin{equation*}\n\t\t\\int_{0}^R\\frac{8s^3}{(1+s^{2})^2}\\frac{1-s^{2}}{1+s^{2}}{\\rm d}s=-4\\Big(\\log (1+R^2)-\\frac{1}{1+R^2}+\\frac{1}{(1+R^2)^2}\\Big),\n\t\\end{equation*}\n\tand\n\t\\begin{equation}\\label{K3-second-3}\n\t\\begin{split}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(\\lambda_{k,j}^{(1)}e^{-\\lambda_{k,j}^{(1)}}).\n\t\\end{split}\n\t\\end{equation}\n\t\n\tFinally, by scaling we immediately observe that\n\t\\begin{equation}\\label{K3-third-3}\n\t\\begin{split}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle O(|x-p_{k,j}^{(1)}|^2),x-p_{k,j}^{(1)}\\rangle{\\rm d}x   \\\\\n\t=&O(e^{-\\frac{3}{2}\\lambda_{k,j}^{(1)}})\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{|z|^{3}}{(1+|z|^{2})^2}{\\rm d}z=O(e^{-\\lambda_{k,j}^{(1)}}).\n\t\\end{split}\n\t\\end{equation}\n\tTherefore, $K_3$ is small in this case as well.\n\t\\begin{equation}\\label{K3-est-2}\n\tK_3=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad \\tau+1\\leq j\\leq m,\n\t\\end{equation}\n\twhere $\\alpha_1>0$ is used. Lemma \\ref{lem-PI1-right} is established.\n\n\\end{proof}\n\nSince $|A_{k,j}|=O(1)$, (\\ref{PI-1}) along with Lemma \\ref{lem-PI1-left} and Lemma \\ref{lem-PI1-right} implies the initial estimate for $A_{k,j}$.\n\n\\begin{cor}\\label{cor-A-kj}\n\t\\begin{equation}\\label{A-kj-est}\n\t|A_{k,j}|=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_1)}}),\\quad 1 \\leq j \\leq m.\n\t\\end{equation}\n\t\\begin{flushright}\n\t\t\\qed\n\t\\end{flushright}\n\\end{cor}\n\n\nBased on (\\ref{A-kj-est}), we can improve the estimates in (\\ref{PI-1-l}) and (\\ref{GRF-est-1}):\n\\begin{equation}\\label{PI-1-l-re}\n\t{\\rm (LHS)}\\ {\\rm of}\\  (\\ref{PI-1})=-4(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}})\\quad 1\\leq j\\leq m.\t\n\\end{equation}\n\\begin{equation}\\label{GRF-est-2}\n\t\\varsigma_k(x)-\\bar{\\varsigma}_k=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2(1+\\alpha_1)}}) \\quad {\\rm in}\\ \\; C^1\\Big(M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)\\Big).\n\\end{equation}\n\nThe identity (\\ref{GRF-est-2}), which is the refined $C^1$-estimate of $\\varsigma_k$ away from the blowup points, will help to improve the estimate of RHS of Pohozaev-type identity (\\ref{PI-1}) and the estimate of $D\\varsigma_k$. The later one will play a part in section \\ref{pf-uni-2}. In order to achieve this goal, we analyse the projections of $\\varsigma_{k,j}$ in more detail.\n\nFor $1\\leq j\\leq \\tau$, we recall the equation of $\\varsigma_k$ in $B(p_{k,j}^{(1)},r_0)$:\n\\begin{equation*}\n\\begin{split}\n\\left\\{\n\\begin{array}{lcl}\n\\Delta \\varsigma_k+\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{U_{k,j}^{(1)}+G_{k,j}^{(1)}-G_{k,j}^{(1)}(p_{k,j}^{(1)})+\\eta_{k,j}^{(1)}}\\varsigma_k\\frac{1-e^{u_k^{(2)}-u_k^{(1)}}}{u_k^{(1)}-u_k^{(2)}}=0, \\\\\n|\\varsigma_k|\\leq 1,\n\\end{array}\n\\right.\n\\end{split}\n\\end{equation*}\nand set the following quantities for convenience:\n\\begin{align*}\n    &a_{k,j}=\\nabla(\\log h_j+G_{k,j}^{(1)})(p_{k,j}^{(1)}), \\quad d_k=\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)},  \\\\\n\t&n_0=\\max\\big\\{n\\in\\mathbb{N}:n\\leq\\frac{1}{2\\gamma} \\big\\},\\quad U_{j}(r)=\\log\\frac{8(1+\\alpha_j)^2}{(1+r^{2(1+\\alpha_j)})^2}.\n\\end{align*}\nThen the equation for $\\varsigma_k$ becomes\n\\begin{equation*}\n\\Delta \\varsigma_k+\\frac{\\rho_kh_j(p_{k,j}^{(1)})e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{g_k(x)}}{\\big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\big)^2}\\Big\\{\\sum_{n=0}^{n_0}\\frac{(-d_k)^{n}}{(n+1)!}\\varsigma_k^{n+1}+O(d_k^{n_0+1})\\Big\\}=0,\n\\end{equation*}\nwhere\n\\begin{align*}\n\tg_k(x)=&\\langle a_{k,j},x-p_{k,j}^{(1)}\\rangle\\Big\\{1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\Big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\Big)^{-1}\\Big\\}  \\\\\n\t&+d_j\\log\\Big(2+e^{\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}|x-p_{k,j}^{(1)}|\\Big)e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}}+O(|x-p_{k,j}^{(1)}|^2)+O(\\epsilon_{k,1}^2)\n\\end{align*}\nAfter scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$, we have\n\\begin{align}\\label{equ-varsigma_kj}\n\\Delta \\varsigma_{k,j}(z)+\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\\varsigma_{k,j}(z)=E_{k,j}(z),\n\\end{align}\nwhere\n\\begin{align*}\n\t&E_{k,j}(z)=\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\n\t\\bigg\\{-\\epsilon_{k,j}\\langle a_{k,j},z\\rangle\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+|z|^{2(1+\\alpha_j)}}\\big)\\varsigma_{k,j}(z)  \\\\\n\t&+\n\t\\Big[1+\\epsilon_{k,j}\\langle a_{k,j},z\\rangle\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+|z|^{2(1+\\alpha_j)}}\\big)\\Big]\\sum_{n=1}^{n_0}\\frac{(-1)^{n+1}d_k^{n}}{(n+1)!}\\varsigma_{k,j}^{n+1}+o(\\epsilon_{k,1})\\bigg\\}.\n\\end{align*}\n\nFor each integer $l\\ge 0$ we define the projections of frequency $l$ as\n\\begin{align*}\n\t&\\xi_l(r)=\\frac{1}{2\\pi}\\int_{0}^{2\\pi}\\varsigma_{k,j}(r\\cos\\theta,r\\sin\\theta)\\cos(l\\theta){\\rm d}\\theta,   \\\\\n\t&\\tilde{\\xi}_l(r)=\\frac{1}{2\\pi}\\int_{0}^{2\\pi}\\varsigma_{k,j}(r\\cos\\theta,r\\sin\\theta)\\sin(l\\theta){\\rm d}\\theta.\n\\end{align*}\nObviously the study of $\\xi_l$ is representative enough. (\\ref{equ-varsigma_kj}) shows that $\\xi_l$ satisfies\n\\begin{align*}\n\t\\xi_l^{''}+\\frac{1}{r}\\xi_l^{'}+\\Big(r^{2\\alpha_j}e^{U_j}-\\frac{l^2}{r^2}\\Big)\\xi_l=\\tilde{E}_{l}(r),\\quad l\\ge 1,\n\\end{align*}\nwhere\n\\begin{align*}\n\t&\\tilde{E}_{1}(r)=r^{2\\alpha_j}e^{U_j}\\Big\\{-\\frac{a_{k,j}^1}{4}\\epsilon_{k,j}r\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+r^{2(1+\\alpha_j)}}\\big)\\xi_0+O(d_k\\xi_1)+o(\\epsilon_{k,1})\\Big\\},  \\\\\n\t&\\tilde{E}_{2}(r)=r^{2\\alpha_j}e^{U_j}\\Big\\{-\\frac{a_{k,j}^1}{4}\\epsilon_{k,j}r\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+r^{2(1+\\alpha_j)}}\\big)\\xi_1+O(d_k\\xi_2)+o(\\epsilon_{k,1})\\Big\\},  \\\\\n\t&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\Big\\{O(d_k\\xi_l)+o(\\epsilon_{k,1})\\Big\\},\\qquad l\\ge 3,\n\\end{align*}\nand $a_{k,j}^1$ is the first component of $a_{k,j}$. Moreover, from (\\ref{GRF-est-2}) we obtain that $\\xi_l(0)=o(1)$ for all $l\\ge 1$ and\n\\begin{equation}\\label{ode-boundary}\n\\begin{split}\n&|\\xi_l(r)|\\leq 1,\\  \\ \\quad r\\in(0,\\frac{r_0}{\\epsilon_{k,j}}),\\ \\ \\quad l\\ge 0,  \\\\\n&\\xi_l(r)=o(\\epsilon_{k,1}),\\quad r\\sim e^{\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}},\\quad l\\ge 1.\n\\end{split}\n\\end{equation}\nFrom the equation of $\\xi_l$ and the maximum principle, we only need to consider the finite $l$. Without loss of generality, we consider $1\\leq l\\leq l_0$ in the following analysis. Let us denote $\\delta_{l,j}=\\frac{l}{1+\\alpha_j}$ and consider the homogeneous ordinary differential equation\n\\begin{equation}\\label{ode}\n\t\\xi_l^{''}+\\frac{1}{r}\\xi_l^{'}+\\Big(r^{2\\alpha_j}e^{U_j}-\\frac{l^2}{r^2}\\Big)\\xi_l=0.\n\\end{equation}\nBy direct computation, we can verify that the following two functions are two fundamental solutions of (\\ref{ode})\n\\begin{align*}\n&\\xi_{l,1}(r)=\\frac{(\\delta_{l,j}+1)r^l+(\\delta_{l,j}-1)r^{2(1+\\alpha_j)+l}}{1+r^{2(1+\\alpha_j)}},\\\\\n&\\xi_{l,2}(r)=\\frac{(\\delta_{l,j}+1)r^{2(1+\\alpha_j)-l}+(\\delta_{l,j}-1)r^{-l}}{1+r^{2(1+\\alpha_j)}}.\n\\end{align*}\nUsing $|\\xi_l|\\leq 1$ we have $C_{l,2}=0$, that is\n\\begin{equation*}\n\t\\xi_l(r)=C_{l,1}\\xi_{l,1}(r)+\\xi_{l,p}(r)\n\\end{equation*}\nwhere $C_{l,1}$ is a constant, and\n\\begin{equation}\\label{solution-p}\n\\xi_{l,p}(r)=\\Big(\\int\\frac{w_1}{w}{\\rm d}r\\Big)\\xi_{l,1}(r)+\\Big(\\int\\frac{w_2}{w}{\\rm d}r\\Big)\\xi_{l,2}(r)\n\\end{equation}\nfor\n\\begin{equation*}\nw=\n\\begin{vmatrix}\n\\xi_{l,1} & \\xi_{l,2} \\\\\n\\xi_{l,1}^{'} & \\xi_{l,2}^{'}\n\\end{vmatrix},\\quad\nw_1=\\begin{vmatrix}\n0 & \\xi_{l,2} \\\\\n\\tilde{E}_l & \\xi_{l,2}^{'}\n\\end{vmatrix},\\quad\nw_2=\\begin{vmatrix}\n\\xi_{l,1} & 0 \\\\\n\\xi_{l,1}^{'} & \\tilde{E}_l\n\\end{vmatrix}.\\quad\n\\end{equation*}\n\nIt is easy to see that $w^{'}=(\\xi_{l,1}\\xi_{l,2}^{'}-\\xi_{l,1}^{'}\\xi_{l,2})^{'}=-\\frac{1}{r}w$, which means $w(r)\\sim \\frac{1}{r}$. Next, let us estimate $\\xi_{l}$ in $(0,\\frac{r_0}{\\epsilon_{k,j}})$ for $l\\ge 1$.\n\n\\smallskip\nFor $1\\leq j\\leq t$, the assumption $D(\\mathbf{p})=0$ implies $a_{k,j}=O(\\epsilon_{k,1}^2)$. Furthermore, for $t+1\\leq j\\leq \\tau$, it is easy to see that $\\epsilon_{k,j}=o(\\epsilon_{k,1})$. Therefore, for all $1\\leq j\\leq \\tau$, we estimate $\\tilde{E}_l$ as follows\n\\begin{equation}\\label{E-1}\n\\begin{split}\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{o(\\epsilon_{k,1})r+O(d_k\\xi_l)+o(\\epsilon_{k,1})\\big\\},\\quad l=1,2;  \\\\\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{O(d_k\\xi_l)+o(\\epsilon_{k,1})\\big\\},\\, \\qquad\\quad\\qquad l\\ge 3.\n\\end{split}\n\\end{equation}\nRoughly,\n\\begin{equation}\\label{E-2}\n\\begin{split}\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{o(\\epsilon_{k,1})r+O(d_k)+o(\\epsilon_{k,1})\\big\\},\\quad l=1,2;  \\\\\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{O(d_k)+o(\\epsilon_{k,1})\\big\\},\\, \\qquad\\quad\\qquad l\\ge 3.\n\\end{split}\n\\end{equation}\nBy using the above estimates (\\ref{E-2}) for $\\tilde{E}_l$ and (\\ref{solution-p}), we have\n\\begin{align*}\n\\xi_{l,p}(r)=&\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\bigg\\{\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}(s+1)\\xi_{l,2}(s){\\rm d}s\\Big)\\xi_{l,1}(r)\\\\\n&+\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}(s+1)\\xi_{l,1}(s){\\rm d}s\\Big)\\xi_{l,2}(r)\\bigg\\},\\quad l=1,2.\\\\\n\\xi_{l,p}(r)=&\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\bigg\\{\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}\\xi_{l,2}(s){\\rm d}s\\Big)\\xi_{l,1}(r)\\\\\n&+\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}\\xi_{l,1}(s){\\rm d}s\\Big)\\xi_{l,2}(r)\\bigg\\},\\quad l\\ge 3.\n\\end{align*}\nDirect computation shows, for $0<r<1$\n\\begin{align*}\n\t\\xi_{l,p}(r)=\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\big(r^l+r^{2\\alpha_j+2}\\big),\\quad l\\ge 1;\n\\end{align*}\nand for $1<r<\\frac{r_0}{\\epsilon_{k,j}}$\n\\begin{align*}\n\\xi_{l,p}(r)=\\left\\{\n\\begin{array}{lcl}\n\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\big(r^{-l}+r^{-(2\\alpha_j+1)}\\big),\\quad l=1,2, \\\\\n\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\big(r^{-l}+r^{-(2\\alpha_j+2)}\\big),\\quad l\\ge 3.\n\\end{array}\n\\right.\n\\end{align*}\n\nConsequently\n\\begin{equation}\\label{xi-est-1}\n\t\\parallel \\xi_{l,p}\\parallel_{L^{\\infty}((0,\\frac{r_0}{\\epsilon_{k,j}}))}=O(d_k)+o(\\epsilon_{k,1}),\\quad 1\\leq l\\leq l_0.\n\\end{equation}\nCombining (\\ref{E-1}) and (\\ref{xi-est-1}), we rewrite $\\tilde{E}_l$ as\n\\begin{equation}\\label{E-3}\n\\begin{split}\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{o(\\epsilon_{k,1})r+O(d_k^2)+o(\\epsilon_{k,1})\\big\\},\\quad l=1,2;  \\\\\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{O(d_k^2)+o(\\epsilon_{k,1})\\big\\},\\, \\qquad\\quad\\qquad l\\ge 3.\n\\end{split}\n\\end{equation}\nThen repeating the above argument $n_0$ times, we obtain\n\\begin{equation}\\label{xi-est-2}\n\\parallel \\xi_{l,p}\\parallel_{L^{\\infty}((0,\\frac{r_0}{\\epsilon_{k,j}}))}=o(\\epsilon_{k,1}),\\quad 1\\leq l\\leq l_0.\n\\end{equation}\nThen, from (\\ref{ode-boundary}), we have\n\\begin{equation*}\n   C_{l,1}=o(\\epsilon_{k,1})O(\\epsilon_{k,j}^l).\n\\end{equation*}\nand\n\\begin{equation}\\label{xi-cos}\n\t\\parallel \\xi_l\\parallel_{L^{\\infty}((0,\\frac{r_0}{\\epsilon_{k,j}}))}=o(\\epsilon_{k,1}),\\quad  l\\ge 1.\n\\end{equation}\nSimilarly,\n\\begin{equation}\\label{xi-sin}\n\\parallel \\tilde{\\xi}_l\\parallel_{L^{\\infty}((0,\\frac{r_0}{\\epsilon_{k,j}}))}=o(\\epsilon_{k,1}),\\quad  l\\ge 1.\n\\end{equation}\n\nIn other words, all projections of high frequency of $\\varsigma_{k,j}$ $(1\\leq j\\leq \\tau)$ are $o(\\epsilon_{k,1})$.\n\n\\medskip\nUsing (\\ref{xi-cos}) and (\\ref{xi-sin}), we now obtain an important sharper estimate of the right hand side of (\\ref{PI-1}).\n\n\\begin{lem}\\label{lem-PI1-right-re}\n\t\\begin{equation}\\label{PI-1-r-4}\n\t{\\rm (RHS)}\\ {\\rm of} (\\ref{PI-1})=-2(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad t+1\\leq j\\leq \\tau.\n\t\\end{equation}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tIn view of (\\ref{K1})$\\sim$(\\ref{expansion}) and (\\ref{K3-first-2})$\\sim$(\\ref{K3-third-2}), we only need to improve the estimate in (\\ref{K3-first-2}). In other words, it is enough to prove the following estimate\n\t\\begin{equation}\\label{K3-first-2-re}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x =o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation}\n\tIn fact, by the change of variable $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$, we have\n    \\begin{align*}\n    &\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x  \\\\\n    =&\\int_{B(p_{k,j}^{(1)},r)}\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{u_k^{(1)}}\\varsigma_k\\frac{1-e^{u_k^{(2)}-u_k^{(1)}}}{u_k^{(1)}-u_k^{(2)}}\\langle a_{k,j},x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n    =&\\int_{B(p_{k,j}^{(1)},r)}\\frac{\\rho_kh_j(p_{k,j}^{(1)})e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{g_k(x)}}{\\big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\big)^2} \\langle a_{k,j},x-p_{k,j}^{(1)}\\rangle  \\\\\n    &\\times\\Big\\{\\sum_{n=0}^{n_0}\\frac{(-d_k)^{n}}{(n+1)!}\\varsigma_k^{n+1}+O(d_k^{n_0+1})\\Big\\}{\\rm d}x  \\\\\n    =&\\epsilon_{k,j}\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\\sum_{n=0}^{n_0}\\frac{(-d_k)^{n}}{(n+1)!}\\varsigma_{k,j}^{n+1}<a_{k,j},z>{\\rm d}x+o(\\epsilon_{k,1}^2).  \\\\\n    \\end{align*}\n    where we used the fact $\\alpha_j<\\alpha_1$ and the definition of $n_0$.\n\tThen from symmetry and the estimates of high frequency of $\\varsigma_{k,j}$, which are (\\ref{xi-cos}) and (\\ref{xi-sin}), we have the following estimate\n\t\\begin{align*}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(\\epsilon_{k,j})o(\\epsilon_{k,1})+o(\\epsilon_{k,1}^2)\n\t=o(\\epsilon_{k,1}^2).\n\t\\end{align*}\nTherefore, (\\ref{K3-first-2-re}) holds. Finally, combining (\\ref{K3-first-2-re}) with the proof of Lemma \\ref{lem-PI1-right}, we obtain the esstimate (\\ref{PI-1-r-4}).\n\t\n\\end{proof}\n\nBased on the Pohozaev-type identity (\\ref{PI-1}) and its refined estimates, which are (\\ref{PI-1-l-re}) (\\ref{PI-1-r-2}) (\\ref{PI-1-r-3}) and (\\ref{PI-1-r-4}), we can improve the estimate for $A_{k,j}$ and prove $b_0=0$.\n\\begin{cor}\\label{cor-A-kj-re}\n\t\\begin{equation}\\label{A-kj-est-re}\n\t|A_{k,j}|=O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad 1 \\leq j \\leq m.\n\t\\end{equation}\n\t\\begin{flushright}\n\t\t\\qed\n\t\\end{flushright}\n\\end{cor}\n\\begin{prop}\\label{prop-b0}\n\t$b_0=0$. In particular, $b_{j,0}=0$, for $1\\leq j\\leq m$.\n\\end{prop}\n\\begin{proof}[\\textbf{Proof}]\n\tNow the global cancellation property of $f_k$ plays a crucial role:\n\t\\begin{equation*}\n\t\t\\sum_{j=1}^{m}A_{k,j}=\\int_{M}f_k{\\rm d}\\mu=0.\n\t\\end{equation*}\n\tFrom (\\ref{PI-1}) (\\ref{PI-1-l-re}) (\\ref{PI-1-r-2}) (\\ref{PI-1-r-3}) and (\\ref{PI-1-r-4}), we can see\n\t\\begin{align*}\n\t\tb_0e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}\\sum_{j=1}^t\\big[\\Delta h(p_j)+\\rho_*-N^*-2K(p_j)\\big]\\big(\\rho_kh_j(p_{k,j}^{(1)})\\big)^{\\frac{1}{1+\\alpha_1}}=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}).\n\t\\end{align*}\n\tOn the other hand, from (\\ref{uk-ave-2}), it holds\n\t\\begin{equation*}\n\t\th_j^2(p_j)=h_1^2(p_1)e^{G_1^*(p_1)}e^{-G_j^*(p_j)}+o(1),\\quad 1\\leq j\\leq t.