diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjhgc" "b/data_all_eng_slimpj/shuffled/split2/finalzzjhgc"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjhgc"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nHolographic principle  relates boundary CFT to bulk theory of gravity, through the correspondence one can study the problems of strong coupling CFT on the boundary  through studying weak coupling gravity in the bulk. Remarkable progress has been made in applications of holographic principle in recent years, including applications of holography to study low energy QCD,  hydrodynamics, condensed matter theory\\cite{0501128,1101.2451,1103.3022,0803.3295,0810.1563}, etc.\n\nRecently, the combination study of holography and quantum information shed light on understanding of quantum gravity. In the initial work\\cite{1306.0533}, Maldacena and Susskind found any pair of entangled black holes are connected by some kind of Einstein-Rosen bridge, i.e., ER=EPR. However, the  ER=EPR duality does not tell how it difficult to transmit information through Einstein-Rosen bridge. Therefore, the concept complexity was introduced. Complexity is the minimal number of simple gates needed to prepare a target state from a reference state. Complexity was originally conjectured to be proportional to the maximum volume of codimension one surface bounded by the CFT slices, $\\mathcal{C}=\\frac{V}{Gl}$, which is called CV duality\\cite{CV,CV2,1711.10887,1803.06680,1807.06361,1803.08627,1808.08719,1808.10169}. The length scale $l$ is chosen according to situations. In order to eliminate the ambiguities in CV duality, CA duality was proposed\\cite{1509.07876,1512.04993}, which states that complexity is proportional to the action in Wheeler-DeWitt(WDW) patch, $\\mathcal{C}=\\frac{I}{\\pi\\hbar}$. CA duality does not involve any ambiguities in CV duality and preserves all the nice features of CV duality. CA duality have passed the tests of shock wave and tensor network.\n\n\nAccording to the definition, complexity growth rate is the speed of quantum computations. Considering black holes are  the densest memory\\cite{9310026,9409089,Bekenstein}, it is conjectured that black holes are the fastest computers in nature. There exist a bound for the speed of quantum computation. Inspired by Margolus-Levitin bound\\cite{9710043}\n\\begin{align}\n\\textmd{orthogonality}\\; \\textmd{time}\\geq \\frac{\\pi \\hbar}{2\\langle E\\rangle},\\nonumber\n\\end{align}\nwhich gives the minimal time needed for a state evolving to an orthogonal state, Lloyd proposed a bound on the speed of computation\\cite{Lloyd}. Brown and collaborators generalized Lloyd's bound and conjectured that there exists a bound on the growth rate of complexity, which is\n\\begin{align}\n\\frac{d\\mathcal{C}}{dt}\\leq \\frac{2E}{\\pi\\hbar}.\\label{complexitybound}\n\\end{align}\nCalculations show that static neutral black holes saturate the bound.\n\nIt is natural to study complexity growth of black holes in different gravity systems to examine CA duality and  Lloyd's bound.\nFor the progresses in this subject please refer to\\cite{1512.04993,1606.08307,1702.06766,1703.10006,1703.06297,1712.09826,1703.10468,1612.03627,1801.03638,1806.06216,1702.06796,1806.10312,1706.03788,1803.02795,1710.05686,1808.09917,1610.05090,1804.07410,1805.07262,\n1810.00758,1708.01779,1808.00067}.\nIn this paper, we intend to study complexity growth of BI black holes, since we are interested in the effects of nonlinearity of BI theory on complexity growth. The inner horizon of a BI black hole may turn into a curvature singularity due to perturbatively unstability\\cite{1702.06766}, which implies a BI black hole may possess a single horizon\\cite{1311.7299,1712.08798,1804.10951}. It's interesting to study the differences in complexity between BI black holes with single horizon and  AdS-Schwarzschild  black holes, although the casual structures of them are identical. Since the magnetic black holes have been studied rarely, in this paper we will pay much attention to the magnetic black holes, and make a comparison of the effects between electric and magnetic charge. As we will see in the following, action growth of magnetic BI black holes exhibit some specific properties that are not found in the electric ones. We  are also interested in studying action growth of BI black holes in massive gravity and study the effects of graviton mass.\n\nRecently, the authors of refs.\\cite{1901.00014,1905.06409,1905.07576} found that, action growth rates vanish for purely magnetic black holes in four dimensions. Which is unexpected since the expected late-time result $\\frac{dI}{dt}\\sim TS$ and electric-magnetic duality cannot be restored. Similar results were also found in the previous versions of this paper. In order to ameliorate the situation, a boundary term of matter field was proposed to be included to the action. In this paper, we add the boundary term proper to the gravity systems we considered and discuss the outcomes of the addition of the boundary term.\n\nThe paper is organized as, we study action growth of dyonic black holes in Einstein-Maxwell gravity in section \\ref{section2}, BI black holes in massive gravity in section \\ref{section3}, and EBI black holes in section \\ref{section4}. In all the gravity systems we considered, we add the boundary term of matter field and discuss the outcomes of the addition. We summarize our calculations in the last section.\n\n\n\\section{Dyonic black holes of Einstein-Maxwell gravity\\label{section2}}\nAs a comparison with the BI black holes in the next two sections, in this section we study action growth of dyonic black holes in Einstein-Maxwell gravity in general dimensions.\nThe action of Einstein-Maxwell theory reads\n\\begin{align}\nI=\\frac{1}{16\\pi}\\int d^dx\\sqrt{-g}\\left[R-2\\Lambda-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\\right].\\label{EinsteinMax}\n\\end{align}\nAfter taking variations of the metric and electromagnetic field, the field equations are given by\n\\begin{align}\nG_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}=\\frac{1}{2}F_{\\mu\\lambda}&F_{\\nu}^{\\;\\;\\lambda}-\\frac{1}{8}F_{\\alpha\\beta}F^{\\alpha\\beta}g_{\\mu\\nu},\\label{eommaxgrav}\\\\\n\\nabla_\\mu F^{\\mu\\nu}&=0.\\label{eommax}\n\\end{align}\nWe take the metric and field strength ansatz for AdS planar black holes in $d=2n+2$ dimensions as\n\\begin{align}\nds^2&=-f(r)dt^2+\\frac{dr^2}{f(r)}+r^2\\left(dx_1^2+dx_2^2+\\cdots+dx_{2n-1}^2+dx_{2n}^2\\right),\\nonumber\\\\\nF&=\\Phi'(r)dr\\wedge dt+p(dx_1\\wedge dx_2+\\cdots+dx_{2n-1}\\wedge dx_{2n}).\\label{strengthansatz}\n\\end{align}\nSolving equations of motion (e.o.m.) of the Maxwell field (\\ref{eommax}) we obtain\n\\begin{align}\n\\Phi(r)=\\int_r^\\infty \\frac{q dr}{r^{2n}}.\n\\end{align}\nNow we are able to solve the e.o.m. of metric (\\ref{eommaxgrav}), and obtain the dyonic black hole solution\n\\begin{align}\nf(r)=-\\frac{\\mu}{r^{d-3}}-\\frac{2\\Lambda}{(d-1)(d-2)}r^2+\\frac{q^2}{2(d-2)(d-3)r^{2(d-3)}}-\\frac{p^2}{4(d-5)r^2}.\\label{dyonicBHmax}\n\\end{align}\n\nAccording to CA duality, complexity is proportional to action in WDW patch. Since WDW patch is in general non-smooth, as shown by Fig.\\ref{fig1}, we employ the method proposed in \\cite{1609.00207,9403018} to calculate the action, which is given by\n\\begin{align}\nI_{tot}=\\int_{\\mathcal{V}}(R-2\\Lambda+\\mathcal{L}_{mat})\\sqrt{-g}dV+2\\Sigma_{T_i}\\int_{\\partial\\mathcal{V}_{T_i}}Kd\\Sigma+2\\Sigma_{S_i}sign(S_i)\\int_{\\partial\\mathcal{V}_{S_i}}\nKd\\Sigma\\nonumber\\\\\n+2\\Sigma_{N_i}sign(N_i)\\int_{\\partial\\mathcal{V}_{N_i}}\\kappa dSd\\lambda+2\\Sigma_{j_i}sign(j_i)\\oint \\eta_{j_i}dS+2\\Sigma_{m_i}sign(m_i)\\oint a_{m_i}dS.\\label{Itot}\n\\end{align}\nWhere $S_i$, $T_i$ and $N_i$ labels spacelike, timelike and null boundary respectively. $K$ is the Gibbons-Hawking term. $\\kappa$ measures the failure of $\\lambda$ to be an affine parameter on the null generators. $\\eta_{j_i}$ is the joint term between non-null hypersurfaces. $a_{m_i}$ is the joint term between null and other types of surfaces. The signatures $sign(N_i), sign(j_i), sign(m_i)$ are determined through the requirement that the gravitational action is additive.\n\n\n\nTo proceed the calculations, it's convenient to introduce  null coordinates\n\\begin{align}\ndu\\equiv dt+f^{-1}dr,\\;\\;\\;\\;\\;\\;dv\\equiv dt-f^{-1}dr.\\label{nullcoord}\n\\end{align}\nUnder the null coordinates the metric becomes\n\\begin{align}\nds^2&=-fdu^2+2dudr+r^2h_{ij}dx^idx^j,\n\\end{align}\nor\n\\begin{align}\nds^2&=-fdv^2-2dvdr+r^2h_{ij}dx^idx^j.\n\\end{align}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.40\\textwidth]{WdWone}\n\\includegraphics[width=.40\\textwidth]{WdWtwo}\n\\end{center}\n\\caption{Wheeler-DeWitt (WDW) patch of  black holes. The left panel represents WDW patch of black holes with single  horizon, the right panel represents WDW patch of black holes with double horizons.}\n\\label{fig1}\n\\end{figure}\n\n\nSince  the dyonic black holes (\\ref{dyonicBHmax})  possess both inner and outer horizons, we only need the right panel of Fig.\\ref{fig1} to calculate action growth.\nFrom the right panel of Fig.\\ref{fig1} we obtain the bulk contribution to $\\delta I$\n\\begin{align}\nI_{\\mathcal{V}_1}&=\\frac{1}{16\\pi}\\omega_2^n\\int_u^{u+\\delta t}du\\int_{r_{-}}^{r_{max}} dr r^{d-2}\\mathcal{L}_{bulk},\\nonumber\\\\\nI_{\\mathcal{V}_2}&=\\frac{1}{16\\pi}\\omega_2^n\\int_v^{v+\\delta t}dv\\int_{r_{+}}^{r_{max}} dr r^{d-2}\\mathcal{L}_{bulk},\n\\end{align}\nwhere $r_{max}$ is the UV cutoff, and $\\omega_2\\equiv\\int dx_1dx_2=\\cdots=\\int dx_{2n-1}dx_{2n}$. With the e.o.m. (\\ref{eommaxgrav}), we have\n\\begin{align}\n\\mathcal{L}_{bulk}\n=\\frac{4\\Lambda}{d-2}+\\frac{q^2}{(d-2)r^{2(d-2)}}-\\frac{p^2}{2r^4}.\n\\end{align}\nTherefore, the bulk contribution to total action growth is\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}&=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\bigg[\\frac{4\\Lambda r_{+}^{d-1}}{(d-1)(d-2)}-\\frac{p^2r_{+}^{d-5}}{2(d-5)}\n-\\frac{q^2}{(d-2)(d-3)r_{+}^{d-3}}\\bigg]\\bigg|_{r_{-}}^{r_{+}}.\n\\end{align}\nAs shown by the right panel of Fig.\\ref{fig1}, there are four joints between null surfaces contributing to $\\delta I$. Actions of the joints are given by\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}&=\\frac{1}{16\\pi}\\left[2\\oint_{\\mathcal{B}'}adS-2\\oint_{\\mathcal{B}}adS\\right]\\nonumber\\\\\n&=\\frac{1}{8\\pi}\\omega_2^n\\left[h(r_{\\mathcal{B}'})-h(r_{\\mathcal{B}})\\right],\n\\end{align}\nwhere $a=\\ln\\left(-\\frac{1}{2}k\\cdot\\bar{k}\\right)$, with $k$ being the null normal to the hypersurface  $v=const$ and  $\\bar{k}$ being the null normal to the hypersurface $u=const$. For the affinely parametrized expressions $k_\\alpha=-c\\partial_\\alpha v$ and $\\bar{k}_\\alpha=\\bar{c}\\partial_\\alpha u$, we have $a=-\\ln\\left(-f\/(c\\bar{c})\\right)$, therefore $h(r)=-r^{d-2}\\ln\\left(-f\/(c\\bar{c})\\right)$. Using $dr=-\\frac{1}{2}f\\delta t$, we obtain\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[r^{d-2}f'+(d-2)r^{d-3}f\\ln\\left(\\frac{-f}{c\\bar{c}}\\right)\\right]\\bigg|_{r=r_{\\mathcal{B}}}.\n\\end{align}\nAt late times, $r_{\\mathcal{B}}\\rightarrow r_{+}$, $r_{\\mathcal{C}}\\rightarrow r_{-}$, we have\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}=\\frac{1}{16\\pi}\\omega_2^nr^{d-2}f'(r)\\big|_{r_{+}}\\delta t,\\;\\;\\;\\;I_{\\mathcal{C}'\\mathcal{C}}=-\\frac{1}{16\\pi}\\omega_2^nr^{d-2}f'(r)\\big|_{r_{-}}\\delta t.\\label{4joints}\n\\end{align}\nThe two terms in (\\ref{4joints}) together give rise to\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}+I_{\\mathcal{C}'\\mathcal{C}}=&\\frac{1}{16\\pi}\\omega_2^n\\delta t\\bigg[-\\frac{4\\Lambda r_{+}^{d-1}}{(d-1)(d-2)}+\\frac{p^2r_{+}^{d-5}}{2(d-5)}\n-\\frac{q^2}{(d-2)r_{+}^{d-3}}\\bigg]\\bigg|_{r_{-}}^{r_{+}}.\n\\end{align}\n\n\nThe total action growth is obtained by sum of all the contributions\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=\\frac{1}{16\\pi}\\left[-\\frac{q^2}{(d-3)r_{+}^{d-3}}+\\frac{q^2}{(d-3)r_{-}^{d-3}}\\right]\\nonumber\\\\\n&=\\left[(M-Q_e\\Phi_e)_{+}-(M-Q_e\\Phi_e)_{-}\\right].\\label{IgdyonicMax}\n\\end{align}\nWhere $Q_e=\\frac{q}{16\\pi}, \\Phi_e=\\frac{q}{(d-3)r_{\\pm}^{d-3}}$ are electric charge and potential respectively.  