\n\t\\end{equation*}\n\tAs a consequence, we obtain\n\t\\begin{equation}\\label{b0-est}\n\t\\begin{split}\n\te^{-\\frac{G_1^*(p_1)}{1+\\alpha_1}}\\big(\\rho_*h_1^2(p_1)\\big)^{-\\frac{1}{1+\\alpha_1}}L(\\mathbf{p})b_0e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}),\n\t\\end{split}\n\t\\end{equation}\n\twhich together with the assumption $L(\\mathbf{p})\\neq 0$ implies $b_0=0$. In particular, $b_{j,0}=0$ for $1\\leq j\\leq m$.\t\n\t\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{main-theorem}}\\label{pf-uni-1}\n\n\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{main-theorem} }]\nLet $p_k^*$ be a maximum point of $\\varsigma_k$, which says\n\\begin{equation}\\label{max=1}\n|\\varsigma_k(p_k^*)|=1\n\\end{equation}\nIn view of Lemma \\ref{lem-limit-2} and Proposition \\ref{prop-b0}, we obtain the fact\n\\begin{equation*}\n\t\\varsigma_k\\to 0 \\quad{\\rm in}\\ \\; C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\}).\n\\end{equation*}\n Therefore,\n\\begin{equation}\\label{pk*}\n\\lim\\limits_{k\\rightarrow\\infty}p_k^*=p_j,\n\\end{equation}\nfor some $p_j\\in\\{p_1,\\cdots,p_m\\}$. Moreover, denoting $s_k=|p_k^*-p_{k,j}^{(1)}|$, by Lemma \\ref{lem-limit-1} and Proposition \\ref{prop-b0}, it holds\n\\begin{equation*}\n\\varsigma_{k,j}\\rightarrow  0 \\quad{\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2).\n\\end{equation*}\nThus,\n\\begin{equation}\\label{sk}\n\\lim\\limits_{k\\rightarrow\\infty}\\epsilon_{k,j}^{-1}s_k=+\\infty\n\\end{equation}\nSetting $\\tilde{\\varsigma_k}(x)=\\varsigma_k(s_kx+p_{k,j}^{(1)})$, $|x|<s_k^{-1}r$, where $r>0$ small enough, then $\\tilde{\\varsigma_k}$ satisfies\n\\begin{align*}\n0=&\\Delta\\tilde{\\varsigma_k}(x)+\\rho_k\\tilde{h}_j(s_kx+p_{k,j}^{(1)})s_k^{2(1+\\alpha_j)}|x|^{2\\alpha_j}c_k(s_kx+p_{k,j}^{(1)})\\tilde{\\varsigma_k}(x) \\\\\n=& \\Delta\\tilde{\\varsigma_k}(x)+\\frac{8(1+\\alpha_j)^2(\\epsilon_{k,j}^{-1}s_k)^{2(1+\\alpha_j)}|x|^{2\\alpha_j}}{\\big(1+(\\epsilon_{k,j}^{-1}s_k)^{2(1+\\alpha_j)}|x|^{2(1+\\alpha_j)}\\big)^2}\\big(1+O(s_k|x|)+o(1)\\big).\n\\end{align*}\n\nOn the other hand, by (\\ref{max=1}), we also have\n\\begin{equation}\\label{scale-max=1}\n\\Big|\\tilde{\\varsigma_k}\\big(\\frac{p_k^*-p_{k,j}^{(1)}}{s_k}\\big)\\Big|=|\\varsigma_k(p_k^*)|=1.\n\\end{equation}\nIn view of (\\ref{sk}) and $|\\tilde{\\varsigma_k}|\\leq 1$, we see that $\\tilde{\\varsigma_k}\\rightarrow\\tilde{\\varsigma_0}$ in $C_{loc}(\\mathbb{R}^2\\backslash\\{0\\})$, where $\\tilde{\\varsigma_0}$ satisfies $\\Delta\\tilde{\\varsigma_0}=0$ in $\\mathbb{R}^2\\backslash\\{0\\}$. Since $|\\tilde{\\varsigma_0}|\\leq 1$, we have $\\Delta\\tilde{\\varsigma_0}=0$ in $\\mathbb{R}^2$. Hence $\\tilde{\\varsigma_0}$ is a constant.\n\nRecalling that $\\frac{|p_k^*-p_{k,j}^{(1)}|}{s_k}=1$ and (\\ref{scale-max=1}), we find that $\\tilde{\\varsigma_0}\\equiv 1$ or $\\tilde{\\varsigma_0}\\equiv -1$. Therefore, we obtain that for $ k $ large enough\n\\begin{equation}\\label{contra1}\n|\\varsigma_k(x)|\\geq\\frac{1}{2},\\quad |x-p_{k,j}^{(1)}|\\in\\big(\\frac{s_k}{2},2s_k\\big).\n\\end{equation}\n\nBy using Lemma \\ref{lem-limit-2}, we have\n\\begin{equation}\\label{contra2}\n\\varsigma_k(x)=o(1)+o(1)\\log R+O(R^{-2(1+\\alpha_j)}),\\quad |x-p_{k,j}^{(1)}|\\in(R\\epsilon_{k,j},d).\n\\end{equation}\nfor fixed $d>0$ small enough and arbitrary $R>0$ large enough.\n\n\\smallskip\nHowever, by (\\ref{sk}), $\\epsilon_{k,j}\\ll s_k$. Thus, $|\\varsigma_k(s_k)|<\\frac{1}{4}$ for $ k $ large enough, which contradicts with (\\ref{contra1}). Theorem \\ref{main-theorem} is established.\n\t\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{main-theorem-2}}\\label{pf-uni-2}\n\nIn this section, we will analyse the behavior of $u_k^{(1)}$ and $u_k^{(2)}$ whose common blowup points include singular source(s) and regular point(s). So in this section $\\tau<m$, $0<\\alpha_j< 1$ for $1\\leq j\\leq \\tau$ and $\\alpha_j=0$ for $\\tau+1\\leq j\\leq m$. Our argument is similar to the approach in \\cite{bart-4} where all blowup points are regular points. The fact that $\\varsigma_{k,j}\\rightarrow 0$ in $C_{loc}(\\mathbb{R}^2)$ for all $1\\leq j \\leq m$ plays a vital role.\n\n\\medskip\nIn \\cite{lin-yan-uniq}, Lin-Yan obtained the following Pohozaev-type identity:\n\\begin{lemA}\\label{Pohozave identity-2}\n\\cite{bart-4,lin-yan-uniq}\n\nFor $\\tau+1\\leq j\\leq m$, it holds\n\\begin{align}\\label{PI-2}\n\\begin{split}\n& \\int_{\\partial B(p_{k,j}^{(1)},r)}\\Big(<\\nu,\\nabla\\varsigma_k>\\nabla_iv_{k,j}^{(1)}+<\\nu,\\nabla v_{k,j}^{(2)}>\\nabla_i\\varsigma_k \\Big) {\\rm d}\\sigma \\\\\n& \\quad -\\frac{1}{2}\\int_{\\partial B(p_{k,j}^{(1)},r)}<\\nabla(v_{k,j}^{(1)}+v_{k,j}^{(2)}),\\nabla\\varsigma_k>\\frac{(x-p_{k,j}^{(1)})_i}{|x-p_{k,j}^{(1)}|} {\\rm d}\\sigma, \\\\\n=\\ & -\\int_{\\partial B(p_{k,j}^{(1)},r)}\\rho_k\\tilde{h}_j(x)\\frac{e^{u_k^{(1)}}-e^{u_k^{(2)}}}{\\parallel u_k^{(1)}-u_k^{(2)} \\parallel_{L^{\\infty}(M)} }\\frac{(x-p_{k,j}^{(1)})_i}{|x-p_{k,j}^{(1)}|} {\\rm d}\\sigma \\\\\n& \\quad + \\int_{B(p_{k,j}^{(1)},r)} \\rho_k\\tilde{h}_j(x)\\frac{e^{u_k^{(1)}}-e^{u_k^{(2)}}}{\\parallel u_k^{(1)}-u_k^{(2)} \\parallel_{L^{\\infty}(M)}} \\nabla_i\\big(\\log \\tilde{h}_j+\\phi_{k,j}\\big) {\\rm d}x.\n\\end{split}\n\\end{align}\n\n\\end{lemA}\n\nBy Lemma 4.6 in \\cite{bart-4} and Appendix D in \\cite{lin-yan-uniq}, we have:\n\\begin{align}\\label{PI-2-r}\n{\\rm (RHS)} \\ {\\rm of}\\ (\\ref{PI-2})=e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\Big(\\sum_{h=1}^2 D_{h,i}^2(\\log \\tilde{h}_j+\\phi_{k,j} )(p_{k,j}^{(1)})b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}).\n\\end{align}\nfor $i=1,2$ and $\\tau+1\\le j\\le m$. The detail of this proof can be found in \\cite{bart-4}.\n\nThe LHS of (\\ref{PI-2}) boils down to sharp estimates of $\\nabla v_{k,j}^{(i)}$ and $\\nabla\\varsigma_k$ on $\\partial B(p_{k,j}^{(1)},r)$. The estimate for $\\nabla v_{k,j}^{(i)}$ is established in   Lemma \\ref{lem-Dv-kj}, and the following lemma provides the estimates for $\\nabla\\varsigma_k$ (see (\\ref{Dsigma_k-1}) for comparison).\n\n\\begin{lem}\\label{lem-C1-est-re}\n\t\n\tFor any $\\theta\\in(0,r)$ small enough, it holds\n\t\\begin{align}\\label{GRF-est-re}\n\t\\begin{split}\n\t\\varsigma_k-\\bar{\\varsigma}_k=\\sum_{j=\\tau+1}^m e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}&\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}) \\\\\n\t &{\\rm in}\\ \\; C^1\\Big(M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)\\Big).\n\t\\end{split}\n\t\\end{align}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\tUsing the same notations in (\\ref{J1+J2+J3}) and (\\ref{J3}), now we only need to show\n\t\\begin{equation*}\n\t\tJ_1=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}),\\quad J_2=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\t\\end{equation*}\n\t\n\tIndeed, from (\\ref{A-kj-est-re}) and the assumption $0< \\alpha_1<1$, we have\n\t\\begin{equation}\\label{J1-re}\n\t\tJ_1=\\sum_{j=1}^m A_{k,j}G(p_{k,j}^{(1)},x)=O(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}})=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}})\n\t\\end{equation}\n\t\n\tRecall that\n\t\\begin{align*}\n\t\tJ_2=&\\sum_{j=1}^\\tau\\int_{M_j}\\big(G(y,x)-G(p_{k,j}^{(1)},x)\\big)f_k(y){\\rm d}\\mu(y)   \\\\\n\t\t=&\\sum_{j=1}^\\tau\\int_{B(p_{k,j}^{(1)},r_0)}f_k(y)e^{\\phi_j(y)}\\langle\\partial _yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle {\\rm d}y +O(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}})\n\t\\end{align*}\n\tBased on (\\ref{xi-cos}) and (\\ref{xi-sin}), by the method similar to the proof of (\\ref{K3-first-2-re}) in Lemma \\ref{lem-PI1-right-re}, we have\n\t\\begin{equation*}\n\t\t\\int_{B(p_{k,j}^{(1)},r_0)}f_k(y)e^{\\phi_j(y)}\\langle\\partial _yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle {\\rm d}y=O(\\epsilon_{k,j})o(\\epsilon_{k,1})+o(\\epsilon_{k,1}^2)=O(\\epsilon_{k,1}^2).\n\t\\end{equation*}\n\tTherefore,\n\t\\begin{equation}\\label{J2-re}\n\t\tJ_2=O(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}})=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\t\\end{equation}\n\t\n\tConsequently (\\ref{GRF-est-re}) holds in $C^1\\big(M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)\\big)$ and the gradient estimate is\n\\begin{equation}\\label{Dsigma_k-2}\n\\begin{split}\n\\nabla\\varsigma_k(x)=\\sum_{j=\\tau+1}^m e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\nabla_x\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\\end{split}\n\\end{equation}\n\t\n\\end{proof}\n\n\n\nBy the improved estimates of $\\nabla v_{k,j}^{(i)}$ and $\\nabla\\varsigma_k$ in  (\\ref{Dv-kj-est}) and (\\ref{Dsigma_k-2}), we can estimate the left hand of (\\ref{PI-2}) just like Lemma 4.7 in \\cite{bart-4} or Appendix D in \\cite{lin-yan-uniq} and the result is:\n\\begin{equation}\\label{PT-l}\n\\begin{split}\n{\\rm (LHS)} \\ {\\rm of}\\ (\\ref{PI-2})=&-8\\pi\\bigg\\{\\sum_{l\\neq j}^{\\tau+1,\\cdots,m}e^{-\\frac{\\lambda_{k,l}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}b_{l,h}\\Big)B_l\\\\\n&\\ \\;+e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}R(y,x)\\big|_{x=y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j\\bigg\\}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}).\n\\end{split}\n\\end{equation}\n\nFinally we prove $b_{j,1}=b_{j,2}=0$ for all $j$.\n\\begin{prop}\\label{prop-b1b2}\n\n$b_{j,1}=b_{j,2}=0$, for all $j=\\tau+1,\\cdots,m$. In particular,\n\\begin{equation*}\n\\varsigma_{k,j}\\rightarrow 0\\quad {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2),\\quad {\\rm for\\ \\, all} \\ \\, j=1,\\cdots,m.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\nObviously, (\\ref{PI-2}) together with (\\ref{PI-2-r}) and (\\ref{PT-l}) implies, for all $i=1,2$, and $j=\\tau +1,\\cdots,m$,\n\\begin{align}\\label{PI-final}\n\\begin{split}\n&e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\Big(\\sum_{h=1}^2 D_{h,i}^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})b_{j,h}\\Big)B_j\\\\\n=&-8\\pi\\sum_{l\\neq j}^{\\tau+1,\\cdots,m}e^{-\\frac{\\lambda_{k,l}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}b_{l,h}\\Big)B_l\\\\\n&-8\\pi e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}R(y,x)\\big|_{x=y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}).\n\\end{split}\n\\end{align}\n\nSet $\\vec{b}=(\\tilde{b}_{\\tau+1,1}B_{\\tau+1},\\tilde{b}_{\\tau+1,2}B_{\\tau+1},\\cdots,\\tilde{b}_{m,1}B_m,\\tilde{b}_{m,2}B_m)$, where\n\\begin{equation*}\n\\tilde{b}_{l,h}=\\lim\\limits_{k\\to +\\infty}\\big(e^{\\frac{\\lambda_{k,j}^{(1)}-\\lambda_{k,1}^{(1)}}{2}}b_{l,h}\\big).\n\\end{equation*}\nThen by (\\ref{p_kj-location}) and letting $k\\to+\\infty$, we obtain that\n\\begin{equation}\\label{b-vector}\nD^2f^*(p_{\\tau+1},\\cdots,p_m)\\cdot \\vec{b}=0\n\\end{equation}\nBy using the non-degeneracy assumption $\\det \\big(D^2f^*(p_{\\tau+1},\\cdots,p_m)\\big)\\neq 0$, we conclude that\n\\begin{equation}\\label{b=0}\nb_{j,1}=b_{j,2}=0,\\quad j=\\tau+1,\\cdots,m.\n\\end{equation}\nProposition \\ref{prop-b1b2} is established.\n\n\\end{proof}\n\n\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{main-theorem-2} }]\nFrom\nLemma \\ref{lem-limit-2} and Proposition \\ref{prop-b0} $\\varsigma_k$ tends to $0$ in\n$C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\})$.\nBy Lemma \\ref{lem-limit-1} and Proposition \\ref{prop-b1b2}, we have\n\\begin{equation*}\n\t\\varsigma_{k,j}\\rightarrow  0\\quad  {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2),\\quad 1\\leq j\\leq m.\n\\end{equation*}\nTheorem \\ref{main-theorem-2} follows just like the last step of the proof of  Theorem \\ref{main-theorem}.\n\n\\end{proof}\n\nFinally, we finish to prove Theorem \\ref{main-theorem-3} and Theorem \\ref{main-theorem-4} about Dirichlet problems.\n\n\\begin{proof}[\\textbf{Proof of Theorems \\ref{main-theorem-3}, \\ref{main-theorem-4} }]\n\tFor the blowup solutions to (\\ref{equ-flat}), the corresponding estimates as in section \\ref{preliminary} have been also obtained in \\cite{chen-lin}\\cite{zhang2} for $\\alpha_j\\in\\mathbb{R}^+\\setminus\\mathbb{N}$ and in \\cite{chen-lin-sharp}\\cite{zhang1}\\cite{gluck} for $\\alpha_j=0$. Those preliminary estimates have almost the same form except for $\\phi_j=0$ and $K\\equiv 0$, where $\\phi_j$ are the conformal factor at $p_j$ and $K$ is the Gaussian curvature of $M$.\n\t\n\tThen, under the assumption of regularity about $\\partial\\Omega$ and $q_j\\in \\Omega$ $(1\\leq j\\leq N)$, \\cite{ma-wei} has showed that the blowup points of (\\ref{equ-flat}) are far away from $\\partial\\Omega$ via the moving plane method and the Pohozaev identities. Consequently, the terms coming from the boundary of domain are included in the error term. In other words, those boundary terms do not affect our argument.\n\t\n\tOn the other hand, the vital part of estimates obtained in section \\ref{difference}, \\ref{anal-pohozaev} and \\ref{pf-uni-2} only come from local analysis, Therefore, such results still work for the Dirichlet problem (\\ref{equ-flat}).\n\t\n\tThus, Theorem \\ref{main-theorem-3} and Theorem \\ref{main-theorem-4} can be proved as Theorem \\ref{main-theorem} and Theorem \\ref{main-theorem-4}, respectively. \n\t\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tThe COVID-19 pandemic disrupted many lives and businesses \\citep{UNcovid}. Governments around the world imposed unprecedented measures to contain the spread of the virus, which often took the form of full lockdown.\n\tThe policies implemented to contain the pandemic as well as the volume and the unbalanced shares of home production were likely to have consequences for the gender distribution of work. On one hand, women are over\u2010represented in sectors that have been defined as essential and in occupations that cannot be performed from home \\citep{OECD2021gender}. On the other hand, women tend to be over\u2010represented in service industries, such as retail, tourism, and hospitality, which have been either subject to lockdown or to strict restrictions for some time \\citep{hupkau2020work, farre2020covid}. Women are also more likely than men to be employed in the informal sector, compensated in cash with no official oversight, and no eligibility to benefits, such as the furlough scheme \\citep{EP2021gender}. Finally, even when working from home, women on average perform most of home production tasks, e.g., childcare, and more in general they bear a large share of the earning penalty associated with childbearing \\citep{sevilla2020baby, kleven2019children, yildirim2021differential}. The overall perception is that women have been hurt by the pandemic disproportionately more compared to men, but there seems to be evidence of  large heterogeneity across countries \\citep{bluedorn2021gender}.\n\t\n\tWe investigate the labour market dynamics in Italy in the period 2013-2020, i.e., before and during the pandemic, using longitudinal quarterly labour force data, with a focus on gender, age and geographical differences. Italy was the first country in Europe to be hit by the COVID-19 pandemic and the first to implement a national lockdown in the beginning of March, 2020 \\citep{saglietto2020covid}. To mitigate its effect on the labour market, the Italian Government pro-actively implemented two aggressive policies: a ban on layoffs and the extension of a pre-existing furlough scheme \\citep{barbieri2021italian}. The Italian case is also particularly interesting as the labour market is largely heterogeneous, with vast, well-known and persistent regional disparities:  industrial activities are mostly concentrated in the North and in the Center, while food industry and tourism are mainly concentrated in Southern regions \\citep{OECDItaly}. In addition, Italy ranks among the weakest of OECD countries regarding job quantity, defined as employment, unemployment and underemployment \\citep{OECD2018}, reflecting persistently large gender employment gaps and a remarkably low female labour force participation rate, particularly in the South \\citep{agovino2019local}. Due to these very different conditions at the outburst of the pandemic, we find that the COVID-19 shock had sizable asymmetric effects across categories of individuals but, less expected, are their size and persistence over time.