It can be seen that, similar to the four-dimensional case\\cite{1901.00014}, magnetic charge does not contribute to action growth, which implies action growth rates vanish for purely magnetically charged black holes. In order to restore the late-time result $\\frac{dI}{dt}\\sim TS$ for purely magnetic black holes and electric-magnetic duality in four dimensions, the action of Einstein-Maxwell theory may be modified by a boundary term of Maxwell field.\n\nThe Maxwell boundary term that considered to be included to the action reads\\cite{1901.00014}\n\\begin{align}\nI_{\\mu Q}&=\\frac{\\gamma}{16\\pi}\\int_{\\partial\\mathcal{M}} d\\Sigma_\\mu F^{\\mu\\nu}A_\\nu.\\label{boundaryMax}\n\\end{align}\nThis term does not affect the field equations but only alter the boundary conditions in the variational principle of Maxwell field. A well-posed variational principle requires Dirichlet boundary condition $\\delta A_a=0$ for the original Maxwell action in (\\ref{EinsteinMax}), while after adding the boundary term (\\ref{boundaryMax}), it requires Neumann boundary condition $n^\\mu\\partial_\\mu\\delta A_a=0$ for $\\gamma=1$ and mixed boundary conditions for  general $\\gamma$. Addition of the boundary term (\\ref{boundaryMax})  is natural when studying thermodynamics or Euclidean action, it produces the Legendre transformation from a grand canonical ensemble with fixed chemical potential to a canonical one with fixed charge. Using the field equations $\\nabla_\\mu F^{\\mu\\nu}=0$ and Stokes' theorem the boundary term (\\ref{boundaryMax}) can be rewritten as\n\\begin{align}\nI_{\\mu Q}=\\frac{\\gamma}{32\\pi}\\int_{\\mathcal{M}} d^dx\\sqrt{-g}F^{\\mu\\nu}F_{\\mu\\nu}.\n\\end{align}\nAction growth for the dyonic black holes now becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=\\left[M-(1-\\gamma)Q_e\\Phi_e-\\gamma Q_m\\Phi_m\\right]_{+}-\\left[M-(1-\\gamma)Q_e\\Phi_e-\\gamma Q_m\\Phi_m\\right]_{-}.\\label{dyonMaxadd}\n\\end{align}\nThis result takes the identical form with the four-dimensional one\\cite{1901.00014}. One sees that, if we take $\\gamma=\\frac{1}{2}$, then electric and magnetic charges contribute to action growth on equal footing, this agrees with electric-magnetic duality in four dimensions. If we take $\\gamma=1$, contrary to the $\\gamma=0$ case, only magnetic charge contributes to action growth, i.e., action growth rates vanish at late times for purely electrically charged black holes in this case.\n\n\\section{BI black holes in massive gravity\\label{section3}}\nIn this section, we study action growth of purely electrically charged BI black holes in massive gravity and discuss the effects of graviton mass.\nThe action of Einstein massive gravity is given by\\cite{1508.01311}\n\\begin{align}\nI=\\frac{1}{16\\pi}\\int d^dx\\sqrt{-g}\\left[R-2\\Lambda+\\mathcal{L}(\\mathcal{F})+m^2\\sum_{i=1}^4c_i\\mathcal{U}_i(g,f)\\right],\n\\end{align}\nwhere $f$ is a fixed symmetric rank-2 tensor. $c_i$ are constants and $\\mathcal{U}_i$ are symmetric polynomials of the eigenvalues of matrix $\\mathcal{K}^\\mu_{\\;\\nu}\\equiv\\sqrt{g^{\\mu\\alpha}f_{\\alpha\\nu}}$\n\\begin{align}\n&\\mathcal{U}_1=[\\mathcal{K}],\\;\\;\\;\\mathcal{U}_2=[\\mathcal{K}]^2-[\\mathcal{K}^2],\\;\\;\\;\\mathcal{U}_3=[\\mathcal{K}]^3-3[\\mathcal{K}][\\mathcal{K}^2]+2[\\mathcal{K}^3],\\nonumber\\\\\n&\\mathcal{U}_4=[\\mathcal{K}]^4-6[\\mathcal{K}^2][\\mathcal{K}]^2+8[\\mathcal{K}^3][\\mathcal{K}]+3[\\mathcal{K}^2]^2-6[\\mathcal{K}^4].\n\\end{align}\n$\\mathcal{L}(\\mathcal{F})$ is the Lagrangian density of BI  theory\n\\begin{align}\n\\mathcal{L}(\\mathcal{F})=4\\beta^2\\left(1-\\sqrt{1+\\frac{F^{\\rho\\sigma}F_{\\rho\\sigma}}{2\\beta^2}}\\right).\\label{LFm}\n\\end{align}\nTaking variation of the metric and electromagnetic field one obtains the field equations\n\\begin{align}\nG_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}-&\\frac{1}{2}g_{\\mu\\nu}\\mathcal{L}(\\mathcal{F})-\\frac{2F_{\\mu\\lambda}F_\\nu^{\\;\\lambda}}{\\sqrt{1+\\frac{F^{\\rho\\sigma}F_{\\rho\\sigma}}{2\\beta^2}}}+m^2\\chi_{\\mu\\nu}=0,\\label{eommg}\\\\\n&\\partial_\\mu\\left(\\frac{\\sqrt{-g}F^{\\mu\\nu}}{\\sqrt{1+\\frac{F^{\\rho\\sigma}F_{\\rho\\sigma}}{2\\beta^2}}}\\right)=0,\\label{eomma}\n\\end{align}\nwhere\n\\begin{align}\n\\chi_{\\mu\\nu}=&-\\frac{c_1}{2}\\left(\\mathcal{U}_1g_{\\mu\\nu}-\\mathcal{K}_{\\mu\\nu}\\right)-\\frac{c_2}{2}\\left(\\mathcal{U}_2g_{\\mu\\nu}-2\\mathcal{U}_1\\mathcal{K}_{\\mu\\nu}\n+2\\mathcal{K}^2_{\\mu\\nu}\\right)-\\frac{c_3}{2}\\big(\\mathcal{U}_3g_{\\mu\\nu}-3\\mathcal{U}_2\\mathcal{K}_{\\mu\\nu}\\nonumber\\\\\n&6\\mathcal{U}_1\\mathcal{K}^2_{\\mu\\nu}-6\\mathcal{K}^3_{\\mu\\nu}\\big)-\\frac{c_4}{2}\\left(\\mathcal{U}_4g_{\\mu\\nu}-4\\mathcal{U}_3\\mathcal{K}_{\\mu\\nu}+12\\mathcal{U}_2\\mathcal{K}^2_{\\mu\\nu}\n-24\\mathcal{U}_1\\mathcal{K}^3_{\\mu\\nu}+24\\mathcal{K}^4_{\\mu\\nu}\\right).\n\\end{align}\n\n\nWe take the static metric ansatz\n\\begin{align}\nds^2=-f(r)dt^2+\\frac{dr^2}{f(r)}+r^2h_{ij}dx^idx^j,\n\\end{align}\nwhere $h_{ij}dx^idx^j$ is the line element of codimension-two hypersurface with constant curvature.\nUsing the reference metric\n\\begin{align}\nf_{\\mu\\nu}=diag(0,0,h_{ij}),\n\\end{align}\nthe $\\mathcal{U}_i$'s can be expressed as\n\\begin{align}\n\\mathcal{U}_1=\\frac{d_2}{r},\\;\\;\\;\\;\\;\\;\n\\mathcal{U}_2=\\frac{d_2d_3}{r^2},\\;\\;\\;\\;\\;\\;\n\\mathcal{U}_3=\\frac{d_2d_3d_4}{r^3}, \\;\\;\\;\\;\\;\\;\n\\mathcal{U}_4=\\frac{d_2d_3d_4d_5}{r^4},\n\\end{align}\nwhere $d_i\\equiv d-i$ is introduced for convenience. Under the assumption of electrostatic potential $A_\\mu=\\Phi(r)\\delta_\\mu^0$,\nwe solve the e.o.m of $A_\\mu$ and find the only non-vanishing component of strength tensor is\n\\begin{align}\nF_{tr}=\\frac{\\sqrt{d_2d_3}q}{r^{d_2}\\sqrt{1+\\frac{d_2d_3q^2}{\\beta^2r^{2d_2}}}}.\n\\end{align}\nSubstituting the above results into (\\ref{eommg}), one obtains the black hole solution\n\\begin{align}\nf(r)=&k-\\frac{m_0}{r^{d_3}}+\\frac{4\\beta^2-2\\Lambda}{d_1d_2}r^2-\\frac{4\\beta^2r^2}{d_1d_2}\\sqrt{1+\\Gamma}+\\frac{4d_2q^2}{d_1r^{2d_3}}\\mathcal{H}\\nonumber\\\\\n&+m^2\\left(\\frac{c_1r}{d_2}+c_2+\\frac{d_3c_3}{r}+\\frac{d_3d_4c_4}{r^2}\\right),\\label{massiveBH}\n\\end{align}\nwith\n\\begin{align}\n\\Gamma=\\frac{d_2d_3q^2}{\\beta^2r^{2d_2}},\\;\\;\\;\\;\\;\\;\\mathcal{H}=\\sideset{_2}{_1}{\\mathop{F}}\\left[\\frac{1}{2},\\frac{d_3}{2d_2},\\frac{3d-7}{2d_2},-\\Gamma\\right].\\label{gammaH}\n\\end{align}\n\nThe details of geometry and thermodynamics of the black hole (\\ref{massiveBH}) can be found in \\cite{1508.01311}. The black hole may possess single or double horizons.\nLet's first calculate action growth of the single-horizoned black hole. The left panel of Fig.\\ref{fig1} shows us the Penrose diagram of this type of black holes. From the panel it is easy to see that, the $\\eta$ terms in (\\ref{Itot}) vanish because there are no joints between non-null hypersurfaces in WDW patch. It is natural to require all null segments to be affine parametrized, therefore all $\\kappa$ terms in (\\ref{Itot}) vanish too. Considering time transition symmetry, the left contributions to $\\delta I=I(t_0+\\delta t)-I(t_0)$ are\n\\begin{align}\n\\delta I=I_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}-2\\int_{\\mathcal{S}}Kd\\Sigma+2\\oint_{\\mathcal{B}'}adS-2\\oint_{\\mathcal{B}}adS\n\\end{align}\n\n\n\n\nWith the e.o.m (\\ref{eommg}), we obtain the expression of bulk Lagrangian\n\\begin{align}\n\\mathcal{L}_{bulk}=\\frac{4\\Lambda}{d_2}-\\frac{8\\beta^2}{d_2}\\left(1-\\sqrt{1+\\frac{d_2d_3q^2}{\\beta^2r^{2d_2}}}\\right)-\\frac{m^2}{d_2}\n\\left[\\frac{d_2c_1}{r}-\\frac{d_2d_3d_4c_3}{r^3}-2\\frac{d_2d_3d_4d_5c_4}{r^4}\\right].\n\\end{align}\nThus  action of the bulk region $\\mathcal{V}_1$  is given by\n\\begin{align}\nI_{\\mathcal{V}_1}&=\\frac{1}{16\\pi}\\Omega_{d_2}\\int_u^{u+\\delta t}du\\int_\\epsilon^{\\rho(u)} dr r^{d_2}\\mathcal{L}_{bulk},\\nonumber\\\\\n&=\\frac{1}{16\\pi}\\Omega_{d_2}\\delta t\\left[\\frac{4\\Lambda}{d_1d_2}r^{d_1}-2F(r)-m^2\\left(\\frac{c_1}{d_2}r^{d_2}-c_3d_3r^{d_4}-2c_4d_3d_4r^{d_5}\\right)\\right]\\bigg|_\\epsilon^{\\rho(u)}.\n\\end{align}\nwhere $\\Omega_{d_2}$ is the volume of the codimension-two hypersurface, $r=\\rho(u)$ represents the $v=v_0+\\delta t$ surface, and $F(r)\\equiv \\int dr r^{d_2}\\frac{4\\beta^2}{d_2}\\left(1-\\sqrt{1+\\frac{d_2d_3q^2}{\\beta^2r^{2d_2}}}\\right)$ is introduced for convenience.\nSimilarly, action of the  bulk region $\\mathcal{V}_2$ is given by\n\\begin{align}\nI_{\\mathcal{V}_2}&=\\frac{1}{16\\pi}\\Omega_{d_2}\\int_v^{v+\\delta t}dv\\int_{\\rho_1(v)}^{\\rho_0(v)} dr r^{d_2}\\mathcal{L}_{bulk},\\nonumber\\\\\n&=\\frac{1}{16\\pi}\\Omega_{d_2}\\delta t\\left[\\frac{4\\Lambda}{d_1d_2}r^{d_1}-2F(r)-m^2\\left(\\frac{c_1}{d_2}r^{d_2}-c_3d_3r^{d_4}-2c_4d_3d_4r^{d_5}\\right)\\right]\\bigg|_{\\rho_1(v)}^{\\rho_0(v)},\n\\end{align}\nwhere $r=\\rho_{0(1)}(v)$ represents the $u=u_{0(1)}$ surface. In the late-time limit, $\\rho_{1}(v)\\rightarrow r_{+}$, we have\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}=\\frac{1}{16\\pi}\\Omega_{d_2}\\delta t\\left[\\frac{4\\Lambda}{d_1d_2}r_{+}^{d_1}-2F(r_{+})+2F(0)-m^2\\left(\\frac{c_1}{d_2}r_{+}^{d_2}-c_3d_3r_{+}^{d_4}-2c_4d_3d_4r_{+}^{d_5}\\right)\\right],\\label{bulkm}\n\\end{align}\nwhere we have taken $\\epsilon\\rightarrow0$. Action of the spacelike surface $\\mathcal{S}$  is\n\\begin{align}\nI_{\\mathcal{S}}&=-\\frac{1}{8\\pi}\\int Kd\\Sigma\\nonumber\\\\\n&=\\frac{1}{16\\pi}\\Omega_{d_2}\\delta td_1\\left[m_0-F(0)\\right].\\label{Km}\n\\end{align}\nwhere we have used the expression $K=\\frac{-1}{r^{d_2}}\\frac{d}{dr}\\left(r^{d_2}\\sqrt{-f}\\right)$ for extensive curvature and taken the $r\\rightarrow0$ limit. At late times, $r_{\\mathcal{B}}\\rightarrow r_{+}$, the joint term is given by\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}\n=&\\frac{1}{16\\pi}\\Omega_{d_2}\\delta t\\bigg[d_3m_0-\\frac{4\\Lambda}{d_1d_2}r_{+}^{d_1}-d_3F(r_{+})+r_{+}F'(r_{+})\\nonumber\\\\\n&+m^2\\left(\\frac{c_1}{d_2}r_{+}^{d_2}-c_3d_3r_{+}^{d_4}-2c_4d_3d_4r_{+}^{d_5}\\right)\\bigg].\\label{jointm}\n\\end{align}\n\nCollecting our calculations, adding up all the contributions  to $\\delta   I$ (\\ref{bulkm}), (\\ref{Km}) and (\\ref{jointm}), we finally arrive at\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=2M-Q\\Phi-C,\\label{onemassive}\n\\end{align}\nwhere $M, Q, \\Phi$ are respectively mass, electric charge and potential of the black hole\n\\begin{align}\n&M=\\frac{d_2 m_0\\Omega_{d-2}}{16\\pi},\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\nQ=\\frac{\\sqrt{d_2d_3}q}{4\\pi},\\nonumber\\\\\n\\Phi&=\\sqrt{\\frac{d_2}{d_3}}\\frac{q}{r_{+}^{d_3}}\\sideset{_2}{_1}{\\mathop{F}}\\left[\\frac{1}{2},\\frac{d_3}{2d_2},\\frac{3d-7}{2d_2},-\\frac{d_2d_3q^2}{\\beta^2r_{+}^{2d_2}}\\right],\n\\end{align}\nand\n\\begin{align}\nC=\\frac{\\beta^2d_3}{8\\pi d_1d_2}\\left(d_2d_3\\right)^{\\frac{d_1}{2d_2}}\\left(\\frac{q}{\\beta}\\right)^{\\frac{d_1}{d_2}}\\frac{\\Gamma\\left(1\/(2d_2)\\right)\\Gamma\\left(d_3\/(2d_2)\\right)}{\\Gamma(1\/2)}\n\\end{align}\n\nNote that, action growth  of BI black holes in massive gravity (\\ref{onemassive}) takes the identical form with the one of EBI black holes\\cite{1702.06766}.  However, graviton mass affects action growth through back-reaction on the geometry. The $m\\rightarrow0$ limit of (\\ref{massiveBH}) leads to black holes of EBI gravity\n\\begin{align}\n\\hat{f}(r)=k-\\frac{m_0}{r^{d_3}}+\\frac{4\\beta^2-2\\Lambda}{d_1d_2}r^2-\\frac{4\\beta^2r^2}{d_1d_2}\\sqrt{1+\\Gamma}+\\frac{4d_2q^2}{d_1r^{2d_3}}\\mathcal{H}.\\label{EBIBH}\n\\end{align}\n When  mass and charge of the black holes are fixed, we have $f(r)-\\hat{f}(r)<0$ (given $c_i$ negative\\cite{1409.2369}) due to the graviton mass, which implies $r_{+}>\\hat{r}_{+}$,  therefore graviton mass leads BI black hole in massive gravity to complexitify faster than their Einstein gravity counterparts.\n\nIt is easy to note also that, when $q\\rightarrow0$, action growth rate (\\ref{onemassive}) reduces to the one of AdS-Schwarzschild black holes, for which Lloyd's bound is saturated. Although the causal structure of single-horizoned BI black holes is identical with the one of AdS-Schwarzschild black holes, the manners of action growth  differ between the two types of black holes due to the presence of electromagnetic field. Therefore, BI electromagnetic field slows down complexification of the black holes.\n\n\nFor black holes with inner and outer horizons, the causal structure of which is shown by the right panel of Fig.\\ref{fig1}. Bulk contribution to $\\delta I$ is given by\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}=\\frac{1}{16\\pi}\\Omega_{d_2}\\delta t\\left[\\frac{4\\Lambda}{d_1d_2}r^{d_1}-2F(r)-m^2\\left(\\frac{c_1}{d_2}r^{d_2}-c_3d_3r^{d_4}-2c_4d_3d_4r^{d_5}\\right)\\right]\\bigg|_{r_{-}}^{r_{+}},\\label{bulkm2}\n\\end{align}\nThe joint terms are\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}+I_{\\mathcal{C}'\\mathcal{C}}=&\\frac{1}{16\\pi}\\Omega_{d_2}\\delta t\\bigg[-\\frac{4\\Lambda}{d_1d_2}r^{d_1}-d_3F(r)+rF'(r)\\nonumber\\\\\n&+m^2\\left(\\frac{c_1}{d_2}r^{d_2}-c_3d_3r^{d_4}-2c_4d_3d_4r^{d_5}\\right)\\bigg]\\bigg|_{r_{-}}^{r_{+}}.\\label{jointm2}\n\\end{align}\nThe total variation of action is then given by the sum of eqs. (\\ref{bulkm2}) and (\\ref{jointm2}), which yields\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=\\left(M-Q\\Phi\\right)_{+}-\\left(M-Q\\Phi\\right)_{-}.\\label{twomassive}\n\\end{align}\nJust as the single horizon case, this result is formally identical with the one of EBI black holes. Similarly, we also have $f(r)-\\hat{f}(r)<0$ for fixed mass and charge parameters, which implies $r_{-}<\\hat{r}_{-}$ and $r_{+}>\\hat{r}_{+}$ ($\\hat{r}_{\\pm}$ are the inner and outer horizons of the EBI black hole (\\ref{EBIBH})), i.e., action growth rates of the double-horizoned BI black holes  in Einstein massive gravity are superior to the ones of the Einstein gravity counterparts too.\n\nNow, we add a boundary term of electromagnetic field to the total action\n\\begin{align}\nI_{\\mu Q}&=\\frac{\\gamma}{16\\pi}\\int d\\Sigma_\\mu\\frac{4F^{\\mu\\nu}}{\\sqrt{1+\\frac{F^2}{2\\beta^2}}}A_\\nu\\nonumber\\\\\n&=\\frac{\\gamma}{8\\pi}\\int d^dx\\sqrt{-g}\\frac{1}{\\sqrt{1+\\frac{F^2}{2\\beta^2}}}F^{\\mu\\nu}F_{\\mu\\nu},\n\\end{align}\nwhich does not alter the field equations. In the second equality we have used Stokes' theorem and the field equations $\\nabla_\\mu\\big(\\frac{F^{\\mu\\nu}}{\\sqrt{1+\\frac{F^2}{2\\beta^2}}}\\big)=0$. For BI black holes with single horizon, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=2M-(1-\\gamma)Q\\Phi-C_1,\\label{onemassiveadd}\n\\end{align}\nwith\n\\begin{align}\nC_1=\\frac{\\beta^2d_3^2}{8\\pi d_1}\\left(\\frac{d_2}{d_3}\\right)^{\\frac{d_1}{2d_2}}\\left(\\frac{q}{\\beta}\\right)^{\\frac{d_1}{d_2}}\\left(d_3^{\\frac{1}{d_2}}+\\gamma\\frac{d_1}{d_2}\\right)\\frac{\\Gamma\\left(1\/(2d_2)\\right)\\Gamma\\left(d_3\/(2d_2)\\right)}{\\Gamma(1\/2)}.\n\\end{align}\n For BI black holes with double horizons, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=\\left[M-(1-\\gamma)Q\\Phi\\right]_{+}-\\left[M-(1-\\gamma)Q\\Phi\\right]_{-}.\\label{twomassiveadd}\n\\end{align}\nNote that, if we set $\\gamma=1$, $Q\\Phi$ does not appear in the expressions of action growth (\\ref{onemassiveadd}) and (\\ref{twomassiveadd}). In this case, for BI black holes with single horizon, electric charge affects action growth only through  the constant $C_1$, for BI black holes with double horizons, action growth rates vanish. This agrees with the result obtained for dyonic black holes in Einstein-Maxwell gravity.\n\n\n\\section{EBI black holes\\label{section4}}\nThe Lagrangian (\\ref{LFm}) is only suitable for constructing BI black holes with electric charge. To construct black holes with both electric and magnetic charges, the Lagrangian (\\ref{LFm}) should be replaced with the general one\\cite{1606.02733}\n\\begin{align}\n\\mathcal{L}=-\\beta^2\\sqrt{-\\det\\left(g_{\\mu\\nu}+\\frac{F_{\\mu\\nu}}{\\beta}\\right)}+\\beta^2\\sqrt{-\\det(g_{\\mu\\nu})}.\\label{BIlag}\n\\end{align}\nIn the large $\\beta$ limit, (\\ref{BIlag}) reduces to Maxwell theory. The Lagrangian of EBI theory is now given by\n\\begin{align}\n\\mathcal{L}=\\sqrt{-g}(R-2\\Lambda_0)-\\beta^2\\sqrt{-\\det\\left(g_{\\mu\\nu}+\\frac{F_{\\mu\\nu}}{\\beta}\\right)},\\label{EBI}\n\\end{align}\nwhere $\\Lambda_0=\\Lambda-\\beta^2\/2$ is the bare cosmological constant, and $\\Lambda$ is the effective cosmological constant. Variation of the action of EBI theory give rise to  field equations\n\\begin{align}\nG^{\\mu\\nu}+&\\Lambda_0g^{\\mu\\nu}+\\frac{\\beta^2}{2}\\frac{\\sqrt{-h}}{\\sqrt{-g}}\\left(h^{-1}\\right)^{(\\mu\\nu)}=0,\\label{eomdyong}\\\\\n&\\nabla_\\mu\\left[\\frac{\\sqrt{-h}}{\\sqrt{-g}}\\beta\\left(h^{-1}\\right)^{[\\mu\\nu]}\\right]=0,\\label{eomem}\n\\end{align}\nwhere $h_{\\mu\\nu}=g_{\\mu\\nu}+F_{\\mu\\nu}\/\\beta$, $h\\equiv\\det(h_{\\mu\\nu})$, and $\\left(h^{-1}\\right)^{\\mu\\nu}$ denotes the inverse of $h_{\\mu\\nu}$, i.e.,\n\\begin{align}\n\\left(h^{-1}\\right)^{\\mu\\rho}h_{\\rho\\nu}=\\delta^\\mu_\\nu,\\;\\;\\;\\;\\;\\;\\;\\;h_{\\nu\\rho}\\left(h^{-1}\\right)^{\\rho\\mu}=\\delta^\\mu_\\nu.\n\\end{align}\n\n\nWe still take the metric and strength ansatz given in eq.(\\ref{strengthansatz}). Solving e.o.m of $A_\\mu$ (\\ref{eomem}), we have\n\\begin{align}\n\\Phi'(r)=\\frac{q}{\\sqrt{\\left(r^4+\\frac{p^2}{\\beta^2}\\right)^n+\\frac{q^2}{\\beta^2}}},\n\\end{align}\nwhere \"$'$\" denotes derivative with respect to $r$. The Einstein equations (\\ref{eomdyong}) imply the dyonic  black hole solution in general dimensions\\cite{1606.02733}\n\\begin{align}\nf(r)=-\\frac{\\mu}{r^{2n-1}}-\\frac{\\Lambda_0}{n(2n+1)}r^2-\\frac{\\beta^2}{2n r^{2n-1}}G(r),\\label{generalBH}\n\\end{align}\nwhere $\\mu$ is the mass parameter, and\n\\begin{align}\nG(r)\\equiv \\int dr \\sqrt{\\left(r^4+\\frac{p^2}{\\beta^2}\\right)^n+\\frac{q^2}{\\beta^2}}\\label{G0}\n\\end{align}\n is introduced for convenience. The details of geometry and thermodynamics of the black hole (\\ref{generalBH}) can be found in \\cite{1606.02733}. The integral (\\ref{G0}) can not be integrated out for general $n$, let's consider the following special cases.\n\n\\subsection{Purely electric EBI black holes\\label{section41}}\nIf we take the $p\\rightarrow 0$ limit of the dyonic black holes (\\ref{generalBH}), we obtain the  purely electric EBI black holes.\nFor  electric EBI black holes with single horizon,  with e.o.m (\\ref{eomdyong}) and the null coordinates introduced in (\\ref{nullcoord}), at late times the bulk contribution to action growth is given by\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}&=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[\\frac{2\\Lambda_0}{n(2n+1)}r_{+}^{2n+1}+\\frac{\\beta^2}{n}\\hat{G}(r_{+})-\\frac{\\beta^2}{n}\\hat{G}(0)\\right].\\label{elecV1V2}\n\\end{align}\nwhere $\\hat{G}(r)\\equiv\\int dr \\sqrt{r^{4n}+\\frac{q^2}{\\beta^2}}$\nis $G(r)$ in (\\ref{G0}) with the magnetic charge parameter $p=0$.\nAction of the $r=0$ surface is\n\\begin{align}\nI_{\\mathcal{S}}\n=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[(2n+1)\\mu-q^{\\frac{2n+1}{2n}}\\beta^{\\frac{2n-1}{2n}}\\frac{\\Gamma\\left(1\/2-1\/(4n)\\right)\\Gamma\\left(1+1\/(4n)\\right)}{\\Gamma\\left(1\/2\\right)}\\right].\\label{elecK}\n\\end{align}\nThe joint term is given by\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}\n=\\frac{\\omega_2^n\\delta t}{16\\pi}\\left[(2n-1)\\mu-\\frac{2\\Lambda_0}{n(2n+1)}r_{+}^{2n+1}+\\frac{2n-1}{2n}\\beta^2\\hat{G}(r_{+})-\\frac{\\beta^2r_{+}}{2n}\\sqrt{r_{+}^{4n}+\\frac{q^2}{\\beta^2}}\\right].\\label{elecjoint}\n\\end{align}\nBy sum of eqs. (\\ref{elecV1V2}), (\\ref{elecK}) and (\\ref{elecjoint}) we have the total variation of action\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=\\frac{\\omega_2^n}{16\\pi}\\left[4n\\mu+\\frac{2n+1}{2n}\\beta^2\\hat{G}(r_{+})-\\frac{\\beta^2r_{+}}{2n}\\sqrt{r_{+}^{4n}+\\frac{q^2}{\\beta^2}}-\\frac{\\beta^2}{n}\\hat{G}(0)\\right]\\nonumber\\\\\n&=2M-Q_e\\Phi_e-\\hat{C},\\label{onepureelec}\n\\end{align}\nwith\n\\begin{align}\nM=\\frac{n\\omega_2^n}{8\\pi}\\mu,\\;\\;\\;\\;\\;Q_e=\\frac{q}{16\\pi}\\omega_2^n,\\;\\;\\;\\;\\;\\Phi_e=\\int_{r_{+}}^\\infty\\frac{qdr}{\\sqrt{r^{4n}+\\frac{q^2}{\\beta^2}}}\n\\end{align}\nare  mass, electric charge and  potential respectively, and the constant\n\\begin{align}\n\\hat{C}=\\frac{(2n-1)\\omega_2^n q^{\\frac{2n+1}{2n}}\\beta^{\\frac{2n-1}{2n}}}{16\\pi(2n+1)}\\frac{\\Gamma\\left(1\/2-1\/(4n)\\right)\\Gamma\\left(1+1\/(4n)\\right)}{\\Gamma\\left(1\/2\\right)}.\\label{Celectric}\n\\end{align}\nWe see that, action growth of electric EBI black holes is in the same manner as the one of the massive gravity counterparts.  Electromagnetic field does not change casual structure of EBI black holes compared with AdS-Schwarzschild black holes, but it affects the manners of action growth, which leads  Lloyd's bound to be unsaturated.\n\nFor electric EBI black holes  with double horizons,  the bulk contribution to $\\delta I$ is\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[\\frac{2\\Lambda_0}{n(2n+1)}r^{2n+1}+\\frac{\\beta^2}{n}\\hat{G}(r)\\right]\\bigg|_{r_{-}}^{r_{+}}.\\label{elec2bulk}\n\\end{align}\nThe joint terms are given by\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}+I_{\\mathcal{C}'\\mathcal{C}}=\\frac{\\omega_2^n\\delta t}{16\\pi}\\left[(2n-1)\\mu-\\frac{2\\Lambda_0}{n(2n+1)}r^{2n+1}+\\frac{2n-1}{2n}\\beta^2\\hat{G}(r)-\\frac{\\beta^2r}{2n}\\sqrt{r^{4n}+\\frac{q^2}{\\beta^2}}\\right]\\bigg|_{r_{-}}^{r_{+}}.\n\\label{elec2joint}\n\\end{align}\nCombining (\\ref{elec2bulk}) and (\\ref{elec2joint}) we have the total action growth rate\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=\\left(M-Q_e\\Phi_e\\right)_{+}-\\left(M-Q_e\\Phi_e\\right)_{-}.\\label{twopureelec}\n\\end{align}\nIn this case, action growth of electric EBI black holes takes the identical form with that of the AdS-RN black holes, however, electromagnetic field affects action growth through the nonlinearity of BI theory.\n\nWe add a boundary term of electromagnetic field to the  action\n\\begin{align}\nI_{\\mu Q}&=\\frac{\\gamma}{16\\pi}\\int d\\Sigma_\\mu\\beta\\frac{\\sqrt{-h}}{\\sqrt{-g}}\\left(h^{-1}\\right)^{[\\nu\\mu]}A_\\nu\\nonumber\\\\\n&=\\frac{\\gamma}{32\\pi}\\int d^dx\\beta\\sqrt{-h}\\left(h^{-1}\\right)^{[\\nu\\mu]}F_{\\mu\\nu},\\label{boundaryBI}\n\\end{align}\nwhich does not affect the field equations. The field equations $\\nabla_\\mu\\big(\\frac{\\sqrt{-h}}{\\sqrt{-g}}\\left(h^{-1}\\right)^{[\\nu\\mu]}\\big)=0$ and Stoke's theorem have been used in the second equality. For electric EBI black holes with single horizon, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=2M-(1-\\gamma)Q_e\\Phi_e-\\hat{C}_1,\\label{onepureelecadd}\n\\end{align}\nwith\n\\begin{align}\n\\hat{C}_1=\\left(\\frac{2n-1}{2n+1}+\\gamma\\right)\\frac{\\omega_2^n q^{\\frac{2n+1}{2n}}\\beta^{\\frac{2n-1}{2n}}}{16\\pi}\\frac{\\Gamma\\left(1\/2-1\/(4n)\\right)\\Gamma\\left(1+1\/(4n)\\right)}{\\Gamma\\left(1\/2\\right)}.\\label{Celectricadd}\n\\end{align}\nFor electric EBI black holes with double horizons, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=\\left[M-(1-\\gamma)Q_e\\Phi_e\\right]_{+}-\\left[M-(1-\\gamma)Q_e\\Phi_e\\right]_{-}.\\label{twopureelecadd}\n\\end{align}\nAfter the addition of the BI boundary term,  action growth rates of the electric EBI black holes take the identical form with that of the massive gravity counterparts too. As we will see in the following,  action growth of the electric EBI black holes with $\\gamma=1$ is in the same manner as that of the magnetic EBI black holes with $\\gamma=0$.\n\n\n\n\\subsection{Pure magnetic EBI black holes\\label{section42}}\nIn this subsection, we calculate action growth of purely magnetic EBI black hole in general dimensions and make a comparison between the effects of electric and magnetic charges on action growth.\n\n\nFor magnetic EBI black holes with single horizon, with e.o.m (\\ref{eomdyong}) and the null coordinates, we obtain the bulk contribution to $\\delta I$\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}&=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[\\frac{2\\Lambda_0}{n(2n+1)}r^{2n+1}+\\frac{\\beta^2}{n}\\bar{G}(r)-p^2\\bar{H}(r)\\right]\\bigg|_\\epsilon^{r_{+}},\\nonumber\\\\\n&=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[\\frac{2\\Lambda_0}{n(2n+1)}r_{+}^{2n+1}+\\frac{\\beta^2}{n}\\bar{G}(r_{+})-p^2\\bar{H}(r_{+})-\\frac{\\beta^2}{n}\\bar{G}(0)+p^2\\bar{H}(0)\\right].\\label{magV1V2}\n\\end{align}\nwhere $\\bar{G}(r)\\equiv\\int dr\\left(r^4+\\frac{p^2}{\\beta^2}\\right)^{n\/2},\\;\\bar{H}(r)\\equiv \\int dr\\left(r^4+\\frac{p^2}{\\beta^2}\\right)^{n\/2-1}$, and we have taken  the $\\epsilon\\rightarrow0$ limit in the second equality.