\n\t\n\tIn particular, we first document how the shares of individuals across seven labour market states (permanent, temporary and self-employment, furlough scheme, unemployment, education and inactivity) before the pandemic differed across ages, gender and geographical areas. The shares of individuals in education was comparable in the Northern-Center and the Southern part of Italy among the 15-24 age cohort, but while in the North and Center the transitions from education were predominantly towards the labour force, in the South the transitions towards inactivity were already much higher. This phenomenon was particularly large among females in the South, leading to an inactivity share of above 50\\% among the 40-49 age cohort. \n\tThe COVID-19 pandemic shock lead to an increased flow of discouraged workers moving from unemployment to inactivity, across age cohorts, gender and geographical areas. We also show evidence of a large outflow of individuals who left permanent (and temporary) employment to become inactive and, more importantly, persisted outside the labour market until the fourth quarter of 2020. Females aged 30-39 with at least one young child, living in the North and Center of Italy, previously hired on a permanent contract show the largest and most persistent outflows. We find similar patterns in the South, but surprisingly the size of the impact is shown to be much smaller.\n\t\n\tThis paper fits in the growing literature which analyzes the asymmetric effects of the pandemic on different categories of individuals \\citep{caselli2021mobility}. A number of papers show that the impact of the shock has been disproportional high among vulnerable workers \\citep{chetty2020economic}. In particular, while there is quite a large consensus regarding the fact that younger low-income workers were more likely to lose their jobs, findings are more controversial regarding other demographic dimensions, such as gender. Some studies have provided evidence that the pandemic is largely affecting women's labour market outcomes. Specifically, \\cite{alon2021mancession} finds higher employment losses for women compared to men in the US; these are confirmed by \\cite{albanesi2021gendered}, who provide evidence of a substantial and persistent drop for women not only in employment, but also in labor force participation.  \\cite{fabrizio2021covid} and \\cite{zamarro2020gender} find that less educated women with young children were the most adversely affected, while \\cite{shibata2020distributional} shows that women and Hispanics are the two categories who lost the most. Finally, \\cite{adams2020inequality} show that women and workers without a college degree are significantly\n\tmore likely to have lost their jobs. However, other contributions point to no gender difference in labour market outcomes as a consequence of the pandemic. \\citep{casarico2020heterogeneous} show that gender is a non-significant predictor of job loss in the aggregate, while \\cite{hupkau2020work} find no difference in outcomes between men and women at the extensive margin, and if anything smaller losses for women at the intensive margin. Overall there is evidence of large heterogeneity across countries \\citep{bluedorn2021gender, adams2020inequality, dang2021gender}.\n\t\n\tThe paper is organized as follows. Section \\ref{sec:italianPolicies} describes the policies implemented in Italy in the outburst of the pandemic, while Section \\ref{sec:ItalianLabourMarket} described the dynamics of the Italian labour market before and during the pandemic. Specifically, Section \\ref{sec:Data} describes the data, Section \\ref{sec:Pre} illustrates the shares and transition probabilities across states before the pandemic, Section \\ref{sec:Post} shows the dynamics during the pandemic and Section \\ref{sec:logit} presents the results of the parametric estimation of the probability to remain active in the labour market by gender and age. Finally, Section \\ref{sec:concludingRemarks} concludes the paper. \n\t\n\t\t\n\\section{COVID-19 policies in Italy}\\label{sec:italianPolicies}\n\t\nThe first cases of COVID-19 in Italy were registered on January 31, 2020, but the virus began to spread exponentially in the second half of February. At the beginning, the virus circulated predominantly in Northern regions but by the beginning of March, it had reached all regions. On  March 10, the whole country went into a full lockdown. On  March 11 the government prohibited nearly all commercial activity except for supermarkets and pharmacies and on March 21 it restricted the movement of people and closed all non-essential businesses and industries. Sectors identified as essential, which could continue operating, include mainly agriculture, some manufacturing, energy and water supply, transports and logistics, ICT, banking and insurance, professional and scientific activities, public administration, education, health care and some service activities. Non-essential sectors which were completely shut include most manufacturing, wholesale and retail trade, hotels, restaurants and bars, entertainment and sport activities \\cite{casarico2020heterogeneous}.\n\nSubsequently, on March 17 the Italian government implemented  two new labor market policies to protect workers: (i) a COVID-19 furlough scheme  and (ii) a ban on layoffs. The former was implemented for an initial duration of 9 weeks, and it applied retroactively starting from February 23. It represents an extension of the regular furlough scheme to all firms, independently on size. This measure aimed at preserving employment and allowed firms to cut labor costs during the lockdown period, by reducing hours of work thanks to a wage subsidy granted by the government. Firms using the COVID-19 furlough scheme could renew temporary contracts, waiving to the norms of the standard regulation. Upon completion of the furlough period, firms were allowed to dismiss employees for redundancy. The ban on layoffs prevented firm to fire workers for 60 days, starting from  March 17; this ban could be applied retroactively to pending, but already validated layoffs from February 23. Two later decrees extended the validity of these measures, which were still in place until the end of 2021.\n\n\\section{The Italian labour market pre and during the COVID-19 pandemic}\\label{sec:ItalianLabourMarket}\n\n\nThe Italian labour market pre-COVID-19 presented specific characteristics, which we deem as crucial to understand the asymmetric impact of the shock on different categories of individuals. The key features are the differentials in labour market participation by gender, males versus females, and by geographical area, North and Center versus South of Italy.\nThe literature has highlighted significant gender differences, and relevant geographical differentials are also reported as a structural feature of the Italian labour market \\citep{bertola2003structure}. Women, on average, show a lower attachment to the labour force  together with a lower commitment to labour market activity compared to men \\citep{schiattarella2018old}. While the North-South divide characterizes many elements of the economic and cultural life in Italy,  it is particularly  striking in women's work, with women from the Southern regions (and the Islands) being much less likely to work and much more likely to end up in unemployment or outside the labour force. A specific characteristic of women in\nSouthern Italy is that they  are comparatively more likely not to work and not to return to the labour after marriage (or childbearing). On average in Italy 30\\% of  Italian mothers in employment  stop working  to care for children or other relatives, and of these only about 12\\% go back to work at some point, but this number is much lower in the Italian South, due to the predominant role of the male breadwinner model \\citep{SOAS2021gender, baussola2014disadvantaged}.\n\n\n\n\nFinally, to gender and geographical characteristics, we add age as another dimension of analysis. We therefore split individuals according to 6 cohorts: the first 15-19 age cohort includes individuals who either decided to drop out high school or keeps on studying, the 20- 24 age cohorts identifies individuals who either decided to stop studying or attend university, individuals in the 25-29 age cohort are in transition between tertiary education and the labour market, the 30-39 age cohort is made of individuals who are likely to have a family with small children, while the 40-49 age cohort comprises individuals with older children. Finally, the 50- 64 age cohort includes mature adults moving towards retirement. The next section (Section \\ref{sec:Data}) describes in detail the data and the methodology used in the analysis.\n\n\n\n\\subsection{Data and methodology\\label{sec:Data}}\n\t\n\tWe use Italian quarterly longitudinal labour force data as provided by the Italian Institute of Statistics (ISTAT) for the period 2013 (quarter I) to 2020 (quarter IV).\\footnote{Data for the period 2013 (quarter I) to 2020 (quarter IV) are available upon request at: https:\/\/www.istat.it\/it\/archivio\/185540.} The Italian Labour Force Survey (LFS) follows a simple rotating sample design where households participate for two consecutive quarters, exit for the following two quarters, and come back in the sample for other two consecutive quarters. As a result, 50\\% of the households, interviewed in a quarter, are re-interviewed after three months, 50\\% after twelve months, 25\\% after nine and fifteen months. This rotation scheme allows to obtain 3 months longitudinal data, which include almost 50\\% of the original sample.\n\nThe longitudinal feature of these data is essential for achieving a complete picture of significant economic phenomena of labour market mobility. Per each individual who has been interviewed we observe a large number of individual and labour market characteristics at the time of the interview and three months before. Taking into account the structure of this database, we compute the labour market flows by calculating the quarter-on-quarter transitions made by background individuals between different labour market states. \n\nOn average approximately 70.000 individuals are interviewed each quarter, of which 45.000 are part of the working age population. The average quarterly inflow of younger individuals in the working age population is 0.3\\%, while the average quarterly outflow of older individuals from the working age population is 0.4\\%,  backing our hypothesis of a (almost) constant working age population within quarters.\n\n\nThe dynamics of the labour market can be efficiently described by Markov Chains with discrete states in discrete time. Our dataset allows to consider quarters as unit of time and to define seven labour market states: permanent (PE), temporary (TE), self-employment (SE), unem- ployment (U), the furlough scheme (FS), education (EDU) and inactivity (NLFET).  The NLFET state collects the working age individuals who are not in the labour force, in education or in training, therefore representing an accurate measure of inactivity \\citep{ose2017youth}. The dynamics are therefore represented through a Transition Probability Matrix (TPM), which shows both permanence in each labour market state and the probability of transition from one state to another in a given period of time, and fully characterizes the dynamics of the shares of the whole population in each state. In particular, the shares of individuals in different states provide a picture of the long-term trends, as they take longer to react to shocks, while the transition probabilities inform about the sudden impact of the (pandemic) shock. Taking into account the structure of the available database, we compute the labour market flows by calculating the quarter-on-quarter transitions made by individuals between different labour market states. In the analysis we take the first quarter of 2020, which marks the time of the initial spread of the virus, as the period when the dynamics of the Italian labour market are expected to change. The inferential analysis on the shares and \n transition probabilities is computed via bootstrap using 1000 draws from the original sample.\n\n\nImportant data limitations are to be mentioned. First, the point-in-time measurement of the worker's labour market state fails to capture transitions within the period (quarter). For instance, if an employed worker becomes unemployed and finds a new job within a quarter, we do not observe those transitions in our data. Second, the available data stop at quarter IV of 2020, while it would be desirable to have data also for 2021 to explore the further persistence of pandemic shock. Second, we do not have information about the household composition of individuals, as we only observe the household size. For this reason, in the last part of our analysis, we use data from the European Labour Force, which contains detailed information about the number and age of children. Finally, another important limit of our analysis is\nthe short longitudinal span, as we have observations about the same individuals only in two consecutive quarters, thus forcing our analysis to be based on a Markovian process of order one, which is a further limit in the study of persistence.\n\n\t\n\\subsection{Pre-COVID-19 pandemic} \\label{sec:Pre}\n \n \nIn this section we study the dynamics of the labour market before the pandemic (2013 quarter II - 2019 quarter IV), with a special attention to the long-run trends caused by the labour market reforms implemented during the period of observation. For this reason, we consider annual transition probabilities, calculated per each quarter as the product of the last four quarterly transition probabilities.\n \n \\begin{figure}[h!]\n \t\\caption{Shares of individuals aged 15-19 in NLFET and EDU in the North and Center and South of Italy.}\n \t\\label{fig:shares1519text}\n \t\\centering\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North1519EDU.eps}\n \t\t\\caption{North and Center - EDU.}\n \t\t\\label{fig:actualAverageMass_North1519EDU}\n \t\t\\vspace{0.1cm}\n \t\\end{subfigure}\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_South1519EDU.eps}\n \t\t\\caption{South - EDU.}\n \t\t\\label{fig:actualAverageMass_South1519EDU}\n \t\\end{subfigure}\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North1519NLFET.eps}\n \t\t\\caption{North and Center - NLFET.}\n \t\t\\label{fig:actualAverageMass_North1519NLFET}\n \t\\end{subfigure}\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_South1519NLFET.eps}\n \t\t\\caption{South - NLFET.