\nAction of the spacelike surface is given by\n\\begin{align}\nI_{\\mathcal{S}}\n=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[(2n+1)\\mu+\\frac{p^{n+1\/2}}{2 n\\beta^{n-3\/2}}\\frac{\\Gamma\\left(1\/4\\right)\\Gamma\\left(3\/4-n\/2\\right)}{\\Gamma\\left(-n\/2\\right)}\\right].\\label{magK}\n\\end{align}\nThe joint term is\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}\n=\\frac{\\omega_2^n\\delta t}{16\\pi}\\left[(2n-1)\\mu-\\frac{2\\Lambda_0}{n(2n+1)}r_{+}^{2n+1}+\\frac{2n-1}{2n}\\beta^2\\bar{G}(r_{+})-\\frac{\\beta^2r_{+}}{2n}\\sqrt{(r_{+}^4+p^2\/\\beta^2)^n}\\right].\\label{magjoint}\n\\end{align}\nWe have the total variation of action by sum of eqs. (\\ref{magV1V2}), (\\ref{magK}) and (\\ref{magjoint})\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=\\frac{\\omega_2^n}{16\\pi}\\left[4n\\mu+\\frac{2n+1}{2n}\\beta^2\\bar{G}(r_{+})-\\frac{\\beta^2r_{+}}{2n}\\sqrt{(r_{+}^4+p^2\/\\beta^2)^n}-p^2\\bar{H}(r_{+})-\\frac{\\beta^2}{n}\\bar{G}(0)\n+p^2\\bar{H}(0)\\right]\\nonumber\\\\\n&=2M-\\bar{C},\\label{onepuremag}\n\\end{align}\nwith\n\\begin{align}\nM=\\frac{n\\omega_2^n}{8\\pi}\\mu,\n\\end{align}\nand\n\\begin{align}\n\\bar{C}=-\\frac{\\omega_2^np^{n+1\/2}}{8\\pi(2n+1)\\beta^{n-3\/2}}\\frac{\\Gamma\\left(1\/4\\right)\\Gamma\\left(3\/4-n\/2\\right)}{\\Gamma\\left(-n\/2\\right)}.\\label{Cmag}\n\\end{align}\nNote that, if we don't consider the BI boundary term,  action growth of magnetic EBI black holes with single horizon depends only on mass and some constant involving $p$, it does not depend on the product of magnetic charge and potential $Q_m\\Phi_m$.  This differs from the pure electric case (\\ref{onepureelec}), where the action growth depends on mass, some constant  and the product $Q_e\\Phi_e$.\nBy comparison of the first line of (\\ref{onepuremag}) and the first line of (\\ref{onepureelec}), we see that, for electric EBI black holes, the term corresponding to $\\bar{H}(r_{+})$ disappears since $p$ vanishes, therefore the term $Q_e\\Phi_e$ in action growth remain. The calculation details of simplifying eq.(\\ref{onepuremag}) from the first line to the second line can be found in the appendix.\n\nIt's interesting to note that, $\\bar{C}$ is a positive constant for $n$ to be odd, while  $\\bar{C}$ vanishes for  $n$ to be even. For odd $n$, according to CA duality we have $\\frac{d\\mathcal{C}}{dt}<\\frac{2M}{\\pi\\hbar}$,  Lloyd's bound is satisfied. For even $n$, we have $\\frac{d\\mathcal{C}}{dt}=\\frac{2M}{\\pi\\hbar}$, Lloyd's bound is saturated. Therefore, if we don't consider the boundary term of electromagnetic field, action growth of magnetic EBI black holes is in the same manner as the one of AdS-Schwarzschild black holes in some dimensions. In this case, magnetic charge affects action growth through back-reaction on the geometry.\n\nFor magnetic EBI black holes  with double horizons,  the bulk contribution to $\\delta I$ is\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}=\\frac{1}{16\\pi}\\omega_2^n\\delta t\\left[\\frac{2\\Lambda_0}{n(2n+1)}r^{2n+1}+\\frac{\\beta^2}{n}\\bar{G}(r)-p^2\\bar{H}(r)\\right]\\bigg|_{r_{-}}^{r_{+}}.\\label{mag2bulk}\n\\end{align}\nThe joint terms are\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}+I_{\\mathcal{C}'\\mathcal{C}}=\\frac{\\omega_2^n\\delta t}{16\\pi}\\left[(2n-1)\\mu-\\frac{2\\Lambda_0}{n(2n+1)}r^{2n+1}+\\frac{2n-1}{2n}\\beta^2\\bar{G}(r)-\\frac{\\beta^2r}{2n}\\sqrt{(r^4+p^2\/\\beta^2)^n}\\right]\\bigg|_{r_{-}}^{r_{+}}.\n\\label{mag2joint}\n\\end{align}\nCombining (\\ref{mag2bulk}) and (\\ref{mag2joint}) we find that\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=0.\\label{twopuremag}\n\\end{align}\nThis result agrees with that of the magnetic black holes in Einstein-Maxwell theory. Although action growth rates do not vanish for the magnetic EBI black holes with single horizon, action growth rates vanish for the magnetic black holes with double horizons both in Einstein-Maxwell gravity and in EBI gravity.\n\n\nWe now add the boundary term of electromagnetic field (\\ref{boundaryBI}) to the  action. For magnetic EBI black holes with single horizon, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=2M-\\gamma Q_m\\Phi_m-\\bar{C}_1,\\label{onepuremagadd}\n\\end{align}\nwith $Q_m=\\frac{np}{16\\pi}\\omega_2,\\;\\Phi_m=\\frac{\\partial M}{\\partial Q_m}$ are magnetic charge and potential respectively, and\n\\begin{align}\n\\bar{C}_1=-\\frac{\\omega_2^np^{n+1\/2}}{8\\pi\\beta^{n-3\/2}}\\left(\\frac{1}{2n+1}-\\frac{\\gamma}{4}\\right)\\frac{\\Gamma\\left(1\/4\\right)\\Gamma\\left(3\/4-n\/2\\right)}{\\Gamma\\left(-n\/2\\right)}.\\label{barC1}\n\\end{align}\nAfter the addition of the BI boundary term, we see that, magnetic charge affects action growth in a similar manner to electric charge. Here, $\\bar{C}_1$ also vanishes for even $n$. Therefore, in the dimensions with even $n$, action growth of magnetic EBI black holes takes the specific form $\\frac{\\delta I}{\\delta t}=2M-\\gamma Q_m\\Phi_m$. Note from eqs. (\\ref{onepuremagadd}) and (\\ref{barC1}) that,  after the addition of the BI boundary term,  Lloyd's bound may be violated for certain values of $\\gamma$. The late-time violations of Lloyd's bound have also been found in Einstein-dilaton system, in Einstein-Maxwell-Dilaton system\\cite{1808.09917,1712.09826}, etc.\n\nFor magnetic BI black holes with double horizons, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=\\left[(M-\\gamma Q_m\\Phi_m)\\right]_+-\\left[(M-\\gamma Q_m\\Phi_m)\\right]_-.\\label{twopuremagadd}\n\\end{align}\nNow, the rate of action growth does not vanish due to addition of the BI boundary term. If we set $\\gamma=1$, magnetic EBI black holes complexify in the same manner as the electric ones with $\\gamma=0$. If we set $\\gamma=\\frac{1}{2}$, the manners of  action growth are similar for magnetic and electric EBI black holes, which implies electric and magnetic charges may contribute to action growth on equal footing in this case. As we will see from the calculating results of the dyonic black hole in the following, this is indeed the case.\n\n\n\n\n\\subsection{Four-dimensional dyonic EBI black hole\\label{section43}}\nSince the integral (\\ref{G0}) can not be integrated out for general $n$, for simplicity we only consider the $n=1$ case, i.e., the dyonic black hole in four dimensions.\n\nFor the dyonic EBI black hole with single horizon, bulk contribution to $\\delta I$ is given by\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}=\\frac{1}{16\\pi}\\omega_2\\delta t\\left[\\frac{2\\Lambda_0}{3}r_{+}^{3}+\\beta^2\\tilde{G}(r_{+})-p^2\\tilde{H}(r_{+})-\\beta^2\\tilde{G}(0)+p^2\\tilde{H}(0)\\right].\\label{dyonV1V2}\n\\end{align}\nwith\n\\begin{align}\n\\tilde{G}(r)\\equiv\\int dr\\sqrt{r^4+\\frac{p^2+q^2}{\\beta^2}},\\;\\;\\;\\;\\;\\tilde{H}(r)\\equiv \\int dr\\frac{1}{\\sqrt{r^4+\\frac{p^2+q^2}{\\beta^2}}}.\n\\end{align}\nAction of the $r=0$ spacelike surface  is\n\\begin{align}\nI_{\\mathcal{S}}=\\frac{1}{16\\pi}\\omega_2\\delta t\\left[3\\mu-\\frac{(p^2+q^2)^{3\/4}\\sqrt{\\beta}}{\\sqrt{\\pi}}\\Gamma\\left(\\frac{1}{4}\\right)\\Gamma\\left(\\frac{5}{4}\\right)\\right].\\label{dyonK}\n\\end{align}\nThe joint term  is\n\\begin{align}\nI_{\\mathcal{B}'\\mathcal{B}}=\\frac{1}{16\\pi}\\omega_2\\delta t\\left[\\mu-\\frac{2\\Lambda_0}{3}r_{+}^{3}+\\frac{1}{2}\\beta^2\\tilde{G}(r_{+})-\\frac{\\beta^2r_{+}}{2}\\sqrt{r_{+}^4+(p^2+q^2)\/\\beta^2}\\right].\\label{dyonjoint}\n\\end{align}\nAdding up all the contributions, the growth rate of action can be written as\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=2M-Q_e\\Phi_e-\\tilde{C},\\label{onedyon}\n\\end{align}\nwhere\n\\begin{align}\n\\tilde{C}=\\frac{(4p^2+q^2)\\omega_2\\sqrt{\\beta}}{48\\pi\\left(q^2+p^2\\right)^{1\/4}}\\frac{\\Gamma\\left(\\frac{1}{4}\\right)\\Gamma\\left(\\frac{5}{4}\\right)}{\\Gamma\\left(\\frac{1}{2}\\right)},\n\\end{align}\nand\n\\begin{align}\nM&=\\frac{\\omega_2}{8\\pi}\\mu,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;Q_e=\\frac{q}{16\\pi}\\omega_2,\\nonumber\\\\\n\\Phi_e&=\\frac{q}{r_{+}}\n\\sideset{_2}{_1}{\\mathop{F}}\\left[\\frac{1}{4},\\frac{1}{2},\\frac{5}{4},-\\frac{p^2+q^2}{r_{+}^4\\beta^2}\\right].\n\\end{align}\nare mass, electric charge and potential  respectively.\n\nThe result (\\ref{onedyon}) matches the ones of purely electric and magnetic EBI black holes with single horizon in general dimensions above, i.e., only the product of electric charge and potential appears in the expression of action growth, magnetic charge affects action growth only through some constant.\n\n\nFor the dyonic EBI black hole with inner and outer horizons,  we have\n\\begin{align}\nI_{\\mathcal{V}_1}-I_{\\mathcal{V}_2}&=\\frac{1}{16\\pi}\\omega_2\\delta t\\left[\\frac{2\\Lambda_0}{3}r^{3}+\\beta^2\\tilde{G}(r)-p^2\\tilde{H}(r)\\right]\\bigg|_{r_{-}}^{r_{+}},\\\\\nI_{\\mathcal{B}'\\mathcal{B}}+I_{\\mathcal{C}'\\mathcal{C}}&=\\frac{1}{16\\pi}\\omega_2\\delta t\\left[\\mu-\\frac{2\\Lambda_0}{3}r^{3}+\\frac{1}{2}\\beta^2\\tilde{G}(r)-\\frac{\\beta^2r}{2}\\sqrt{r^4+(p^2+q^2)\/\\beta^2}\\right]\\bigg|_{r_{-}}^{r_{+}}.\n\\end{align}\nThus the growth rate of action is\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=\\left(M-Q_e\\Phi_e\\right)_{+}-\\left(M-Q_e\\Phi_e\\right)_{-}.\\label{twodyon}\n\\end{align}\nThis result agrees with the ones  of purely electric and magnetic EBI black holes, and takes the identical form with that of dyonic black holes in Einstein-Maxwell gravity. It seems that, independent of the linearity or  nonlinearity of electromagnetic theory, only electric charge affects action growth of the  black holes with double horizons, action growth rates vanish for the purely magnetically charged black holes.\n\n\nAfter the addition of  the boundary term (\\ref{boundaryBI}), for dyonic EBI black holes with single horizon, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}=2M-(1-\\gamma)Q_e\\Phi_e-\\gamma Q_m\\Phi_m-\\tilde{C}_1,\\label{onedyonadd}\n\\end{align}\nwhere $Q_m$, $\\Phi_m$ are magnetic charge and potential\n\\begin{align}\nQ_m=\\frac{p}{16\\pi}\\omega_2,\\;\\;\\;\\Phi_m=\\frac{p}{r_{+}}\n\\sideset{_2}{_1}{\\mathop{F}}\\left[\\frac{1}{4},\\frac{1}{2},\\frac{5}{4},-\\frac{p^2+q^2}{r_{+}^4\\beta^2}\\right],\n\\end{align}\nand\n\\begin{align}\n\\tilde{C}_1=\\frac{\\omega_2\\sqrt{\\beta}}{16\\pi^{3\/2}\\left(q^2+p^2\\right)^{1\/4}}\\left[\\frac{5}{6}(p^2+q^2)+\\left(\\frac{1}{2}-\\gamma\\right)(p^2-q^2)\\right]\\frac{\\Gamma\\left(\\frac{1}{4}\\right)\\Gamma\\left(\\frac{5}{4}\\right)}\n{\\Gamma\\left(\\frac{1}{2}\\right)}.\\label{tildeC1}\n\\end{align}\nFor dyonic BI black holes with double horizons, action growth becomes\n\\begin{align}\n\\frac{\\delta I}{\\delta t}&=\\left[M-(1-\\gamma)Q_e\\Phi_e-\\gamma Q_m\\Phi_m\\right]_{+}-\\left[M-(1-\\gamma)Q_e\\Phi_e-\\gamma Q_m\\Phi_m\\right]_{-}.\\label{twodyonadd}\n\\end{align}\nThe calculating outcomes (\\ref{onedyonadd}) and (\\ref{twodyonadd}) agree with the ones of purely electric and magnetic EBI black holes.\nIf we set $\\gamma=1$,  the product $Q_e\\Phi_e$ in the expressions of action growth vanishes, only product $Q_m\\Phi_m$ remains, electric charge only affects action growth of the dyonic EBI black hole with single horizon through some constant. If we set $\\gamma=\\frac{1}{2}$, the asymmetric part under $p\\leftrightarrow q$ in (\\ref{tildeC1}) vanishes, then electric and magnetic charges contribute to action growth on equal footing for the dyonic EBI black holes with both single and double horizons.\n\n\n\n\n\\section{Summary and Discussion}\nIn this paper, we study action growth of BI black holes. As a comparison, we first review action growth of dyonic black holes in Einstein-Maxwell gravity in general dimensions, and notice that action growth takes the identical form with the four-dimensional case.  Action growth rates vanish for purely magnetic black holes if we don't consider the Maxwell boundary term. After the inclusion of Maxwell boundary term, if we set $\\gamma=\\frac{1}{2}$, then electric and magnetic charges contribute to action growth on equal footing, which is in accord with electric-magnetic duality. If we set $\\gamma=1$, then action growth rates vanish for purely electric black holes.