}\n \t\t\\label{fig:actualAverageMass_South1519NLFET}\n \t\\end{subfigure}\n \t\\vspace{0.2cm}\n \t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n \\end{figure}\n \n Specifically, a number of pension reforms increased retirement age and lead to a raise in the share of individuals aged 50-64 in permanent employment \\citep{de2017dynamics}. Moreover, a number of reforms\\footnote{The \\textit{Decreto Poletti} in 2014 reduced the hiring costs and increased the number of extensions within the same duration, the \\textit{Decreto Dignita'} in 2018 reduced the maximum length, increased the hiring costs and reduced the number of possible extensions} changed the rules for the utilization of temporary contracts, creating from time to time incentives (or disincentives) for firms to hire workers on such contract, leading to increasing or decreasing shares of workers on temporary employment. Finally, in 2015, strong fiscal incentives for the hiring of permanent employees and the introduction of a new permanent contract, with firing costs increasing with tenure, lead to a significant increase in the transitions from unemployment and temporary employment to permanent employment \\citep{boeri2019tale}. \n Net of these trends due to labour market reforms, through the analysis of the shares of individuals across states and the transition probabilities, we find large differences in the labour market choices of individuals in the North and the South of Italy. We also observe very different behaviors between males and females, across different age cohorts. We report all figures by gender, age and geographical location and detailed comments in Appendix \\ref{app:dynamicsprecovid}, while we summarize here the main findings. The different patterns between these four categories of individuals (males and females in the North and in the South) start as early as when they are in secondary education (15-19 age group). On average females stay longer in education in both geographical areas compared to males, however already in this age cohort  more individuals drop out school and enter the NLFET state in the South (Figure \\ref{fig:shares1519text}). This pattern is similar among males and females, with approximately 4\\% of individuals aged 15-19 being inactive in the North and already approximately 9\\% being inactive in the South.\n \n \n  \n\\begin{figure}[!h]\n\t\\caption{Annual transition probabilities of individuals aged 20-24 from education to temporary employment and the NLFET state in the North and Center and South of Italy.}\n\t\\label{fig:transprob2024femalestext}\n\t\\caption*{\\scriptsize{\\textbf{Females}}.}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleNorthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - North and Center.}\n\t\t\\label{fig:femaleNorthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleSouthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - South.}\n\t\t\\label{fig:femaleSouthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleNorthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET - North and Center.}\n\t\t\\label{fig:femaleNorthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleSouthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET  - South.}\n\t\t\\label{fig:femaleSouthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\caption*{\\scriptsize{\\textbf{Males}}.}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleNorthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - North and Center.}\n\t\t\\label{fig:maleNorthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleSouthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - South.}\n\t\t\\label{fig:maleSouthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleNorthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET - North and Center.}\n\t\t\\label{fig:maleNorthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleSouthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET  - South.}\n\t\t\\label{fig:maleSouthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\nThe percentage of individuals (both males and females) who go to university is similar in the two geographical areas, but while in the North and Center, those who leave education  are much more likely to enter the labour market, mainly with a temporary contract, in the South they are more likely to join the NLFET state (Figure \\ref{fig:transprob2024femalestext}). Among females approximately 14\\% transit from education to a temporary contract in the North and Center, compared to less than 6\\% in the South; at the same time approximately 8\\% transit to the NLFET state in the North and Center, compared to 12\\% in the South. Among males, 14\\% move from education to temporary employment and 6\\% to the NLFET state in the North and Center, compared to 8\\% and 12\\%, respectively in the South.\nThis worrying bleeding of individuals in the South from education to NLFET state, across both females and males, seems like an irreversible process: persistence in the NLFET state remains very high across all cohorts. Over time more and more individuals in the South, mostly females, keep joining the NLFET state from all other states, particularly from unemployment and temporary employment. This leads to a dramatic situation in which 35\\% of females in the 25-29 age category in the South is in the NLFET state. This percentage keeps growing as they get older, reaching 45\\% among the 30-39 age category (Figure \\ref{fig:shares3039text}) and more than 50\\% among the 40-49 age category. Hence, the percentage of 30-39 years old in permanent employment is dramatically lower in the South, with less than 40\\% of males and 22\\% of females hired on a permanent contract compared to more than 60\\% of males and 50\\% of females in the North and Center, respectively. \n\n\\begin{figure}[!ht]\n\t\\caption{Shares of individuals aged 30-39 in PE and NLFET in the North and Center and South of Italy.}\n\t\\label{fig:shares3039text}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North3039PE.eps}\n\t\t\\caption{PE - North and Center.}\n\t\t\\label{fig:actualAverageMass_North3039PE}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{actualAverageMass_South3039PE.eps}\n\t\\caption{PE - South.}\n\t\\label{fig:actualAverageMass_South3039PE}\n\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North3039NLFET.eps}\n\t\t\\caption{NLFET - North and Center.}\n\t\t\\label{fig:actualAverageMass_North3039NLFET}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{actualAverageMass_South3039NLFET.eps}\n\t\\caption{NLFET - South.}\n\t\\label{fig:actualAverageMass_South3039NLFET}\n\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\n\n\\subsection{Dynamics during the COVID-19 pandemic} \\label{sec:Post}\n\nWe analyse the impact of the pandemic on the labour market of the four categories of individuals (females and males in the North and Center and South of Italy), by age groups. Specifically, in Section \\ref{sec:dynamicsshares} we analyse the dynamics of the shares, while in Section \\ref{sec:dynamicstrans} we focus on the transition probabilities. As we are interested in the sudden impact of the pandemic and its short-run impact, we compute quarterly transition probabilities.\n\n\\subsubsection{The dynamics of shares}\\label{sec:dynamicsshares}\n\nTo investigate the way the pandemic has affected the distribution of individuals across the seven labour market states, we compare the shares of individuals by age category in quarter IV of 2020 with the same quarter one year before (Table \\ref{sharechangesqIVtext}).\\footnote{In Appendix \\ref{app:dynamicssharesduring} we report the same statistics comparing the shares of individuals by age group in quarter III of 2020 with the same quarter one year before.}\n\nIn quarter IV of 2020, the shares of males living in the North and Center of Italy in the NLFET state increased across all age groups, except for the 30-39 cohort, compared to the same quarter one year before, with larger changes among the 20-24 and 25-29 age groups. When we compare those shares with the ones of males living in the South, we notice similar patterns: across (almost) all age categories the share of males in the NLFET state is higher. We also observe a higher share of 25-29 years old in education in the South, while no significant change in the education state in the North and Center. Among females in the North and Center of Italy we also have a larger presence in the NLFET state across all age categories (except for the 15-19 and 50-64 age groups). The increase is particularly important for the 20-24 and 30-39 age categories. However, quite surprisingly, the share of females living in the South in the NFLET state is larger only among the 25-29 age group. Finally, there is an increase in the share of 15-19 and 20-24 years old females in education in the South compared to the same quarter one year before, while no significant change in education in the North and Center. \n\nOverall, we find a substantial increase in the share of males across all age categories in the NLFET state in the North and Center as well as in the South of Italy. With regards to females, while in the North and Center the NLFET share increased across all age categories, in the South it increased only for the 25-29 age category, while it did not change for all other age groups. We also observe a higher share of younger individuals in education in the South (both males and females), but no significant change in the North and Center.\n\n\\begin{table}[htbp] \\centering \n\\caption{Changes in the shares in different labour market states between quarter IV of 2019 and quarter IV of 2020 by category of individuals.}  \n\\label{sharechangesqIVtext} \n\\scriptsize{\n\t\\begin{tabular}{@{\\extracolsep{5pt}} cccccccc} \n\t\\\\[-1.8ex]\\hline \\\\[-1.8ex] \n\t\\multicolumn{8}{c}{ Males - North and Center}\\\\\\\\[-1.8ex] \n\t\t\\hline \\\\[-1.8ex] \n\t\t& SE & TE & PE & U & NLFET & EDU & FS \\\\ \n\t\t\\hline \\\\[-1.8ex] \n\t\t15-19 & $0.002$ & $$-$0.012^{***}$ & $0.0004$ & $0.00001$ & $\\textbf{0.007}^{**}$ & $0.002$ & $0.001^{*}$ \\\\ \n\t\t& $(0.113)$ & $(0.002)$ & $(0.424)$ & $(0.482)$ & $(0.045)$ & $(0.383)$ & $(0.053)$ \\\\ \n\t\t20-24 & $0.002$ & $$-$0.011$ & $$-$0.011^{*}$ & $0.005$ & $\\textbf{0.018}^{***}$ & $$-$0.020^{**}$ & $0.016^{***}$ \\\\ \n\t\t& $(0.331)$ & $(0.131)$ & $(0.088)$ & $(0.212)$ & $(0.000)$ & $(0.041)$ & $(0.000)$ \\\\ \n\t\t25-29 & $$-$0.007$ & $$-$0.015^{**}$ & $$-$0.024^{***}$ & $0.003$ & $\\textbf{0.020}^{***}$ & $$-$0.003$ & $0.026^{***}$ \\\\ \n\t\t& $(0.197)$ & $(0.041)$ & $(0.002)$ & $(0.300)$ & $(0.000)$ & $(0.330)$ & $(0.000)$ \\\\ \n\t\t30-39 & $$-$0.004$ & $$-$0.010^{***}$ & $$-$0.016^{***}$ & $$-$0.002$ & $0.003$ & $0.002$ & $0.027^{***}$ \\\\ \n\t\t& $(0.222)$ & $(0.002)$ & $(0.006)$ & $(0.232)$ & $(0.127)$ & $(0.144)$ & $(0.000)$ \\\\\n\t\t40-49 & $$-$0.006$ & $$-$0.008^{***}$ & $$-$0.020^{***}$ & $$-$0.004^{**}$ & $\\textbf{0.013}^{***}$ & $$-$0.0004$ & $0.026^{***}$ \\\\ \n\t\t& $(0.121)$ & $(0.000)$ & $(0.000)$ & $(0.047)$ & $(0.000)$ & $(0.146)$ & $(0.000)$ \\\\\n\t\t50-64 & $$-$0.007^{*}$ & $$-$0.005^{***}$ & $$-$0.009^{**}$ & $$-$0.005^{***}$ & $\\textbf{0.008}^{**}$ & $0.0001$ & $0.018^{***}$ \\\\ \n\t\t& $(0.062)$ & $(0.001)$ & $(0.030)$ & $(0.000)$ & $(0.022)$ & $(0.182)$ & $(0.000)$ \\\\ \n\t\t\\hline \\\\[-1.8ex] \n \\multicolumn{8}{c}{Males - South}\\\\\\\\[-1.8ex] \n \\hline \\\\[-1.8ex] \n & SE & TE & PE & U & NLFET & EDU & FS \\\\ \n \\hline \\\\[-1.8ex] \n15-19 & $0.003$ & $0.010^{***}$ & $$-$0.003$ & $$-$0.019^{***}$ & $$-$0.0002$ & $0.009$ & $0.0003$ \\\\\n & $(0.112)$ & $(0.008)$ & $(0.214)$ & $(0.000)$ & $(0.494)$ & $(0.229)$ & $(0.167)$ \\\\  \n20-24 & $$-$0.009^{**}$ & $$-$0.020^{**}$ & $0.008$ & $$-$0.016^{**}$ & $\\textbf{0.016}^{*}$ & $0.010$ & $0.011^{***}$ \\\\\n& $(0.038)$ & $(0.014)$ & $(0.161)$ & $(0.042)$ & $(0.079)$ & $(0.229)$ & $(0.000)$ \\\\ \n25-29 & $0.002$ & $$-$0.024^{***}$ & $$-$0.018$ & $$-$0.022^{**}$ & $\\textbf{0.017}^{*}$ & $\\textbf{0.034}^{***}$ & $0.012^{***}$ \\\\ \n & $(0.409)$ & $(0.002)$ & $(0.046)$ & $(0.011)$ & $(0.065)$ & $(0.000)$ & $(0.000)$ \\\\ \n30-39 & $$-$0.004$ & $$-$0.014^{***}$ & $$-$0.019^{**}$ & $$-$0.021^{***}$ & $\\textbf{0.027}^{***}$ & $0.004$ & $0.027^{***}$ \\\\\n& $(0.285)$ & $(0.002)$ & $(0.017)$ & $(0.000)$ & $(0.000)$ & $(0.110)$ & $(0.000)$ \\\\ \n40-49 & $0.005$ & $$-$0.012^{***}$ & $$-$0.017^{**}$ & $$-$0.016^{***}$ & $\\textbf{0.016}^{***}$ & $0.0005$ & $0.025^{***}$ \\\\\n  & $(0.236)$ & $(0.001)$ & $(0.018)$ & $(0.000)$ & $(0.003)$ & $(0.227)$ & $(0.000)$ \\\\ \n50-64 & $0.003$ & $0.0005$ & $$-$0.007$ & $$-$0.008^{***}$ & $$-$0.005$ & $0.0001$ & $0.017^{***}$ \\\\ \n& $(0.321)$ & $(0.459)$ & $(0.169)$ & $(0.009)$ & $(0.210)$ & $(0.222)$ & $(0.000)$ \\\\  \n\\hline \\\\[-1.8ex] \n \\multicolumn{8}{c}{Females - North and Center}\\\\\\\\[-1.8ex] \n \\hline \\\\[-1.8ex] \n & SE & TE & PE & U & NLFET & EDU & FS \\\\ \n \\hline \\\\[-1.8ex] \n15-19 & $0.001$ & $$-$0.003$ & $0.0004$ & $$-$0.002$ & $0.002$ & $0.001$ & $0.001^{*}$ \\\\\n &$(0.245)$ & $(0.204)$ & $(0.409)$ & $(0.341)$ & $(0.331)$ & $(0.450)$ & $(0.054)$ \\\\ \n20-24 & $$-$0.002$ & $$-$0.041^{***}$ & $$-$0.011^{*}$ & $0.013^{***}$ & $\\textbf{0.034}^{***}$ & $$-$0.001$ & $0.008^{***}$ \\\\\n & $(0.305)$ & $(0.000)$ & $(0.050)$ & $(0.009)$ & $(0.000)$ & $(0.467)$ & $(0.000)$ \\\\ \n25-29 & $$-$0.0001$ & $$-$0.021^{***}$ & $$-$0.027^{***}$ & $0.009^{*}$ & $\\textbf{0.017}^{**}$ & $$-$0.005$ & $0.027^{***}$ \\\\ \n & $(0.485)$ & $(0.005)$ & $(0.002)$ & $(0.077)$ & $(0.022)$ & $(0.244)$ & $(0.000)$ \\\\ \n30-39 & $$-$0.009^{**}$ & $$-$0.007^{*}$ & $$-$0.029^{***}$ & $$-$0.015^{***}$ & $\\textbf{0.034}^{***}$ & $$-$0.001^{}$ & $0.027^{***}$ \\\\ \n & $(0.012)$ & $(0.052)$ & $(0.000)$ & $(0.000)$ & $(0.000)$ & $(0.225)$ & $(0.000)$ \\\\ \n40-49 & $$-$0.009^{***}$ & $$-$0.010^{***}$ & $$-$0.020^{***}$ & $$-$0.007^{***}$ & $\\textbf{0.020}^{***}$ & $0.0004$ & $0.024^{***}$ \\\\ \n & $(0.008)$ & $(0.000)$ & $(0.000)$ & $(0.001)$ & $(0.000)$ & $(0.247)$ & $(0.000)$ \\\\\n50-64 & $$-$0.006^{**}$ & $$-$0.006^{***}$ & $$-$0.007^{*}$ & $$-$0.001$ & $0.002$ & $0.0002$ & $0.018^{***}$ \\\\  \n& $(0.010)$ & $(0.000)$ & $(0.083)$ & $(0.293)$ & $(0.358)$ & $(0.194)$ & $(0.000)$ \\\\ \n \\hline \\\\[-1.8ex] \n \\multicolumn{8}{c}{Females - South}\\\\\\\\[-1.8ex] \n \\hline \\\\[-1.8ex] \n & SE & TE & PE & U & NLFET & EDU & FS \\\\ \n \\hline \\\\[-1.8ex] \n15-19 & $0.002^{*}$ & $$-$0.002$ & $$-$0.005^{***}$ & $$-$0.015^{***}$ & $0.004$ & $\\textbf{0.017}^{**}$ & $0.0003$ \\\\\n & $(0.058)$ & $(0.229)$ & $(0.001)$ & $(0.001)$ & $(0.315)$ & $(0.049)$ & $(0.213)$ \\\\ \n20-24 & $$-$0.014^{***}$ & $$-$0.024^{***}$ & $$-$0.018^{***}$ & $$-$0.019^{**}$ & $0.013$ & $\\textbf{0.050}^{***}$ & $0.013^{***}$ \\\\\n & $(0.000)$ & $(0.000)$ & $(0.001)$ & $(0.017)$ & $(0.143)$ & $(0.000)$ & $(0.000)$ \\\\ \n25-29 & $$-$0.006$ & $$-$0.020^{***}$ & $$-$0.016^{**}$ & $$-$0.037^{***}$ & $\\textbf{0.058}^{***}$ & $0.007$ & $0.014^{***}$ \\\\\n  & $(0.166)$ & $(0.003)$ & $(0.041)$ & $(0.000)$ & $(0.000)$ & $(0.254)$ & $(0.000)$ \\\\ \n30-39 & $0.001$ & $0.007$ & $$-$0.013^{**}$ & $$-$0.022^{***}$ & $0.010$ & $$-$0.001$ & $0.017^{***}$ \\\\ \n& $(0.418)$ & $(0.109)$ & $(0.042)$ & $(0.000)$ & $(0.154)$ & $(0.424)$ & $(0.000)$ \\\\\n40-49 & $0.003$ & $$-$0.001$ & $$-$0.023^{***}$ & $$-$0.005$ & $0.011$ & $0.002^{*}$ & $0.013^{***}$ \\\\\n & $(0.304)$ & $(0.461)$ & $(0.000)$ & $(0.148)$ & $(0.116)$ & $(0.086)$ & $(0.000)$ \\\\ \n50-64 & $0.001$ & $$-$0.005^{**}$ & $$-$0.013^{***}$ & $0.001$ & $0.009$ & $0.0004$ & $0.007^{***}$ \\\\\n  & $(0.367)$ & $(0.024)$ & $(0.007)$ & $(0.284)$ & $(0.114)$ & $(0.143)$ & $(0.000)$ \\\\ \n \\hline \\\\[-1.8ex] \n\\end{tabular} \n}\n\\end{table} \n\n\\subsubsection{The dynamics of transition probabilities}\\label{sec:dynamicstrans}\n\nTo assess the way the pandemic has affected the transition probabilities across labour market states, we compare the actual data with the counterfactual scenario of no pandemic shock, i.e., the quarterly transition probabilities for the four categories of individuals across demographic groups against the forecasted quarterly transition probabilities for the quarters during the pandemic, i.e., quarter I of 2020 to quarter IV of 2020.\\footnote{The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 quarter I- 2019 quarter IV.