\n\nWe study action growth of electric BI black holes in massive gravity, and find that BI black holes in massive gravity always complexify faster than their Einstein gravity counterparts due to the back-reaction of graviton mass on geometry. If we include the Maxwell boundary term and set $\\gamma=1$, then the term $Q\\Phi$ disappears in the expressions of action growth.\n\nWe study action growth of pure electric and magnetic EBI black holes in general dimensions. Before inclusion of the BI boundary term to the action,  action growth of electric and magnetic EBI black holes with single horizon are\nin different manners,\nwhich are $\\frac{\\delta I}{\\delta t}=2M-Q_e\\Phi_e-\\hat{C}$ and $\\frac{\\delta I}{\\delta t}=2M-\\bar{C}$ respectively.\nWe notice that, for the magnetic black holes, the constant $\\bar{C}$ vanishes for even $n$. Therefore, in the dimensions with even $n$, action growth of magnetic EBI black holes saturates Lloyd's bound, i.e., it takes the identical form with that of the AdS-Schwarzschild black holes. In this case, magnetic charge affects action growth through back-reaction on the geometry. For EBI black holes with double hirizons, action growth of the electric ones takes the identical form with that of AdS-RN black holes, while action growth of the magnetic ones vanishes, this agrees with the result obtained from the dyonic black holes in Einstein-Maxwell gravity.\n\nIf we include the BI boundary term to the action, action growth of the electric (magnetic) EBI black holes with $\\gamma=1$ takes  similar form with that of the magnetic (electric) ones with $\\gamma=0$.\nWe find that,  for the  magnetic EBI black holes with single horizon, in the dimensions with even $n$ the constant in $\\frac{\\delta I}{\\delta t}$ vanishes, then action growth takes the specific form $\\frac{\\delta I}{\\delta t}=2M-\\gamma Q_m\\Phi_m$. In the dimensions with odd $n$, due to the contribution of the BI boundary term, complexity growth of the black holes may violate Lloyd's bound for certain values of $\\gamma$.\n\nThe calculating result of the four-dimensional dyonic EBI black hole agrees with that of the electric and magnetic EBI black holes in general dimensions. When we set $\\gamma=\\frac{1}{2}$, electric and magnetic charges contribute to action growth on equal footing.\n\n\\section*{Acknowledgment}\n\nKM would like to thank Profs. Peng Wang, Haitang Yang, Liu Zhao and Haishan Liu for valuable discussions.\n\n\\section*{Appendix}\n\nThe magnetic potential is given by\n\\begin{align}\n\\Phi_m&=\\frac{\\partial M}{\\partial Q_m}\n=\\frac{r_{+}^{2n+1}\\beta^2\\omega_2^{n-1}}{2np}\\left(\\left(1+\\frac{p^2}{r_{+}^4\\beta^2}\\right)^{n\/2}-\n\\sideset{_2}{_1}{\\mathop{F}}\\left[-\\frac{1}{4}-\\frac{n}{2},-\\frac{n}{2},\\frac{3}{4}-\\frac{n}{2},-\\frac{p^2}{r_{+}^4\\beta^2}\\right]\\right).\n\\end{align}\nThe two terms in eq.(\\ref{onepuremag}) can be simplified to\n\\begin{align}\n\\frac{\\omega_2^n}{16\\pi}\\left(\\frac{2n+1}{2n}\\beta^2\\bar{G}(r_{+})-\\frac{\\beta^2r_{+}}{2n}\\sqrt{(r_{+}^4+p^2\/\\beta^2)^n}\\right)=-\\frac{1}{n}Q_m\\Phi_m.\\label{appendix2term}\n\\end{align}\nFrom the definition of $\\bar{H}(r_{+})$, we have\n\\begin{align}\n\\bar{H}(r_{+})&\\equiv \\int dr_{+}\\left(r_{+}^4+\\frac{p^2}{\\beta^2}\\right)^{n\/2-1}\n=\\frac{r_{+}^{2n-3}}{2n-3}\\sideset{_2}{_1}{\\mathop{F}}\\left[\\frac{3}{4}-\\frac{n}{2},1-\\frac{n}{2},\\frac{7}{4}-\\frac{n}{2},-\\frac{p^2}{r_{+}^4\\beta^2}\\right].\n\\end{align}\nWith the formula $\\sideset{_2}{_1}{\\mathop{F}}\\left[a,b,c,z\\right]=\\frac{c-1}{(a-1)(b-1)}\\frac{d}{dz}\\sideset{_2}{_1}{\\mathop{F}}\\left[a-1,b-1,c-1,z\\right]$, $\\bar{H}(r_{+})$ can be rewritten as\n\\begin{align}\n\\bar{H}(r_{+})\n=-\\frac{r_{+}^{2n+1}\\beta^2}{2np^2}\\left(\\left(1+\\frac{p^2}{r_{+}^4\\beta^2}\\right)^{n\/2}-\n\\sideset{_2}{_1}{\\mathop{F}}\\left[-\\frac{1}{4}-\\frac{n}{2},-\\frac{n}{2},\\frac{3}{4}-\\frac{n}{2},-\\frac{p^2}{r_{+}^4\\beta^2}\\right]\\right).\n\\end{align}\nTherefore\n\\begin{align}\n\\frac{\\omega_2^n}{16\\pi}p^2\\bar{H}(r_{+})=-\\frac{1}{n}Q_m\\Phi_m.\\label{appendixH}\n\\end{align}\nBy comparison of eqs.(\\ref{appendix2term}) and (\\ref{appendixH}), we immediately have\n\\begin{align}\n\\frac{\\omega_2^n}{16\\pi}\\left(\\frac{2n+1}{2n}\\beta^2\\bar{G}(r_{+})-\\frac{\\beta^2r_{+}}{2n}\\sqrt{(r_{+}^4+p^2\/\\beta^2)^n}-p^2\\bar{H}(r_{+})\\right)=0.\n\\end{align}\n\n\n\n\n\\providecommand{\\href}[2]{#2}\\begingrou\n\\footnotesize\\itemsep=0pt\n\\providecommand{\\eprint}[2][]{\\href{http:\/\/arxiv.org\/abs\/#2}{arXiv:#2}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNonlinear dynamics is a topic that not only cover all the disciplines, in both\nnatural and social sciences, but also is now becoming part of introductory level\nundergraduate courses in sciences. Searching for affordable, easy to setup, and\nreconfigurable classroom demonstrations that allow to investigate the physical\nand mathematical nature of nonlinear dynamical systems has been always a matter\nof interest for instructors. This paper describes a simple apparatus used to\nexplore the behavior of one-dimensional chaotic maps, like the Logistic Map,\nwhen different initial conditions are chosen.\n\nThe device consists of an inexpensive open-source microcontroller connected to\nan array of light emitting diodes (LEDs), and programmed to iterate the logistic\nmap in the one-dimensional interval $[0,1]$. This interval is divided in ten\nequal parts and mapped one-to-one to a single LED in the array. When the\nlogistic map produces certain value after an iteration, the corresponding LED\nlights up showing the value in the one dimensional domain approximately. After\nseveral iterations, it is possible to \\textit{visualize} the trajectory of the\nmap by looking at the blinking LEDs sequence. Sensitivity to initial conditions,\ndensity of periodic orbits, strange attractors, and bifurcations are visualized\neasily with this device.\n\n\n\\section{One dimensional maps}\nThe logistic map is, perhaps, the simplest example of how a nonlinear dynamical\nequation can give rise to very complex, chaotic behavior.~\\citealp{Weisstein}\nInitially introduced as a mathematical model of population\ngrowth,~\\citealp{Verhulst} it rapidly found applications in diverse areas like\nmathematical biology, biometry, demography, condense matter, econophysics, and\ncomputation.~\\citealp{Ausloos} The logistic map function is defined as\n\\begin{equation}\\label{Eq:LogisticMap}\n  X_{n+1} = A\\,X_n (1 - X_n) \\equiv f_A(X),\n\\end{equation}\nwhere the factor $A$ is a model-dependent parameter representing\n\\textit{external} conditions to the system, and $X_n$ is the population in the\n$n$th-period cycle, scaled so that its value fits in the interval\n$[0,1]$.~\\citealp{Marion} The function $f_A$ is called an \\textit{iteration\nfunction} because we may find the population fraction $X$ in the following\nperiod cycle by repeating the mathematical operations expressed in\n\\eqref{Eq:LogisticMap}.~\\citealp{Hilborn} It is also one-dimensional since there\nis only a single variable $X$, and the resulting curve is a line.\n\n\n\\section{Setup}\nWe show a photograph of the apparatus setup in Fig.~\\ref{Fig01}, and its\nschematics in Fig.~\\ref{Fig02}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.9]{Figure01}\n\\caption{\\label{Fig01}Setup of the Logistic Map device. An array of ten LEDs are\nconnected to the Arduino microcontroller using the same number of $220\\,\\Omega$\nresistors.}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.23]{Figure02}\n\\caption{\\label{Fig02}Circuit diagram of the ten LEDs array connected to the\nArduino.}\n\\end{figure}\n\nThe apparatus uses an open-source Arduino prototyping platform made up of an\nAtmel AVR processor (microcontroller). It has 14 digital input\/output pins, 6\nanalog inputs, a 16 MHz crystal oscillator, a USB connection, a power jack, an\nICSP header, and a reset button.~\\citealp{Arduino} Arduino is a registered\ntrademark---only the official boards are named ``Arduino''---so clones usually\nhave names ending with ``duino.''~\\citealp{Schmidt} The Arduino can be connected\nto a computer through the USB port and programmed using a language similar to\nC++. The program is uploaded into the microcontroller using an Integrated\nDevelopment Environment (IDE). In the Arduino world, programs are known as\n\\textit{sketches}.\n\nWe connected an array of ten $5\\,\\mbox{mm}$ LEDs to the digital pins of the\nmicrocontroller using $220\\,\\Omega$ resistors (or $1\\,\\mbox{k}\\Omega$ for dimmer\nLEDs). We also used a solderless breadboard to hook up all the electric\ncomponents to the microcontroller.\n\n\n\\section{The sketch}\nListing~\\ref{List01} shows a simple program (sketch) used to operate the\nmicrocontroller, and \\textit{visualize} the logistic map by looking at the array\nof blinking LEDs. A glowing LED represents an iteration of the one-dimensional\nmap, and it is linked with a value in the interval $[0,1]$.\n\nArduino programs require two mandatory functions: \\texttt{setup()} and\n\\texttt{loop()}. Any variable or constant defined outside these two functions is\nconsidered to be \\textit{global}. In the \\texttt{setup()} function, we tell the\nmicrocontroller that there are 10 LEDs connected to the digital pins and that\nthey are intended to be turned on and off. In the \\texttt{loop()} function, the\nlogistic map is iterated, and the visualization process takes place as we\nobserve the LEDs turning on and off, one after another, following the evolution\nof the nonlinear system.\n\nUsing the \\texttt{if()} and \\texttt{else} control structures, we divide the\ninterval $[0,1]$ in ten identical segments and associate an LED to each one of\nthem. For example, the first LED from the left represents the first interval\nsegment $[0,0.1)$, the second LED represents the segment $[0.1,0.2)$, and so on.\nThe last LED represents the final segment $[0.9,1.0]$. When the microcontroller\niterates the logistic map, a value belonging to one of these ten intervals is\nreturned, and the corresponding LED turns on for 500 milliseconds. This process\nis repeated infinitely, so we can observe the \\textit{orbit} followed by the\nlogistic map by watching at the blinking LEDs sequence.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.3]{Figure03}\n\\caption{\\label{Fig03} Cobweb diagrams for the logistic map for different values\nof $A$ and $X_0$. The $45^{\\circ}$ line, $X_{n+1}=X_n$, is used to find the\nfixed points and follow the iteration process. Plots (a) and (b) show the\npresence of attractors at $X=0$ and $X=0.655$, respectively. Plot (c) displays a\nperiod doubling bifurcation, and (d) the onset of chaos.}\n\\end{figure}\n\nThe initial conditions for the logistic map can be changed anytime and uploaded\nagain into the microcontroller. Thus, we can explore the behavior of the chaotic\nmap when different values of the parameters $A$ and $X_0$ are chosen. It is\namusing to study the bifurcations of the logistic map by looking at the blinking\nsequence of LEDs. We start with the parameter values $A=0.9$ and $X_0=0.5$\n($0<A<1$). The LED associated with the interval $[0.5,0.6)$ turns on first. As\nthe iterations of the logistic map take place, we observe that the sequence of\nblinking LEDs moves toward the very first LED, which is associated with the\nsegment $[0,0.1)$. The sequence ends up with this first LED left on\nindefinitely. This evolution is shown in Fig.~\\ref{Fig03}(a), the cobweb diagram\nof the logistic map for that selection of parameters. For this case, $X=0$ is an\n\\textit{attractor}, and the domain $[0,1]$ forms a \\textit{basin of attraction}.\n\nAnother example of an attractor happens when $A=2.9$ and $X_0=0.2$ ($1<A<3$). We\nsee that, after few iterations and blinking LEDs, there is only one LED that\nremains on indefinitely. This LED corresponds to the interval $[0.6,0.7)$. In\nthis case, we have a \\textit{fixed point} occurring at $X=0.655$. As it is shown\nin Fig.~\\ref{Fig03}(b), the orbit of the map follows a square spiral that\nconverges into that fixed point. Now, if $A=3.1$ and $X_0=0.2$ ($3<A<3.44948$),\nwe can observe a 2-\\textit{period cycle}; the value of $X$ oscillates between\n0.558 and 0.765, and the two corresponding LEDs turn on and off intermittently.\nHere, the fixed point $X=0.677$ becomes a \\textit{repeller}.\nFigure~\\ref{Fig03}(c) shows the periodic behavior of the logistic map for this\nset of parameters. Unfortunately, higher periodicity (like 4-period, 8-period\ncycles, or higher) cannot be observed clearly with this device because of the\nway the LEDs are mapped into the $[0,1]$ interval. More LEDs are required to\nimprove the ``resolution'' of the device. However, the microcontroller has a\nlimit of 13 digital pins, and other electric components (like an extra\nmicrochip) need to be incorporated into the circuit if we want more LEDs\ncontrolled by the device.\n\nFinally, when $A=3.7$ and $X_0=0.05$ ($3.56994<A<4$), chaos\nonsets.\\citealp{Sprott} An ``apparently'' randomness of blinking LEDs is\nobserved. After many iterations, there is no single LED that remains on\nindefinitely, and no single LED that has not been turned on at least once. The\norbit of the logistic map covers all the domain of the interval $[0,1]$.\nFigure~\\ref{Fig03}(d) shows all points visited by the logistic map orbit after\n$10\\,000$ iterations.\n\n\\lstset{language=C++,\n        basicstyle=\\scriptsize,\n        commentstyle=\\normalfont,\n        frame=single,\n        captionpos=b,\n        morekeywords={pinMode,digitalWrite,delay}\n}\n\\begin{lstlisting}[float=h,caption={Logistic Map sketch for the Arduino.},\n  numbers=none,label={List01}]\n  \/\/ Blinking Logistic Map\n\n  \/\/ choose the pin for each LED\n  const int NbrLEDs  = 10;\n  const int LEDpin[] = {2,3,4,5,6,7,8,9,10,11};\n\n  const int wait = 500;  \/\/ wait for 500 milliseconds\n\n  \/\/ Logistic Map parameters\n  const double A  = 3.7;  \/\/ Logistic map constant\n\n  double X0 = 0.2;  \/\/ Initial position (0 < X0 < 1.0)\n  double X  = X0;   \/\/ Use X0 as your first calculated point\n\n  \/\/ setup() initializes the LED pins\n  void setup() {\n      for (int i = 0; i < NbrLEDs; i++) {\n          pinMode(LEDpin[i], OUTPUT);\n      }\n  }\n\n  \/\/ loop() iterates the Logistic Map and turn on\/off LEDs\n  void loop()\n  {\n      if (X < 0.1)\n          blinkLED(LEDpin[0]);\n      else if ((X >= 0.1) && (X < 0.2))\n          blinkLED(LEDpin[1]);\n      else if ((X >= 0.2) && (X < 0.3))\n          blinkLED(LEDpin[2]);\n      else if ((X >= 0.3) && (X < 0.4))\n          blinkLED(LEDpin[3]);\n      else if ((X >= 0.4) && (X < 0.5))\n          blinkLED(LEDpin[4]);\n      else if ((X >= 0.5) && (X < 0.6))\n          blinkLED(LEDpin[5]);\n      else if ((X >= 0.6) && (X < 0.7))\n          blinkLED(LEDpin[6]);\n      else if ((X >= 0.7) && (X < 0.8))\n          blinkLED(LEDpin[7]);\n      else if ((X >= 0.8) && (X < 0.9))\n          blinkLED(LEDpin[8]);\n      else\n          blinkLED(LEDpin[9]);\n\n      \/\/ iterates the Logistic Map function\n      X0 = X;\n      X  = A * X0 * (1.0 - X0);\n  }\n\n  \/\/ blinkLED function\n  \/\/ turn on\/off LEDs\n  void blinkLED(const int pin) {\n      digitalWrite(pin, HIGH);  \/\/ turn LED on\n      delay(wait);              \/\/ wait 500 milliseconds\n      digitalWrite(pin, LOW);   \/\/ turn LED off\n  }\n\\end{lstlisting}\n\n\n\\section{Final comments}\nWith this apparatus, students may understand more easily the behavior of\none-dimensional chaotic maps looking at an \\textit{actual} one-dimensional array\nof LEDs; every point in the interval [0,1] produces another point within the\nsame interval after the logistic map is iterated. By controlling the parameter\n$A$ and the initial value $X_0$, the instructor and students can observe under\nwhat conditions periodic and chaotic behavior may occur, providing a better\nunderstanding of periodicity, bifurcations, and the route to chaos of a\nnonlinear dynamical system. The setup described here is inexpensive, easy, and\nfun to assemble, also enhances computer programming and electronics assembly\nskills.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nUnderstanding and predicting crowd behaviour plays a pivotal role in video based surveillance; and as such is becoming essential for discovering public safety risks, and predicting crimes or patterns of interest. Recently, focus has been given to understanding human behaviour at a group level, leveraging observed social interactions. Researchers have shown this to be important as interactions occur at a group level, rather than at an individual or whole of crowd level. \n\nAs such we believe group detection has become a mandatory part of an intelligent surveillance system; however this group detection task presents several new challenges ~\\cite{solera2016socially,solera2013structured}. Other than identifying and tracking pedestrians from video, modelling the semantics of human social interaction and cultural gestures over a short sequence of clips is extremely challenging. Several attempts \\cite{solera2016socially,solera2013structured,pellegrini2010improving,yamaguchi2011you} have been made to incorporate handcrafted  physics based features such as relative distance between pedestrians, trajectory shape and motion based features to model their social affinity. Hall et. al \\cite{hall1966hidden} proposed a proxemic theory for such physical interactions based on different distance boundaries; however recent works \\cite{solera2016socially,solera2013structured} have shown these quantisations fail in cluttered environments. \n\nFurthermore, proximity doesn't always describe the group membership. For instance two pedestrians sharing a common goal may start their trajectories in two distinct source positions, however, meet in the middle. Hence we believe being reliant on a handful of handcrafted features to be sub-optimal \\cite{isola2017image,alahi2016social,fernando2018task}. \n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[width=\\linewidth]{figures\/model.pdf}\n   \\caption{Proposed group detection framework:  After observing short segments of trajectories for each pedestrian in the scene, we apply the proposed trajectory prediction algorithm to forecast their future trajectories. The context representation generated at this step is extracted and compressed using t-SNE dimensionality reduction. Finally, the DBSCAN clustering algorithm is applied to detect the pedestrian groups.}\n  \\label{fig:model}\n\\end{figure}\n\nTo this end we propose a deep learning algorithm which automatically learns these group attributes. We take inspiration from the trajectory modelling approaches of \\cite{fernando2017soft+} and \\cite{fernando2018tracking}, where the approaches capture contextual information from the local neighbourhood.  We further augment this approach with a Generative Adversarial Network (GAN) \\cite{gupta2018social,sadeghian2018sophie,fernando2018task} learning pipeline where we learn a custom, task specific loss function which is specifically tailored for future trajectory prediction, learning to imitate complex human behaviours. \n\nFig. \\ref{fig:model} illustrates the proposed approach. First, we observe short segments of trajectories from 1 to $T_{obs}$ for each pedestrian, $p^{k}$, in the scene. Then, we apply the proposed trajectory prediction algorithm to forecast their future trajectories from $T_{obs +1} -T_{pred}$. This step generates hidden context representations for each pedestrian describing the current environmental context in the local neighbourhood of the pedestrian. We then apply t-SNE dimensionality reduction to extract the most discriminative features, and we detect the pedestrian groups by clustering these reduced features. \n\nThe simplistic nature of the proposed framework offers direct transferability among different environments when compared to the supervised learning approaches of\\cite{solera2016socially,solera2013structured,pellegrini2010improving,yamaguchi2011you}, which require re-training of the group detection process whenever the surveillance scene changes. This ability is a result of the proposed deep feature learning framework which learns the required group attributes automatically and attains commendable results among the state-of-the-art. \n\nNovel contributions of this paper can be summarised as follows:\n\n\\begin{itemize}\n\\item We propose a novel GAN pipeline which jointly learns informative latent features for pedestrian trajectory forecasting and group detection. \n\\item We remove the supervised learning requirement for group detection, allowing direct transferability among different surveillance scenes. \n\\item We demonstrate how the original GAN objective could be augmented with sparsity regularisation to learn powerful features which are informative to both trajectory forecasting and group detection tasks. \n\\item We provide extensive evaluations of the proposed method on multiple public benchmarks where the proposed method is able to generate notable performance, especially among unsupervised learning based methods. \n\\item We present visual evidence on how the proposed trajectory modelling scheme has been able to embed social interaction attributes into its encoding scheme. \n\\end{itemize}\n\n\\section{Related Work}\nRelated literature is categorised into human behaviour prediction approaches (see Sec. \\ref{sec:related_work_human_behaviour_prediction}); and group detection architectures (see Sec. \\ref{sec:related_work_group_detection}).\n\n\\subsection{Human Behaviour Prediction}\n\\label{sec:related_work_human_behaviour_prediction}\nSocial Force models \\cite{helbing1995social,yamaguchi2011you}, which rely on the attractive and repulsive forces between pedestrians to model their future behaviour, have been extensively applied for modelling human navigational behaviour. However with the dawn of deep learning, these methods have been replaced as they have been shown to ill represent the structure of human decision making \\cite{fernando2017soft+,fernando2018tree,gupta2018social}. \n\nOne of the most popular deep learning methods is the social LSTM \\cite{alahi2016social} model which represents the pedestrians in the local neighbourhood using LSTMs and then generates their future trajectory by systematically pooling the relavant information. This removes the need for handcrafted features and learns the required feature vectors automatically through the encoded trajectory representation. This architecture is further augmented in \\cite{fernando2017soft+} where the authors propose a more efficient method to embed the local neighbourhood information via a soft and hardwired attention framework. They demonstrate the importance of fully capturing the context information, which includes the short-term history of the pedestrian of interest as well as their neighbours. \n\nGenerative Adversarial Networks (GANs) \\cite{gupta2018social,sadeghian2018sophie,fernando2018task} propose a task specific loss function learning process where the training objective is a minmax game between the generative and discriminative models. These methods have shown promising results, overcoming the intractable computation of a loss function, in tasks such as autonomous driving \\cite{fernando2018learning,li2017infogail}, saliency prediction \\cite{fernando2018task,pan2017salgan}, image to image translation \\cite{isola2017image} and human navigation modelling \\cite{gupta2018social,sadeghian2018sophie}.\n\nEven though the proposed GAN based trajectory modelling approach exhibits several similarities to recent works in \\cite{gupta2018social,sadeghian2018sophie}, the proposed work differs in multiple aspects. Firstly, instead of using CNN features to extract the local structure of the neighbourhood as in \\cite{sadeghian2018sophie}, pooling out only the current state of the neighbourhood as in \\cite{gupta2018social}, or discarding the available historical behaviour which is shown to be ineffective \\cite{fernando2017soft+,fernando2018tree,sadeghian2018sophie}; we propose an efficient method to embed the local neighbourhood context based on the soft and hardwired attention framework proposed in \\cite{fernando2017soft+}. Secondly, as we have an additional objective of localising the groups in the given crowd, we propose an augmentation to the original GAN objective which regularises the sparsity of the generator embeddings, generating more discriminative features and aiding the clustering processes. \n\n\\subsection{Group Detection}\n\\label{sec:related_work_group_detection}\n\nSome earlier works in group detection \\cite{cristani2011social,setti2013multi} employ the concept of F-formations \\cite{kendon1990conducting}, which can be seen as specific orientation patterns that individuals engage in when in a group. However such methods are only suited to stationary groups. \n\nIn a separate line of work researchers have analysed pedestrian trajectories to detect groups. Pellegrinin et. al \\cite{pellegrini2010improving} applied Conditional Random Fields to jointly predict the future trajectory of the pedestrian of interest as well as their group membership. \\cite{yamaguchi2011you} utilises distance, speed and overlap time to train a linear SVM to classify whether two pedestrians are in the same group or not. In contrast to these supervised methods, Ge et. al \\cite{ge2012vision} proposed using agglomerative clustering of speed and proximity features to extract pedestrian groups.