} \n\nTable \\ref{fig:transProbUtoNLFET} reports the transition probabilities from unemployment to the NLFET state for all categories of individuals, thus capturing whether an increased number of discouraged workers who are pessimistic about the probability to find a job gave up on their search. Across all groups and across all age cohorts these probabilities have significantly increased in quarter II of 2020, compared to the forecasted probabilities. While for some categories the change was temporary and went back to the pre-pandemic rates in quarter III of 2020, for others, such as 20-24 and 40-49 females in the South the increased percentage persisted until quarter IV of 2020. Similar patterns are observed among 30-39 males in the South, while among both males and females in the North and Center, we do not observe persistence. Hence, while in the outburst of the pandemic all unemployed workers across different types got discouraged to some degree and left the labour market, the higher transition rates from unemployment to inactivity persisted mostly for females in the South. \n\n\\begin{figure}[htbp]\n\t\\caption{Transition probabilities from unemployment to the NLFET state by age groups.}\n\t\\label{fig:transProbUtoNLFET}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{15-19}}.}\n\t\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_15}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_15}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_15}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_15}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{20-24}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_20}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_20}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_20}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_20}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{25-29}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_25}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_25}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_25}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_25}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{30-39}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_30}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_30}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{40-49}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{ \\scriptsize{\\textit{Note}: The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 (quarter I)- 2019 (quarter IV). Confidence intervals at 90\\% are computed using 1000 bootstraps and reported in parenthesis. EDU refers to education, TE to temporary employment, PE to permanent employment, U to unemployment. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\nWe then focus on the transition rates from permanent and temporary employment to the NLFET state (Tables \\ref{fig:transProbTEtoNLFET}-\\ref{fig:transProbPEtoNLFET}) to assess the effect of the pandemic on the outflows from permanent and temporary employment. We report statistics for the 30-39 and 40-49 categories for which the results are more striking.\\footnote{Statistics for all other age categories are reported in Appendix \\ref{app:dynamicssharesduring}.} The pandemic significantly increased the transition probabilities from temporary and permanent employment to the NLFET state mostly for females, and among those mainly for the ones living in the North and Center, across both age categories. Data show that females in the North and Center are the ones who are more affected by the pandemic as compared to women in the South, as a larger percentage was active on the labour market at the time of the shock. In quarter III of 2020 females aged 30-39 (40-49) had a probability to transit from temporary employment to the NLFET state of about 25\\% (19\\%) compared to the forecasted 10\\% (12\\%). Similar patterns are found for the probability to transit from permanent employment to the NLFET state: it jumped to 2.5\\% (1.3\\%) compared to a forecasted probability of 1.7\\% (0.5\\%) for women aged 30-39 (40-49). Although these numbers seem low, they correspond to approximately 40.000 women in the age 30-39 cohort and 25.000 women in the age 40-49 cohort moving from employment to inactivity.\n\n\n\\subsubsection{Female labour market participation and household composition}\n\nDuring the pandemic due to the prolonged schools closure, women had to juggle between their jobs and the children care \\citep{qian2020covid}. Particularly women with small children might have been pushed out of the labour market due to caring responsibilities.  However, in the South, where the female inactivity rate in these age cohorts was already much higher, the impact of the pandemic has been less strong. Unfortunately, our data do not provide information about the number and age of children, but only about the household size. Therefore, we use  European Labour Force Survey data for Italy for 2019  to compute the shares of females in the age cohorts 30-39 and 40-49 with at least one child below the age of 11 by employment status, distinguishing between the North and Center and South of Italy (Table \\ref{sharechangesELFS2019}).\\footnote{In the Appendix we report similar statistics for 2020.} We select the age of children to be below 11, which we consider the cutoff age, below which children need the presence of their parents. \nWe find that 47.5\\% of females in the  30-39 age group with at least one young child is  employed on a permanent contract in the North and Center, compared to 19.1\\% in the South. While the share on temporary employment and self-employment is comparable, we find the share in unemployment to be higher in the South (10.2\\% against 6.2\\%) while the share of inactive females is much higher in the South, i.e., 55.7\\% compared to 28.2\\%. When we focus on the 40-49 age group, we observe very similar patterns.\n\n\n\n\\begin{figure}[htbp]\n\t\\caption{Transition probabilities from temporary employment to the NLFET state by age groups.}\n\t\\label{fig:transProbTEtoNLFET}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{30-39}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromTE_age_class_30}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromTE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromTE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromTE_age_class_30}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{40-49}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromTE_age_class_40}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromTE_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromTE_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromTE_age_class_40}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{ \\scriptsize{\\textit{Note}: The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 (quarter I)- 2019 (quarter IV). Confidence intervals at 90\\% are computed using 1000 bootstraps and reported in parenthesis. EDU refers to education, TE to temporary employment, PE to permanent employment, U to unemployment. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\caption{Transition probabilities from permanent employment to the NLFET state by age groups.}\n\t\\label{fig:transProbPEtoNLFET}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{30-39}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromPE_age_class_30}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromPE_age_class_30}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{40-49}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:transProbFromEDUtoSE_APP}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:transProbFromEDUtoTE_APP}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:transProbFromEDUtoPE_APP}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:transProbFromEDUtoU_APP}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{ \\scriptsize{\\textit{Note}: The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 (quarter I)- 2019 (quarter IV). Confidence intervals at 90\\% are computed using 1000 bootstraps and reported in parenthesis. EDU refers to education, TE to temporary employment, PE to permanent employment, U to unemployment. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\n\n\n\\begin{table}[!htbp] \\centering \n\t\\caption{Percentage of females with at least one child below the age of 11 by geographical area and employment status (2019).}  h\n\t\\label{sharechangesELFS2019} \n\t\\scriptsize\n\t\\begin{tabular}{@{\\extracolsep{5pt}} ccccc} \n\t\t\\\\[-1.8ex]\\hline \\\\[-1.8ex] \n\t\t&\\multicolumn{2}{c}{Age 30-39}&\\multicolumn{2}{c}{Age 40-49}\\\\\t\n\t\t\\hline \\\\[-1.8ex]\n\t\t&North and Center&South&North and Center&South\\\\\n\t\t\\hline \\\\[-1.8ex]\n\t\tPermanent  &47.5&19.1&55.0&28.1\\\\\n\t\tTemporary&8.7&7.1&6.4&5.9\\\\\n\t\tSelf-employed&8.6&6.9&12.1&9.2\\\\\n\t\tUnemployed& 6.2&10.2&5.4&8.4\\\\\n\t\tInactive&\t28.2&55.7&20.2&48.0\\\\\n\t\t\\hline \\\\[-1.8ex]\n\t\tTotal (in 000s)& 1270&736&1249&559\\\\\n\t\t\\hline \n\t\t\\multicolumn{5}{l}{\\tiny{\\textit{Source}: ELFS data.}}\n\t\\end{tabular}\t\n\\end{table}\n\nWe then split the sample of females with at least one child below the age of 11 by household size: we compute the share of females living in a household with less or more than two people, as a proxy for the presence of children, by  geographical area and age cohort (Table \\ref{shareHH2ELFS2019}).  \n\n\\begin{table}[!htbp] \\centering \n\t\\caption{Percentage of females with at least one child below the age of 11 by geographical area and household size.}  \n\t\\label{shareHH2ELFS2019} \n\t\\scriptsize\n\t\\begin{tabular}{@{\\extracolsep{5pt}} ccccc} \n\t\t\\\\[-1.8ex]\\hline \\\\[-1.8ex] \n\t\t&\\multicolumn{2}{c}{Age 30-39}&\\multicolumn{2}{c}{Age 40-49}\\\\\t\n\t\t\\hline \\\\[-1.8ex]\n\t\t&North and Center&South&North and Center&South\\\\\n\t\t\\hline \\\\[-1.8ex]\n\t\t$>$ 2 components&81.2& 71.5&55.2&44.0\\\\\n\t\t$\\leq$ 2 components& 6.6&9.3&6.0&4.8\\\\ \n\t\t\n\t\t\\hline \n\t\t\\multicolumn{5}{l}{\\tiny{\\textit{Source}: ELFS data.}}\n\t\\end{tabular}\t\n\\end{table}\t\n\nAmong females in the 30-39 age cohort,  81.2\\% live in a household with more than two components in the North and Center, and 71.5\\% in the South. This implies that among females in the 30-39 age category the household size is a good proxy to assess whether they have children below the age of 11. We use this information to compute the transition probabilities  from permanent and temporary employment to the NLFET state for the two groups of females in the North and Center and in the South using the quarterly longitudinal labour force data (Figure \\ref{fig:transProbHHsize}).\n\n\\begin{figure}[!ht]\n\t\\caption{Transition probabilities of females aged 30-39  from permanent and temporary employment, to the NLFET in the North and Center and South of Italy by household size.}\n\t\\label{fig:transProbHHsize}\n\t\\caption*{\\scriptsize{\\textbf{From permanent employment}}.}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleNorthHHmore2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{North and Center - Household $>$ 2.}\n\t\t\\label{fig:FemaleNorthHHmore2_transProbtoNLFETfromPE_age_class_30}\n\t\t\\vspace{0.1cm}\n\t\t\\vspace{0.1cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleNorthHHless2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{North and Center - Household $\\leq$ 2.}\n\t\t\\label{fig:FemaleNorthHHless2_transProbtoNLFETfromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleSouthHHmore2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{South - Household $>$ 2.}\n\t\t\\label{fig:FemaleSouthHHmore2_transProbtoNLFETfromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleSouthHHless2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{South - Household $\\leq$ 2.}\n\t\t\\label{fig:FemaleSouthHHless2_transProbtoNLFETfromPE_age_class_30}\n\t\\end{subfigure}\n\t\\caption*{\\scriptsize{\\textbf{From temporary employment}}.}\n\t\\centering\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleNorthHHmore2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{North and Center - Household $>$ 2.}\n\t\\label{fig:FemaleNorthHHmore2_transProbtoNLFETfromTE_age_class_30}\n\t\\vspace{0.1cm}\n\t\\vspace{0.1cm}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleNorthHHless2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{North and Center - Household $\\leq$ 2.}\n\t\\label{fig:FemaleNorthHHless2_transProbtoNLFETfromTE_age_class_30}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleSouthHHmore2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{South - Household $>$ 2.}\n\t\\label{fig:FemaleSouthHHmore2_transProbtoNLFETfromTE_age_class_30}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleSouthHHless2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{South - Household $\\leq$ 2.}\n\t\\label{fig:FemaleSouthHHless2_transProbtoNLFETfromTE_age_class_30}\n\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\nWe find that the transition probabilities to the NLFET state have significantly and persistently increased for females in the  30-39 age group living in the North and Center in a household with more than two people from both permanent and temporary employment. We find no statistically significant effect for females living in households with less than two components neither in the North and Center or the South, neither from permanent nor from temporary employment. Interestingly, we also find no significant effect for females living in the South in a household with more than two people from both permanent and temporary employment. These results confirm our previous evidence of a stronger pandemic impact on females  with young children living in the North and Center of Italy. Those women are likely to have been forced out of the labour market to take care of their children during the prolonged school closure. The worrying part of the story is the persistence of this effect until the end of 2020, as the literature has provided large evidence of weaker attachment of females to the labour market compared to males and of high female state dependency in the inactivity state \\citep{duhautois2018state}. As explained by \\cite{honore2000panel}, this state dependency might be of two types: either because lagged decisions play a role (true state dependency) or because of different propensities across individuals to experience the event (spurious state dependency). Distinguishing between these two types is extremely important for policy reasons, however long longitudinal data on individual histories are required, which unfortunately we do not have availability of. This is therefore part of our future agenda.\n\n\\subsection{Estimation of the probability to remain active in the labour market}\\label{sec:logit}\n\nIn this section we provide additional support to our findings by using a parametric estimation technique.\nWe group observations according to whether the individuals are in or out of the labour force, as our analysis so far has shown that this is the most important feature to be attributed to the pandemic. The estimation is carried out using a \\textit{logit} model, in which the dependent variable is the probability to be active  on the labour market in the next quarter conditional on being active the current quarter.\\footnote{Alternatively, we could have estimated a multinomial logit, taking into account the transitions to all seven labour market states considered in our analysis so far. However, the validity of such estimation would have  depended on the fulfillment of the Independence of Irrelevant Alternatives (IIA) condition \\citep{train2009discrete}. In our case the IIA assumes that the odds-ratio is not altered with the addition or deletion of a particular alternative, which is very likely to be violated in our setting.} We split the sample into four sub-samples, according to gender and age. Specifically, we focus on the 30-39 and 40-49 age cohorts as these are the ones who have been more severely affected by the COVID-19 pandemic. Permanence in the initial state of activity is the baseline category for interpretation of results.\nThe estimation of a logit model using design-based longitudinal weights may create severe numerical issues \\citep{train2009discrete}. Hence, we run 1000 bootstraps using the longitudinal sample weights to estimate the model's coefficients and their 95\\% confidence intervals. \n\n\\begin{table}[!htbp] \\centering \n\t\\caption{Odds-ratios of remaining active on the labour market next period conditional on being active.} \n\t\\label{tab:logit} \n\t\\scriptsize{\n\t\t\\begin{tabular}{@{\\extracolsep{5pt}}lcccc} \n\t\t\t\\\\[-1.8ex]\\hline \n\t\t\n\t\t\n\t\t\t\\\\[-1.8ex] & \\multicolumn{4}{c}{Active in the labour market next quarter} \\\\\n\t\t\t\\hline  \t\\\\[-1.8ex]\n\t\t\t\\\\[-1.8ex]  & Female 30-39 & Male 30-39 & Female 40-49 & Male 40-49\\\\ \n\t\t\t\\hline \\\\[-1.8ex] \n\n\t\t\t2020 & \\textbf{0.509} & \\textbf{0.627} & 0.745 & 0.750 \\\\ \n\t\t\t& (0.374-0.701) & (0.460-0.834) & (0.526-1.064) & (0.543-1.022) \\\\ \n\t\t\t& & & & \\\\ \n\n\t\t\t\\\\[-1.8ex] \n\t\t\t\\textbf{2020 $\\times$(North or Center)$\\times$}& \\textbf{0.614 }& 0.761 & 0.953 & 1.272 \\\\ \n\t\t\t\\textbf{$\\times$(Household members >2)}& (0.401-0.892) & (0.473-1.113) & (0.621-1.406) & (0.850-1.854) \\\\ \n\t\t\t& & & & \\\\ \n\t\t\t(Intercept) & 18.228 & 23.247 & 25.899 & 39.220 \\\\ \n\t\t\t& (15.075-21.960) & (18.958-29.223) & (21.354-31.183) & (30.986-48.260) \\\\ \n\t\t\t& & & & \\\\ \n\t\t\t\\hline \\\\[-1.8ex] \n\t\t\tObservations & 62,950 & 78,307 & 95,990 & 120,232 \\\\ \n\t\t\tLog Likelihood & $-$14,065.210 & $-$12,152.510 & $-$16,144.900 & $-$14,945.660 \\\\ \n\t\t\tAkaike Inf. Crit. & 28,172.430 & 24,347.030 & 32,331.800 & 29,933.320 \\\\ \n\t\t\t\n\t\t\t\\hline \\\\[-1.8ex] \n\t\t\t\\multicolumn{5}{l}{\\textit{Note:} Confidence interval at 95\\% are calculated by 1000 bootstrap using sample weights.} \\\\ \n\t\t\\end{tabular} \n\t}\n\\end{table}\n\nTable \\ref{tab:logit} displays the estimated odds-ratios of the probability to remain active on the labour market, conditional on being active the quarter before given a set of explanatory variables. In particular, the odds-ratio represents the ratio between the probability that the event will occur with respect to the probability the event will not occur, conditioned to a given explanatory variable; hence, an odds-ratio grater than one implies an increased occurrence of the event, while an odds-ratio lower than one implies a decreased occurrence of the event with respect to a given explanatory variable. The reference category includes Italian individuals with a tertiary level of education living in the South in a household with less than two people.\n\nThe full regression with all the variables included is reported in Table \\ref{apptab:logit} in Appendix \\ref{app:logit}. Immigrants, both from EU and non-EU countries, are more likely to exit the labour market than Italian citizens across all four categories. Not surprisingly, primary and secondary educated individuals show a lower likelihood to persist in the active state, compared to tertiary educated individuals. The geographical location is important: living in the North and Center increases the likelihood to remain active, compared to living in the South, and the effect is much stronger for males than females. Finally, among females in both age groups, living in a household with more than two people significantly decreases the probability to remain active on the labour market. We also find evidence of seasonality, which means in quarter III among both the 30-39 and 40-49 age cohorts, for males, an increased persistence in the active state, while for females, a significant decrease in the probability to remain active. \n\nRegarding the effect of the pandemic, the probability to remain active either did not change or increased before 2020; however, with the pandemic (year 2020) we estimate a significant decrease in the probability of remaining active among individuals in the  30-39 age cohort (both males and females). Specifically, the odds-ratio is 0.5, meaning that the probability of remaining active in 2020 is half the same probability in previous years.\nImportantly, for females in the 30-39 age cohort resident in the North and Center of Italy in a household with more than two people, the pandemic has significantly reduced the probability to be active on the labour market. The odds-ratio for this category of individuals is 0.6, implying that the probability of remaining active during the pandemic is almost half than it was before the shock.  This result confirms our previous finding that women with small children living in the North and Center were more likely to go out from the labour market during the pandemic. We argue that part of explanation can be found in the heavier caring responsibilities caused by the long-lasting school closure in the lockdown and post-lockdown periods in Italy. Among women in the South we do not observe any significant change, probably due to the already high inactivity rate among this category of individuals before the pandemic.\n\n \n\n\\section{Concluding remarks \\label{sec:concludingRemarks}}\n\t\nIn this paper, we  assess the short-term impact of the COVID-19 pandemic on the Italian labour market by studying how the shares  and transition probabilities of individuals between labour market states have changed after the country entered a full lockdown on March 10, 2020. \nWe find remarkable asymmetric effects across gender, age and geographical location, which we attribute to the very different state of the labour market different categories of individuals faced at the outburst of the pandemic. Specifically, we show evidence of an increased number of discouraged workers who exited the unemployment state and joined the NLFET state across gender, age groups and geographical location, but mostly among individuals in the South. Most worrying, we find evidence of substantial outflows of females in the North and Center of Italy in their 30s with small children  leaving employment, either permanent or temporary, and becoming inactive. To appreciate \nthe magnitude of the shock, in quarter III of 2020 females aged 30-39 (40-49) had a quarterly transition probability from temporary employment to NLFET of about 25\\% (19\\%) compared to the forecasted 10\\% (12\\%) in absence of the pandemic shock. The quarterly transition probability from permanent employment to NLFET jumped to 2.5\\% (1.3\\%) compared to a forecasted probability of 1.7\\% (0.5\\%) for women aged 30-39 (40-49). Although these numbers seem low, they correspond to approximately 40.000 women in the age 30-39 cohort and 25.000 women in the age 40-49 cohort moving from employment to inactivity in a quarter.\nSurprisingly, we do not find the same strong outflows of women in the South from employment to inactivity. We argue that the already high share of females in their 30s and 40s in NLFET could have played a role, i.e., the  women who are active on the labour market in Southern regions are strongly self-selected.\nThe outflow of women in the North and Center in their 30s with young children from employment to inactivity which started in the beginning of the pandemic persisted until the end of 2020, making likely the presence of long-lasting scarring effects on the Italian female labour force participation.\n\n\\newpage\n\t\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn just a few years since its inception, the BitTorrent protocol and similar systems have become the predominant P2P file sharing model~\\cite{bittorrent}. But the recent activities of those seeking to take down P2P infrastructures have forced the file sharing community to adapt to a hostile environment~\\cite{isps}. Operators of global BitTorrent trackers now take two notable measures in order to indemnify themselves from legal action: (1) the trackers are located in countries that are not party to international copyright treaties, and (2) access to trackers is controlled by private, invite-only communities with strict membership requirements~\\cite{bittorrent-communities}. The former allows operators to ignore legal threats to shutdown their services that a law-abiding ISP would normally have to comply with. But this approach can be both prohibitively expensive and difficult to arrange. Additionally, limiting access to only privileged users only temporarily protects a site that has been made private; it takes only a single seditious user to undermine the network and provide damaging evidence to the right parties.\n\nThe aforementioned measures may protect tracker operators but they provide little protection to the average file sharing user. This is because the fundamental principle of the BitTorrent protocol is that users download and upload data directly with other untrusted users, rather than download from a single, central source~\\cite{bittorrent}. Although some P2P clients employ communication encryption and protocol obfuscation enhancements, such measures do not protect a user from malicious clients that harvest file sharing activity information for future litigation. Furthermore, it has been shown that while it may not be possible to easily view encrypted packet contents, a third-party observer can still deduce that file sharing is occurring by identifying network pairs based on a tracker's public peer list~\\cite{sandvine,isps}.\n\nAnother limitation of current BitTorrent-like models is that the networks rely on altruistic users to keep files available for others. This is problematic in an environment where users want to limit their exposure to any traffic logging clients, and thus it is in their interest to disconnect immediately once they have the successfully downloaded their desired files. Content in these networks is unavailable once all of the peers that have the complete file depart. Newly arriving clients may be able to download and share some fraction of the data (if any is available), but they must wait and hope that a client returns to the network with the rest of file. Enhancements to private trackers, such as upload\/download ratios, provide incentives for clients to continue to seed files~\\cite{bittorrent-communities}, but these economic models are difficult to initiate and do little to maintain less popular older files.\n\nIn response to the lack of user anonymity and long-term data persistence in existing P2P systems, some users may seek an alternative. But because traditional data hosting solutions are not a viable option for sharing certain content that may have legal consequences, these users must use more questionable means for sharing data. Motivated by this, we developed the \\textit{Graffiti Network} distributed file sharing protocol that uses multiple third-party storage sites as a data replication and transfer medium between clients. The Graffiti approach is to use publically available web sites to store multiple copies of shared content. We use the term \\textit{graffiti} for our work since we are storing data in a way that non-network participants may regard as unsightly or unwanted vandalism. Our approach presents several new security challenges over other existing P2P systems where clients transmit data directly with each other: (1) a newly arriving peer can still download files even if all other peers have long disconnected, (2) a peer does not need to know about the existence of other peers, and (3) a tracker does not need multiple peers to enforce tit-for-tat policies~\\cite{bittorrent}.\n\nThe layout of this paper is as follows. First, we provide an overview of the Graffiti Network file sharing model. We then discuss our experimental prototype of the Graffiti Network model that is integrated with a BitTorrent system. The results from our one-year study on the efficacy of our prototype in a real-world deployment show that the use of public storage sites in a file sharing system is possible. We then conclude with a discussion about how both administrators and software developers can guard against such a threat.\n\n\\section{Related Work}\n\\label{sec:related}\nWe motivate our work by first discussing the related background research and literature.\n\n\\subsection{BitTorrent}\n\\label{sec:related-bittorrent}\nThe BitTorrent protocol defines the operations of a P2P network that facilitates the efficient sharing of files in a distributed manner~\\cite{bittorrent}. Our model inherits many of the features of BitTorrent, but employs third-party storage sites as an intermediary for data transfers, rather than allowing clients to directly download files from each other. This indirection makes it difficult to discover the identities of users that are participating in a Graffiti Network.\n\nThe overall efficiency and throughput of BitTorrent systems has been shown to scale gracefully to accommodate many users arriving at the same time to download new and popular files~\\cite{bittorrent-modeling}. But while the model works well in the short term, it does not ensure the long term availability of esoteric content or files that become less popular over time. This problem is especially prevalent for content that is released in ``episodes'': new content is shared profusely when it is released, but the number of peers decreases as the file becomes older and newer episodes are released. In a five month study of BitTorrent network activity, it was shown that the average time that a client stays in the network to continue sharing a file after it has received the entire file set was only seven hours~\\cite{bittorrent-lifetime}. These results, however, are based on the sharing activity of copyright-free files, and therefore the clients do not have a vested interest in disconnecting immediately. In contrast, a study \\cite{bittorrent-measure2} explicitly focused on illegal file sharing activity showed that the departure rate of peers is much faster than previously assumed in \\cite{bittorrent-modeling}. The results in \\cite{bittorrent-measure1} show that the average availability of a torrent is less than nine days and that most swarms completely die out in only 13 days. Thus, without the incentives for sharing found in private communities~\\cite{bittorrent-communities}, most BitTorrent content becomes unavailable after just a short amount of time. To overcome the capricious nature of users, Graffiti Networks use storage sites that have the potential to always be available, and thus the shared files are still accessible after the initial interest in the content has subsided. With enough replication, enforced by a strict asynchronous tit-for-tat model, we believe that a Graffiti Network could provide clients with access to files months or years after it was first introduced to the Internet.\n\n\\subsection{Peer-to-Peer Storage Systems}\n\\label{sec:related-p2pstorage}\nMuch of the previous work on developing P2P storage systems that provide block storage across multiple nodes is based on distributed hash tables~\\cite{cfs,oceanstore,past}. These approaches have the same deficiencies as the BitTorrent model: peers download file blocks directly from other peers, thereby losing anonymity, and the systems do not provide mechanisms to provide long term availability for less popular files after peers disconnect from the network. Other systems are focused on providing anonymous and secure P2P data storage~\\cite{publius}. The POTSHARDS system provides secure long-term data storage when the content originator no longer exists using secret splitting and data reconstruction techniques to handle partial losses~\\cite{potshards}; their approach assumes multiple, semi-reliable storage backends that are willing to host a client's data. The Freenet anonymous storage system uses key-based routing to locate files stored on remote peers~\\cite{freenet}. As discussed in \\cite{cfs}, Freenet's anonymity limits both its reliability and performance: files are not associated with any predictable server, and thus unpopular content may disappear since no one is responsible for maintaining replicas.