\n\nMost recently Solera et. al \\cite{solera2013structured} proposed proximity and occupancy based social features to detect groups using a trained structural SVM. In \\cite{solera2016socially} the authors extend this preliminary work with the introduction of sociologically inspired features such as path convergence and trajectory shape. However these supervised learning mechanisms rely on hand labeled datasets to learn group segmentation, limiting the methods applicability. Furthermore, the above methods all utilise a predefined set of handcrafted features to describe the sociological identity of each pedestrian, which may be suboptimal. Motivated by the impressive results obtained in \\cite{fernando2017soft+} with the augmented context embedding, we make the first effort to learn  group attributes automatically and jointly through trajectory prediction. \n\n\\section{Architecture}\n\\subsection{Neighbourhood Modelling}\n\\label{sec:neighbourhood_modelling}\nWe use the trajectory modelling framework of \\cite{fernando2017soft+} (shown in Fig. \\ref{fig:neighbourhood_modelling}) for modelling the local neighbourhood of the pedestrian of interest.  \n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[width=.8\\linewidth]{figures\/neighbourhood_embedding.pdf}\n   \\caption{Proposed neighbourhood modelling scheme \\cite{fernando2017soft+}:  A sample surveillance scene is shown on the left. The trajectory of the pedestrian of interest, $k$ , is shown in green, and has two neighbours (in purple) to the left, one in front and none on right. The neighbourhood encoding scheme shown on the right: Trajectory information is encoded with LSTM encoders. A soft attention context vector $C^{s,k}_{t}$ is used to embed trajectory information from the pedestrian of interest, and a hardwired attention context vector $C^{h,k}_t$ is used for neighbouring trajectories.  In order to generate $C^{s,k}_{t}$ we use a soft attention function denoted $a_t$ in the above figure, and the hardwired weights are denoted by $w$. The merged context vector $C^{*,k}_{t}$ is then generated by merging $C^{s,k}_{t}$ and $C^{h,k}_t$.}\n  \\label{fig:neighbourhood_modelling}\n\\end{figure}\n\nLet the trajectory of the pedestrian $k$, from frame 1 to $T_{obs}$ be given by,\n\\begin{equation}\np^k = [p_1, \\ldots, p_{T_{obs}}],\n\\end{equation}\nwhere the trajectory is composed of points in a Cartesian grid. Then we pass each trajectory through an LSTM \\cite{hochreiter1997long} encoder to generate its hidden embeddings, \n\\begin{equation}\nh^k_t =  LSTM (p_t^k, h_{t-1}^k),\n\\label{eq:lstm_encoding}\n\\end{equation}\ngenerating a sequence of embeddings,\n\\begin{equation}\nh^k =   [h_1^k, \\ldots ,h_{T_{obs}}^k].\n\\end{equation}\n\nFollowing \\cite{fernando2017soft+}, the trajectory of the pedestrian of interest is embedded with soft attention such that, \n\\begin{equation}\nC^{s,k}_{t}=\\sum_{j=1}^{T_{obs}} \\alpha_{tj}h^k_j ,\n\\end{equation}\nwhich is the weighted sum of hidden states. The weight $\\alpha_{tj}$ is computed by,\n\\begin{equation}\n\\alpha_{tj}=\\cfrac{exp(e_{tj})}{\\sum_{l=1}^{T} exp(e_{tl})} , \n\\end{equation}\n\\begin{equation}\ne_{tj}=a(h^{k}_{t-1},h^k_j).\n\\end{equation}\nThe function $a$ is a feed forward neural network jointly trained with the other components. \n\nTo embed the effect of the neighbouring trajectories we use the hardwired attention context vector $C^{h,k}_t$ from \\cite{fernando2017soft+}. The hardwired weight $w$ is computed by, \n\\begin{equation}\nw^n_{j}=\\cfrac{1}{\\mathrm{dist}(n,j)},\n\\label{eq:hw_weight}\n\\end{equation}\nwhere $\\mathrm{dist}(n,j)$ is the distance between the $n^{th}$ neighbour and the pedestrian of interest at the $j^{th}$ time instant. Then we compute $C^{h,k}_t$ as the aggregation for all the neighbours such that,\n\\begin{equation}\nC^{h,k}_t=\\sum_{n=1}^{N}\\sum_{j=1}^{T_{obs}} w^n_{j}h^{n}_{j} ,\n\\end{equation}\nwhere there are $N$ neighbouring trajectories in the local neighbourhood, and $h^{n}_{j}$ is the encoded hidden state of the $n^{th}$ neighbour at the $j^{th}$ time instant. Finally we merge the soft attention and hardwired attention context vectors to represent the current neighbourhood context such that, \n\\begin{equation}\nC_{t}^{*,k}=\\mathrm{tanh}([C^{s,k}_{t}, C^{h,k}]).\n\\label{eq:context_vector}\n\\end{equation}\n\n\\subsection{Trajectory Prediction}\n\\label{sec:trajectory_gen}\nUnlike \\cite{fernando2017soft+}, we use a GAN to predict the future trajectory. There exists a minmax game between the generator (G) and the discriminator (D) guiding the model G to be closer to the ground truth distribution. The process is guided by learning a custom loss function which generates an additional advantage when modelling complex behaviours such as human navigation, where multiple factors such as human preferences and sociological factors influence behaviour. \n\nTrajectory prediction can be formulated as observing the trajectory from time 1 to $T_{obs}$, denoted as $[p_1, \\ldots, p_{T_{obs}}]$, and forecasting the future trajectory for time $T_{obs+1}$ to $T_{pred}$, denoted as $[y_{T_{obs+1}}, \\ldots, y_{T_{pred}}]$. The GAN learns a mapping from a noise vector $z$ to an output vector y, $G: z \\rightarrow y$ \\cite{fernando2018task}. Adding the notion of time, the output of the model $y_t$ can be written as $G: z_t \\rightarrow y_t$.\n\nWe augment the generic GAN mapping to be conditional on the current neighbourhood context $C_{t}^{*}$, $G: ({C_{t}^{*}, z_t}) \\rightarrow y_t$, such that the synthesised trajectories follow the social navigational rules that are dictated by the environment. \n\nThis objective can be written as,\n\\begin{equation}\nV=\\mathbb{E}_{y_t, C^*_t \\sim p_{data}}([ log D(C^*_t, y_t)]) + \\mathbb{E}_{C^*_t \\sim p_{data}, z_t \\sim noise}([ 1- log D(C^*_t, G(C^*_t, z_t))]). \n\\label{eq:gan}\n\\end{equation}\nOur final aim is to utilise the hidden state embeddings from the trajectory generator to discover the pedestrian groups via clustering those embeddings. Hence having a sparse feature vector for clustering is beneficial as they are more discriminative compared to their dense counterparts \\cite{figueroa2017learning}. Hence we augment the objective in Eq. \\ref{eq:gan} with a sparsity regulariser such that,\n\\begin{equation}\nL_1=|| f(G (C^{*}_{t}, z_t)) ||_1 ,\n\\label{eq:sparsity}\n\\end{equation}\nand \n\\begin{equation}\nV^*=V + \\lambda L_1 ,\n\\label{eq:gan_augmented}\n\\end{equation}\nwhere $f$ is a feature extraction function which extracts the hidden embeddings from the trajectory generator $G$, and $\\lambda$ is a weight vector which controls the tradeoff between the GAN objective and the sparsity constraint. \n\n\\begin{figure}[htb]\n  \\centering\n    \\includegraphics[width=\\linewidth]{figures\/GAN.pdf}\n   \\caption{Proposed trajectory prediction framework: The generator model $G$ samples from the noise distribution $z$ and synthesises a trajectory $y_t$, which is conditioned upon the local neighbourhood context $C_t^*$. The discriminator $D$ considers both $y_t$ and $C_t^*$ when classifying the authenticity of the trajectory.} \n  \\label{fig:GAN}\n\\end{figure}\n\nThe architecture of the proposed trajectory prediction framework is presented in Fig. \\ref{fig:GAN}. We utilise LSTMs as the Generator ($G$) and the Discriminator ($D$) models. $G$ samples from the noise distribution, $z$, and synthesises a trajectory for the pedestrian motion which is conditioned upon the local neighbourhood context, $C_t^*$, of that particular pedestrian. Utilising these predicted trajectories, $y_t$, and the context embeddings, $C_t^*$, $D$ tries to discriminate between the synthesised and ground truth human trajectories. \n\n\\subsection{Group Detection}\nFig. \\ref{fig:model} illustrates the proposed group detection framework. We pass each trajectory in the given scene through Eq. \\ref{eq:lstm_encoding} to Eq. \\ref{eq:context_vector} and generate the neighbourhood embeddings, $C_{t}^{*,k}$. Then using the feature extraction function $f$ we extract the hidden layer activations for each pedestrian $k$ such that, \n\\begin{equation}\n\\theta_{t}^{k}=f(G (C^{*,k}_{t}, z_t)) .\n\\label{eq:feature_extractor}\n\\end{equation}\n\nThen we pass the extracted feature vectors through a t-SNE \\cite{maaten2008visualizing} dimensionality reduction step. The authors in \\cite{figueroa2017learning} have shown that it is inefficient to cluster dense deep features. However they have shown the t-SNE algorithm to generate discriminative features capturing the salient aspects in each feature dimension.  Hence we apply t-SNE for the $k^{th}$ pedestrian in the scene such that, \n\\begin{equation}\n\\eta^{k}=\\text{t-SNE}( [\\theta^{k}_{1}, \\ldots, \\theta^{k}_{T_{obs}}]) .\n\\label{eq:t_SNE}\n\\end{equation}\n\nAs the final step we apply DBSCAN \\cite{ester1996density} to discover similar activation patterns, hence segmenting the pedestrian groups. DBSCAN enables us to cluster the data on the fly without specifying the number of clusters. The process can be written as,\n\\begin{equation}\n[\\beta^{1}, \\ldots, \\beta^{N} ]=\\mathrm{DBSCAN}( [\\eta^{1}, \\ldots, \\eta^{N} ]) ,\n\\label{eq:DBSCAN}\n\\end{equation}\nwhere there are $N$ pedestrians in the given scene and $\\beta^n \\in [\\beta^{1}, \\ldots, \\beta^{N} ]$ are the generated cluster identities.\n\n\\section{Evaluation and Discussion}\n\\subsection{Implementation Details}\nWhen encoding the neighbourhood information, similar to \\cite{fernando2017soft+}, we consider the closest 10 neighbours from each of the left, right, and front directions of the pedestrian of interest. If there are more than 10 neighbours in any direction, we take the closest 9 trajectories and the mean trajectory of the remaining neighbours. If a trajectory has less than 10 neighbours, we created dummy trajectories with hardwired weights (i.e Eq. \\ref{eq:hw_weight}) of 0, such that we always have 10 neighbours. \n\nFor all LSTMs, including LSTMs for neighbourhood modelling (i.e Sec. \\ref{sec:neighbourhood_modelling}), the trajectory generator and the discriminator (i.e Sec \\ref{sec:trajectory_gen}), we use a hidden state embedding size of 300 units. We trained the trajectory prediction framework iteratively, alternating between a generator epoch and a discriminator epoch with the Adam \\cite{kingma2014adam} optimiser, using a mini-batch size of 32 and a learning rate of 0.001 for 500 epochs. The hyper parameter $\\lambda = 0.2$, and the hyper parameters of DBSCAN, epsilon$=0.50$, minPts$=1$, are chosen experimentally.\n\n\\subsection{Evaluation of the Trajectory Prediction}\n\\label{sec:ped_eval}\n\\subsubsection{Datasets}\nWe evaluate the proposed trajectory predictor framework on the publicly available walking pedestrian dataset (BIWI) \\cite{pellegrini2009you}, Crowds By Examples (CBE) \\cite{lerner2007crowds} dataset and Vittorio Emanuele II  Gallery (VEIIG) dataset \\cite{bandini2014towards}. The BIWI dataset records two scenes, one outside a university (ETH) and one at a bus stop (Hotel). CBE records a single video stream with a medium density crowd outside a university (Student 003). The VEIIG dataset provides one video sequence from an overhead camera in the Vittorio Emanuele II Gallery (gall). The training, testing and validation splits for BIWI, CBE and VEIIG are taken from \\cite{pellegrini2009you}, \\cite{solera2013structured} and \\cite{solera2016socially} respectively. \n\nThese datasets include a variety of pedestrian social navigation scenarios including collisions, collision avoidance and group movements, hence presenting challenging settings for evaluation. Compared to BIWI which has low crowd densities, CBE and VEIIG contain higher crowd densities and as a result more challenging crowd behaviour arrangements, continuously varying from medium to high densities.