\n\n\\subsection{Steganographic Storage Systems}\n\\label{sec:related-stegstorage}\nAlthough the Graffiti Network model is not a pure steganographic-based storage system, it does share similar properties of this class of systems~\\cite{stegvault,mnemosyne}. The Mnemosyne storage service applies the steganography techniques from a local storage system \\cite{stegfs} to a distributed hash table~\\cite{mnemosyne}. The StegVault proposal uses secret sharing to build a secure P2P storage system on top of reliable multicast~\\cite{stegvault}. One key benefit of these systems is that users have plausible deniability of the existence of hidden data because it is concealed inside covering data~\\cite{steghide}.\n\n\\begin{figure*}[t]\n   \\centering\n   \\includegraphics[width=.75\\textwidth]{images\/graffiti-diagram.eps}\n   \\caption{For a given a fileset, the client communicates with the tracker in the following manner: \\textbf{(1)} the client sends the tracker the list of pieces it already has; \\textbf{(2)} the tracker responds a list of instructions on where the client should download a sub-piece and the location of where to upload a replica; \\textbf{(3)} after downloading the new sub-piece, the client then navigates the target storage site and uploads a new encrypted and encoded sub-piece payload; \\textbf{(4)} the storage site returns an HTML page and the client verifies that the upload was successful. This process repeats until the client has all the pieces of the fileset and has produced enough replicas for the tracker.}\n   \\label{fig:overview}\n\\end{figure*}\n\n\\subsection{Alternative Storage Sites}\n\\label{sec:related-storagesites}\nSince the Graffiti Network model relies on gaining access to and the circumvention of third-party storage sites to host content, we consider the alternative approach of using dedicated storage services that are explicitly designed for the storage and transfer of large files. The Amazon Simple Storage Service provides a well-defined API for writing arbitrary data files, but it currently charges for both the storage space an account uses as well as the network bandwidth used to transfer data~\\cite{amazon-s3}. The Gmail Filesystem enables Google email accounts to be used as a network storage medium, but adopting approach would require users to share account information~\\cite{gmail-fs}. The Usenet news service is another potential storage system, but servers often impose a message retention time and many ISPs have discontinued providing this service to customers for free.\n\nFree web-based file-hosting sites also do not provide the robustness that we seek in our file sharing model~\\cite{rapidshare}. One limitation of these sites is that large files are broken into separate downloads and users must wait for some time period before they are allowed to retrieve the next piece. Furthermore, the user must manually enter each segment URL into their browser and repeatedly pass human-validation tests~\\cite{captcha}. These free hosting sites are also under scrutiny because many of their users post illegal content, and thus the site operators streamline the removal process for files and the disclosure of offending users' information for copyright holders in order to quickly diffuse any legal action that may disrupt the hosting site's revenue stream. Despite this, it is possible to include file-hosting sites as just one of the many options available in a Graffiti Network deployment (see Section~\\ref{sec:sites}).\n\nLastly, another proposed solution is to create a highly-volatile storage site by sending data packets to unsuspecting network entities to leverage network latency as a type of durability~\\cite{juggling}. The idea is to continuously send data to targets that relay the same data back to the source, therefore two copies of the data are always theoretically available. This approach is not practical for the Graffiti Network model because it does not allow the data to be shared amongst multiple peers. Furthermore, it requires that the original data source remain online in order to keep cycling the packets back out over the wire.\n\n\\section{Graffiti Network Model}\n\\label{sec:model}\nWe now describe how a file-sharing system based on the Graffiti Network model would operate. We discuss various measures and techniques that ensure the system is stable, usable, and scalable. Such qualities are necessary to facilitate wide-spread adoption by file-sharing participants, thereby making the threat a real possibility.\n\nTo describe the Graffiti model, we adopt the terminology of the BitTorrent protocol~\\cite{bittorrent}. We define a \\textit{fileset} as a set of one or more files that peers wish to share. The fileset's data is divided into multiple fixed-length \\textit{pieces} of $n$ bytes (the last piece can contain less than $n$ bytes) and are numbered sequentially. Each piece is divided further into fixed-length \\textit{sub-pieces}. A Graffiti Network that is deployed to distribute these pieces is comprised of three distinct components: (1) a \\textit{tracker} coordinates the replication and sharing procedures of a fileset, (2) a \\textit{client} downloads and replicates the fileset data managed by the tracker, and (3) third-party storage \\textit{sites} store and provide access to fileset data for peers. Any client that wishes to download and reconstruct the original fileset is required by the tracker to produce multiple sub-piece \\textit{replicas} on as many storage sites as possible.\n\nA high-level overview of the Graffiti Network protocol is shown in Figure~\\ref{fig:overview}. To connect to the Graffiti Network, the client first announces itself to the tracker and provides it with a list of all the pieces that the peer has already downloaded. The tracker responds with a series of sub-piece \\textit{request pairs} for a new piece that the client is missing. Each request pair consists of (1) a download location where the peer can retrieve a sub-piece and (2) instructions to produce a new replica on a different storage site for the data it just downloaded. Graffiti trackers follow a strict tit-for-tat protocol: for each sub-piece that a peer downloads, that peer is required to generate a replica for a previously downloaded sub-piece on a different storage site and send the location of this new replica back to the tracker before it can receive the next piece.\n\n\\subsection{Central Tracker}\n\\label{sec:tracker}\nThe tracker provides a directory service for peers to retrieve a fileset. For each piece of data in a fileset, the tracker maintains a table of the sub-piece replica locations on sites that were generated by clients. Each replica is annotated with three pieces of meta-data: (1) a unique encryption key for that replica, (2) a checksum for each sub-piece, and (3) the first and last byte sequences of the encrypted data block on the storage site. The tracker uses a different encryption key per entry to ensure that each replica is stored as a unique character sequence to prevent the use of tools to discover other replicas. The checksum and sequence markers also allow peers to determine whether a replica has the proper byte sequence and to locate data boundaries at the storage site location.\n\nFor each connected peer, the tracker maintains an \\textit{active piece set} (APS) of download\/upload replica pairs that are unfulfilled requests for a client. Each pair consists of a sub-piece identifier that the tracker provided for client to download and a storage site location where the tracker instructed the client to make a new replica. Once the client provides the tracker with information about a new replica for a download\/upload pair, the entry is removed from that client's APS and the client is allowed to receive new information. The size of the APS is determined by the tracker's administrator and prevents a client for downloading too many sub-pieces without producing any new replicas. As in the BitTorrent protocol, the Graffiti tracker strives for uniform availability of all data pieces~\\cite{bittorrent}. Since the tracker decrees what pieces the clients must replicate for each request in the APS, it can decide to replicate the ``rarest'' pieces first.\n\nMalicious clients in Graffiti Networks are quite different than malicious clients in BitTorrent networks~\\cite{bitthief}. A rogue Graffiti client may have other ulterior goals: (1) to discover all of the storage site locations used by a tracker in order to contact site administrators and have the replica data removed or (2) to falsely identify valid storage sites and replica locations as invalid in an attempt to disrupt operations. In the first case of trying to discover all of a fileset's replicas, the tracker can use throttling measures to prevent a client from learning too much in a short amount of time. But for the latter problem, the tracker should not actively check whether a client actually uploaded the data at the location it claims it did, due to security and economic reasons. Instead it can employ proxies or other third-party entities to determine whether a client is behaving properly. For example, the tracker can retrieve a page through the Coral Cache or Tor services to determine if the data was stored at the location claimed by a client~\\cite{freedman04,dingledine04}.\n\n\n\n\\subsection{Client}\n\\label{sec:client}\nA Graffiti client allows a user to automatically download a fileset stored on one or more storage sites. A user must first obtain a metadata file for a specific fileset uniquely identified by an ``info hash'' in order to begin downloading~\\cite{bittorrent}.\n\nAfter the client first announces itself to the tracker at the address listed in the metadata file, the tracker places the peer in an ``initialization'' mode. This is always done regardless of whether the client is connecting for the first time or if it is returning with some pieces already downloaded. The tracker sends every new client the same \\textit{initial piece set} (IPS) that will use for the first phase of downloading and replication. This initial set is the same for all clients arriving within a certain time period to prevent a client from initiating multiple new connections without ever creating new replicas. The size of the initial set is the same size as the APS and its information is changed to a different random set of sub-pieces at regular intervals (e.g., hours or days, rather than minutes). Thus, it is possible for a rogue client to retrieve a complete fileset without ever producing a new replica for the network, but it would take several days or weeks to cycle through all of the tracker's IPS combinations if there were a significantly large number of pieces. The client is required to also produce two new replicas for each sub-piece in the IPS, even if the client has already downloaded the pieces previously. This policy is akin to a new tenant paying ``last month's rent'' before moving into an apartment: it ensures that client cannot disconnect from the network without creating new replicas for each piece that it downloads.\n\nOnce the client successfully downloads and generates sufficient replicas for its IPS, it leaves the initialization phase and is then allowed to receive arbitrary pieces. The protocol works the same before: the tracker maintains an APS for each client and only gives new download locations once that particular client has produced a new replica on a storage site.\n\n\\subsection{Storage Sites}\n\\label{sec:sites}\nA potential Graffiti storage site is any accessible network entity that allows for data to be stored and retrieved using a known network protocol. In practice, peers will likely use publically available web sites that provide services that Graffiti clients repurpose to store arbitrary blocks of data. This approach has the distinction that all data movement appears as normal HTTP traffic, and thus is immune to current ISP throttling and tracking techniques~\\cite{isps}.\n\nThe ideal storage site for a Graffiti Network is one that allows for anyone to post data without CAPTCHA protections~\\cite{captcha} and is either unmoderated or has long abandoned by its owner. A popular and high-traffic wiki site, for example, would not be a good storage site candidate as it likely that non-malicious visitors would quickly notice the changes made by Graffiti clients to store replica data. With the rise of many open-source web-publishing platforms, there are many potential targets that allow for anonymous or semi-anonymous data posting. Notable examples include paste-bins, wiki sites, message boards, and blogs. An HTML-based storage site also allows the data to be disseminated to peers through disparate channels once it is online, such as through Coral Cache~\\cite{freedman04} or Tor~\\cite{dingledine04}. The data embedded in the site's pages could also be picked up by search engine caching and archiving services for longer-term storage.\n\nOther potential storage sites include any photo and file hosting sites that allow for automated data uploading. In the case of the former, the data could also be hidden inside of image files using well-known techniques~\\cite{steganography,ramkumar01}. As the Internet evolves, new targets will emerge that can be incorporated into existing networks. The system could also allow clients to use storage sites that are password protected for writing data, but where an account is not required to read back the data. This obviates the need for a client to send the tracker account information, which could then be used improperly by other clients to tamper with or destroy the data.\n\nUsing involuntary web sites as storage dumps seems counterintuitive if the main goal of the network is data persistence and availability, since replicas are promptly removed when site administrators and moderators discover them. The Graffiti model overcomes this challenge and takes advantage of ``free network storage'' through a massive replication and obfuscation process. It is not trivial, however, to automatically store arbitrary data on random web sites nor is it trivial to discover which sites are available with the properties stated above. The prevalence of popular web publishing software means that one only needs to target a small number of platforms in order to circumvent a large portion of the Internet. Furthermore, many sites, such as wikis and message boards, often display the network location of the user responsible for adding new content or making changes to their pages, which makes it difficult to deny responsibility for participating in illegal activities. We argue that by fracturing a fileset's replicas across hundreds of storage sites, it is difficult to be fully implicated when only a fraction of the evidence is available. A distributed effort to probe websites and uncover open storage paths could allow peers to draw on a nearly limitless pool of available storage.\n\n\\section{Experimental Deployment}\n\\label{sec:simulation}\nTo determine whether the Graffiti Network model is a viable and thus is a potential threat, we implemented a prototype Graffiti tracker and client as an extension to the BitTorrent protocol. We then stored a sample data set on a large number of open sites and measured the availability of our data for almost an entire year.\n\nWe built our system on top of the open-source libtorrent~\\cite{libtorrent} BitTorrent library in order to allow clients to participate in torrent swarms concurrently with Graffiti Network activities. When enough peers are available, the client operates strictly in BitTorrent mode. But if the number of distributed copies in the swarm drops below a threshold, the client begins to contact the tracker using the Graffiti protocol in conjunction with its BitTorrent operations. As new pieces are retrieved from storage sites, they are passed to libtorrent's storage manager for seeding to other peers.\n\n\\subsection{Storage Site Discovery}\nIn our experimental prototype, we target the open source MediaWiki~\\cite{mediawiki} platform as the potential storage site for the network. Due to the popularity of sites like Wikipedia that use MediaWiki, we believe that it is the most widely deployed wiki platform with a large number of less-experienced users that install the software without changing the permissive default settings. Another key characteristic is that the MediaWiki platform maintains a complete revision log for each article, which allows Graffiti peers to retrieve data even if the changes are reversed or the content is altered.\n\nWe decided to test our system on open MediaWiki sites that we do not have control over as this allows us to best measure whether our assumptions about how long the data will remain on the sites are correct. We developed a distributed web crawler to discover MediaWiki installations through search engines using keywords that are uniquely indicative of a newly installed site. The crawler purposely ignored well-known MediaWiki sites (e.g., those sites that are part of the Wikipedia Foundation) and the commercialized versions of MediaWiki (e.g., Wikia). For each site that the crawler found, we probed it to determine what kind of protection scheme it utilizes and the last time that it was updated (see Table~\\ref{tab:sites}). Of the 23,156 unique MediaWiki installations that we found, 8,483 sites allowed for anonymous editing and 5,983 allowed users to register accounts without CAPTCHA or email protections in order to make edits~\\cite{captcha}. The default MediaWiki installation provides a primitive arithmetic ``puzzle'' protection countermeasure that we found in use on 1,157 sites; this puzzle is easily broken with just a few lines of code, and thus did not prevent our system from storing data on these sites. Lastly, in order to minimize the impact of our experiments, we only targeted those sites that had not been updated within the last three months, thereby reducing our list to 5,646 sites; lowering the threshold to two months would have yielded a total of 11,987 potential storage sites.\n\n\n\\begin{table}[!t]\n   \\centering\n   \\begin{tabular}{lrr}\n      & Sites Found & Sites Used \\\\\n      Anonymous Edits & 8,483 & 3,161 \\\\\n      Registration Protected & 5,983 & 2,347 \\\\\n      Puzzle Protected & 1,157 & 138 \\\\\n      CAPTCHA Protected & 1,586 & - \\\\\n      Not Publicly Modifiable & 5,946 & - \\\\\n      \\hline\n      \\textbf{Total:} & \\textbf{23,156} & \\textbf{5,646} \\\\\n   \\end{tabular}\n   \\caption{The categories of protection used by the MediaWiki sites discovered during the collection process and the sites used in the experimental deployment.}\n   \\label{tab:sites}\n\\end{table}\n\n\\begin{figure*}[!ht]\n   \\centering\n   \\begin{minipage}{3.2in}\n      \\centering\n      \\includegraphics[width=3.25in]{graphs\/visit_availability.eps}\n      \\caption{Percentage of total replicas removed over time categorized by the type of failure.}\n      \\label{fig:graph-visits}\n   \\end{minipage}\n   \\begin{minipage}{0.3in}\n      \\hspace*{0.3in}\n   \\end{minipage}\n   \\begin{minipage}{3.2in}\n      \\centering\n      \\includegraphics[width=3.2in]{graphs\/total_availability.eps}\n      \\caption{The availability of replicas categorized by its corresponding storage site's protection schemes.}\n      \\label{fig:graph-availability}\n   \\end{minipage}\n   \\vspace{-.15in}\n\\end{figure*}\n\nThe Graffiti client stores data on MediaWiki sites as base64-encoded, Blowfish-encrypted blocks of text that are written in a new article titled with a random word from the dictionary. A more resilient approach would be to modify a popular page on a given site, and then immediately reverse the changes and mark the revision as vandalism. This has two significant implications compared to writing data to a newly created article. Foremost is that removing this data completely from the page's history requires administrators to delete the entire page from the database and restore the latest revision by hand, thereby losing all the previous legitimate revisions. Second, such an attack is more likely to be overlooked by a site's operators since they may only care whether the changes were reversed. We deemed this technique too malevolent for the purpose of our experiments, and thus chose to not implement it.\n\nTo retrieve a sub-piece stored on one of these storage sites, the client downloads the web page and extracts the text surrounded by the byte sequence markers provided by the tracker. The client then reverses the base64 encoding, decrypts the data, and verifies that it matches the checksum provided by the tracker.\n\n\\subsection{System Configuration}\nFor our experimental deployment, we used a Linux ISO split into 512KB pieces and 64KB sub-pieces as our sample data file that the clients want to share. Even though we were able to store up to 512KB payloads on a single MediaWiki page, we choose to use a smaller sub-piece size. Again, another more malicious approach would be to store a payload with the size that can be uploaded and retrieved but causes either a browser or the server to choke if the operator tries to access the page through the MediaWiki administrative interface. For example, we found that it was possible to store 512KB pieces that would exhaust the default 20MB memory limit of PHP if someone tried to remove the data. Thus, the only way to remove the content is to execute the proper SQL commands directly in the database, which is likely too difficult for most users.\n\nWe initiated file sharing activity on April 10th, 2009 using a tracker and five clients deployed in our departmental lab. Each client connects to the tracker and produces a full copy of a sub-piece on one of the 5,600+ MediaWiki sites. We assume that all clients are truthful about whether a replica is available and do not falsify replica URLs. We instrumented the tracker to target each storage site only once (although variations in sub-domains and URL rewriting led to some sites being used more than once).\n\nAlong with the data payload, at the top of each wiki page we stored a small paragraph with an explanation of the seemingly random text. This description also included a unique tracking link back to our web page with further information about the project. Tracking users' click-throughs from these links allows us to measure to some extent whether humans were actually discovering our payload pages before they were deleted.\n\nOnce the clients pushed out all of the data to the sites, we then used a separate tool to check daily whether the data we stored is still in place and has not been modified. We check every replica regardless if it has not been available for some time to ensure that the errors are not transient.\n\n\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=3.2in]{graphs\/domain_availability.eps}\n   \\caption{The cumulative availability of replicas categorized by their domain type: .com (42.5\\%), .edu (3.2\\%), .org (24.1\\%), US-based other (14.0\\%), and Non-US-based other (16.1\\%). }\n   \\label{fig:graph-domain}\n\\end{figure}\n\n\n\\subsection{Results}\n\\label{sec:experiments-results}\nWe now report on the availability of the 5,646 replicas that we stored in our experiments from April 10th to February 28th, 2010. For each missing replica, we categorize the replica as (1) \\textit{removed} if the site is available but the original page is missing, (2) \\textit{changed} if both the site and the original page are available, but the data does not match our stored checksum, or (3) \\textit{not found} if the site is no longer available (e.g., the domain name has expired or MediaWiki was uninstalled). Our investigation found that the missing replicas were only either removed or not found; no replica had its contents altered.\n\nOn the last day of our data collection, roughly 40\\% of the replicas were still available and hosting the original data that the prototype clients uploaded. The graph in Figure~\\ref{fig:graph-visits} shows a timeline of the percentage of replicas that are not available on each day that we checked. The first notable data point is that an initial 20\\% of the replicas were removed within the same week that they were created. The rate in which sites are removed then tapers off as time progresses. We attribute this drop-off in activity to two possible reasons. Foremost is that by default any changes to a MediaWiki site will appear on the first page of revision logs for seven days after the revision is created, and thus our actions are more likely to be discovered soon after the data is posted. The second possible reason is because a story about our project appeared on the front page of a popular technology news website on the third day of our experiment~\\cite{slashdot}. We believe that the ``notoriety'' of the project during this period may have caused administrators to examine their websites to see if they were targeted by our system. Once this initial attention diminished, the slopes of the lines in Figure~\\ref{fig:graph-visits} decrease and it takes another 35 days before another 10\\% of the replicas are removed. After about 100 days, the growth rate of replicas being removed (i.e., the lower portion of the curve in Figure~\\ref{fig:graph-visits}) tapers off and the number of sites that become unavailable begins to rise. This is expected since many of the sites were not actively used by their proprietor, and thus are taken down arbitrarily.\n\nThe graph in Figure~\\ref{fig:graph-availability} shows how the replicas were removed over time in relation to their storage site's protection scheme. The salient aspect of the result is that initially sites that employed some type of protection were faster to remove replicas. This is expected, since many of the sites that employed some protection were still being used by users despite having not been updated recently, whereas many of the completely open sites still displayed the default MediaWiki homepage message and thus were never even used once they were installed. Such sites are likely long forgotten by their owners who may never discover the replicas once they pass the default seven day revision log window. But after approximately 120 days, the percentage of missing replicas stored on sites allowing for anonymous edits surpasses sites using the basic registration protection.\n\nLastly, the graph in Figure~\\ref{fig:graph-domain} charts the availability of replicas with respect to the domain name of the storage site. We attribute greater durability of data stored on .edu and .org sites compared to other domains; such organizations are likely to use open-source software for collaboration and internal sites are often not behind corporate firewalls.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe results presented in the previous section clearly demonstrate the efficacy of the Graffiti Network model as a means for facilitating longer-term file sharing. We therefore argue that the threat of such a system does indeed exist and sites need to take measures to protect themselves from being used in such a manner that we have describe.\n\n\\subsection{Countermeasures}\nMuch of the feedback that we received on the project was from administrators that expressed their desire to provide an open wiki site that allowed anonymous contributions, despite the inevitable exposure to vandalism and spam. We counter that such sites that do not want to require users to register an account should still use CAPTCHA protections, such as before a user is allowed to edit a page. In practice, we found that the \\mbox{reCAPTCHA}~\\cite{vonahn08} project is the most effective protection as it does not require administrators to install special server-side graphics libraries and strikes a proper balance between availability and complexity. More complex CAPTCHA schemes would not deter future Graffiti clients that are able to solve CAPTCHAs (either manually or programmatically) and may only inhibit legitimate visually impaired users. If sites wish to still remain open, the CAPTCHA could be selectively enabled only when an unverified user tries to post data larger than some low default threshold or creates too many new pages in a short time span.\n\nWe also believe that other simple protection measures could be included in popular web applications to prevent abandoned or forgotten sites from being used for unintended purposes. For example, MediaWiki's default behavior could be to lock down the editing features of a site after a certain number of days if it was installed but then never actually used. This approach is similar to the one used by some blogging platforms to disable comments on older posts. Administrators could easily re-enable this functionality by simply logging into the site again. Another technique is to use a page counter that is invoked on the client-side (e.g., through JavaScript) and then compare the results with server-side logs to determine whether there are an unusually large number of users accessing pages through a non-browser client. Web application frameworks, such as Ruby on Rails and Django, could also provide similar features to protect custom-made sites.\n\n\\subsection{Variations \\& Adaptations}\nOther than for P2P activities, the Graffiti model is also of potential use for large-scale distributed systems used by criminal organizations, often referred to as \\textit{botnets}. The goal of most botnet operators is to gain access to a large supply of computational resources for purposes of network communication (e.g., sending emails or DOS attacks). If these goals shift towards more data-centric activities, then systems based on some of the principals of the Graffiti Network model may become prevalent in order to store large amounts of data for the botnet. Alternatively, instead of storing replicated data, the commandeered storage sites could also be used as a control channel for other entities in the botnet.\n\n\n\n\\section{Acknowledgments}\nThe authors would like to thank Arvid Norberg at BitTorrent, Inc. for his assistance with the libtorrent library~\\cite{libtorrent}.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have presented an overview of Graffiti Networks, a new file sharing model that allows peers to subversively use third-party storage sites as an intermediary for transferring files between users. Our client-tracker paradigm is similar to the BitTorrent protocol, but is designed to provide long term file availability to users while preserving their anonymity. We do not intend the Graffiti model to supplant BitTorrent networks, as it will never achieve the same maximum network throughput nor will it ever be as efficient. We believe, however, that our approach can have a symbiotic relationship with existing deployments: peers would use a Graffiti Network-like system to improve the long term availability of shared files, while leveraging the faster initial transfer rates of direct P2P communication for data dissemination. We have implemented a prototype and shown that data can be stored on publically accessible sites for extended periods of time, beyond what is often possible in other existing peer-to-peer systems. After almost an entire year, roughly 40\\% of the data that we stored on sites that are not under our control was still available. These results indicate that malicious users may adopt the Graffiti Network model, and thus site operators should take measures to prevent their sites from being used in this manner.\n\n\n\n\n{\\footnotesize\n \\linespread{0.85}\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}