\n\n\\subsubsection{Evaluation Metrics}\n\\label{sec:track_error_metrics}\nSimilar to \\cite{sadeghian2018sophie,gupta2018social} we evaluated the trajectory prediction performance with the following 2 error metrics: Average Displacement Error (ADE) and Final Displacement Error (FDE). Please refer to \\cite{sadeghian2018sophie,gupta2018social} for details. \n\n\\subsubsection{Baselines and Evaluation}\nWe compared our trajectory prediction model to 5 state-of-the-art baselines. As the first baseline we use the Social Force (SF) model introduced in \\cite{yamaguchi2011you}, where the destination direction is taken as an input to the model and we train a linear SVM model similar to \\cite{fernando2017soft+} to generate this input. We use the Social-LSTM (So-LSTM) model of \\cite{alahi2016social} as the next baseline and the neighbourhood size hyper-parameter is set to 32 px. We also compare to the Soft $+$ Hardwired Attention (SHA) model of \\cite{fernando2017soft+} and similar to the proposed model we set the embedding dimension to be 300 units and consider a 30 total neighbouring trajectories. We also considered the Social GAN (So-GAN) \\cite{gupta2018social} and attentive GAN (SoPhie) \\cite{sadeghian2018sophie} models. To provide fair comparisons we set the hidden state dimensions for the encoder and decoder models of So-GAN and SoPhie to be 300 units. For all models we observe the first 15 frames (i.e 1- $T_{obs}$) and predicted the future trajectory for the next 15 frames (i.e $T_{obs+1}$ - $T_{pred}$). \n\\begin{table}[htb]\n\\centering\n\\caption{Quantitative results for the BIWI \\cite{pellegrini2009you}, CBE \\cite{lerner2007crowds} and VEIIG \\cite{bandini2014towards} datasets. In all methods the forecast trajectories are of length 15 frames. Error metrics are as in Sec. \\ref{sec:track_error_metrics}. `-' refers to unavailability of that specific evaluation. The best values are denoted in bold.}\n\\label{tab:track_prediction}\n\\resizebox{0.980\\textwidth}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n          \nMetric & Dataset                          & SF \\cite{yamaguchi2011you}                            & So-LSTM \\cite{alahi2016social}                      & SHA \\cite{fernando2017soft+}                          & So-GAN \\cite{gupta2018social}  & SoPhie \\cite{sadeghian2018sophie}  & Proposed\\\\ \\hline\n                         & ETH (BIWI)                                         &        1.42                                         &                1.05                                 &          0.90                                       & 0.92                                            & 0.81        &     \\textbf{0.63}                               \\\\ \\cline{2-8} \n                         & Hotel (BIWI)                                       &         1.03                                        &                0.98                                 &         0.71                                        & 0.65                                            & 0.76               &     \\textbf{0.55}                        \\\\ \\cline{2-8} \n                          & Student 003 (CBE)                                  &    1.83                                             &           1.22                                      &        0.96                                         & -                                              & -         &       \\textbf{0.72}                              \\\\ \\cline{2-8} \n\\multirow{-4}{*}{ADE}    &  gall (VEIIG) & 1.72 & 1.14 & 0.91  & -  & -  & \\textbf{0.68}\\\\ \\hline \\hline\n                         & ETH (BIWI)                                         &        2.20                                         &           1.84                                      &          1.43                                       & 1.52                                            & 1.45             &     \\textbf{1.22}                          \\\\ \\cline{2-8} \n                         & Hotel (BIWI)                                       &        2.45                                         &             1.95                                    &             1.65                                    & 1.62                                            & 1.77              &        \\textbf{1.43}                      \\\\ \\cline{2-8} \n                       & Student 003 (CBE)                                  &     2.63                                            &           1.97                                      &           1.80                                      & -                                              & -          &        \\textbf{1.65}                            \\\\ \\cline{2-8}   \n\\multirow{-4}{*}{FDE}    & gall (VEIIG) & 2.55 &1.83  & 1.65  & - & - & \\textbf{1.45} \\\\ \\hline \n\\end{tabular}\n}\n\\end{table}\n\nWhen observing the results tabulated in Tab. \\ref{tab:track_prediction} we observe poor performance for the SF model due to it's lack of capacity to model history. Models So-LSTM and SHA utilise short term history from the pedestrian of interest and the local neighbourhood and generate improved predictions. However we observe a significant increase in performance from methods that optimise generic loss functions such as So-LSTM and SHA to GAN based methods such as So-GAN and SoPhie. This emphasises the need for task specific loss function learning in order to imitate complex human social navigation strategies. In the proposed method we further augment this performance by conditioning the trajectory generator on the proposed neighbourhood encoding mechanism. \n\nWe present a qualitative evaluation of the proposed trajectory generation framework with the SHA and So-GAN baselines in Fig. \\ref{fig:track_pred} (selected based on the availability of their implementations). The observed portion of the trajectory is denoted in green, the ground truth observations in blue and predicted trajectories are shown in red (proposed), yellow (SHA) and brown (So-GAN). Observing the qualitative results it can be clearly seen that the proposed model generates better predictions compared to the state-of-the-art considering the varying nature of the neighbourhood clutter. For instance in Fig. \\ref{fig:track_pred} (c) and (d) we observe significant deviations between the predictions for SHA and So-GAN and the ground truth. \nHowever the proposed model better anticipates the pedestrian motion with the improved context modelling and learning process. It should be noted that the proposed method has a better ability to anticipate stationary groups compared to the baselines, which is visible in Fig. \\ref{fig:track_pred} (c).\n\n\\begin{figure}[htb]\n  \\centering\n   \n    \\subfigure[]{\\includegraphics[width=4.5cm,height=3.0cm]{figures\/track_pred\/myplot_25_with_pred_1}}\n    \\subfigure[]{\\includegraphics[width=4.5cm,height=3.0cm]{figures\/track_pred\/myplot_25_with_pred_2}}\n    \\subfigure[]{\\includegraphics[width=4.5cm,height=3.0cm]{figures\/track_pred\/myplot_25_with_pred_3}}\n    \\subfigure[]{\\includegraphics[width=4.5cm,height=3.0cm]{figures\/track_pred\/myplot_25_with_pred_4}}\n   \\caption{Qualitative results for the proposed trajectory prediction framework for sequences from the CBE dataset. Given (in green), Ground Truth (in blue) and Predicted trajectories from proposed (in red), SHA model (in yellow) crom So-GAN (in brown). For visual clarity, we show only the trajectories for some of the pedestrians in the scene.}\n  \\label{fig:track_pred}\n\\end{figure}\n\n\\subsection{Evaluation of the Group Detection}\n\n\\subsubsection{Datasets}\nSimilar to Sec. \\ref{sec:ped_eval} we use the BIWI, CBE and VEIIG datasets in our evaluation. Dataset characteristics are reported in Tab. \\ref{tab:dataset_charateristics}. \n\n\\begin{table}[htb]\n\\centering\n\\caption{Dataset characteristics for different sequences in BIWI \\cite{pellegrini2009you}, CBE \\cite{lerner2007crowds} and VEIIG \\cite{bandini2014towards} datasets}\n\\label{tab:dataset_charateristics}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nDataset    & ETH (BIWI) & Hotel (BIWI) & Student-003 (CBE) & gall (VEIIG) \\\\ \\hline\nFrames     & 1448       & 1168         & 541               & 7500        \\\\ \\hline\nPedestrian & 360        & 390          & 434               & 630         \\\\ \\hline\nGroups     & 243        & 326          & 288               & 207         \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\subsubsection{Evaluation Metrics}\nOne popular measure of clustering accuracy is the pairwise loss $\\Delta_{pw}$ \\cite{zanotto2012online}, which is defined as the ratio between the number of pairs on which $\\beta$ and $\\hat{\\beta}$ disagree on their cluster membership and the number of all possible pairs of elements in the set. \n\nHowever as described in \\cite{solera2013structured,solera2016socially} $\\Delta_{pw}$ accounts only for positive intra-group relations and neglects singletons. Hence we also measure the Group-MITRE loss, $\\Delta_{GM}$, introduced in \\cite{solera2013structured}, which has overcome this deficiency. $\\Delta_{GM}$ adds a fake counterpart for singletons and each singleton is connected with it's counterpart. Therefore $\\delta_{GM}$ also takes singletons into consideration.\n\n\\subsubsection{Baselines and Evaluation}\n\\label{sec:group_detection_eval}\nWe compare the proposed Group Detection GAN (GD-GAN) framework against 5 recent state-of-the-art baselines, namely \\cite{shao2014scene,zanotto2012online,yamaguchi2011you,ge2012vision,solera2016socially}, selected based on their reported performance in public benchmarks. \n\nIn Tab. \\ref{tab:group_detection} we report the Precision $(P)$ and Recall $(R)$ values for $\\Delta_{pw}$ and $\\Delta_{GM}$ for the proposed method along with the state-of-the-art baselines. The proposed GD-GAN method has been able to achieve superior results, especially among unsupervised grouping methods. It should be noted that methods \\cite{solera2016socially,shao2014scene,zanotto2012online,yamaguchi2011you} utilise handcrafted features and use supervised learning to separate the groups. As noted in Sec. \\ref{sec:intro} these methods cannot adapt to scene variations and require hand labeled datasets for training. Furthermore we would like to point out that the supervised grouping mechanism in \\cite{solera2016socially} directly optimises $\\Delta_{GM}$. However, without such tedious annotation requirements and learning strategies, the proposed method has been able to generate commendable and consistent results in all considered datasets, especially in cluttered environments \\footnote{see the supplementary material for the results for using supervised learning to separate the groups on proposed context features.}. \n\nIn Fig. \\ref{fig:group_pred} we show groups detected by the proposed GD-GAN method for sequences from the CBE and VEIIG datasets. Regardless of the scene context, occlusions and the varying crowd densities, the proposed GD-GAN method generates acceptable results. We believe this is due to the augmented features that we derive through the automated deep feature learning process. These features account for both historical and future behaviour of the individual pedestrians, hence possessing an ability to detect groups even in the presence of occlusions such as in Fig. \\ref{fig:group_pred} (c). \n\n\\begin{table}[htb]\n\\centering\n\\caption{Comparative results on the BIWI \\cite{pellegrini2009you}, CBE \\cite{lerner2007crowds} and VEIIG \\cite{bandini2014towards} datasets using the $\\Delta_{GM}$ \\cite{solera2013structured} and $\\Delta_{PW}$ \\cite{zanotto2012online} metrics. `-' refers to unavailability of that specific evaluation. The best results are shown in bold and the second best results are underlined.}\n\\label{tab:group_detection}\n\\resizebox{1.0\\textwidth}{!}{\n\\begin{tabular}{|l|>{\\centering}m{1cm}|>{\\centering}m{1cm}|>{\\centering}m{1.25cm}|>{\\centering}m{1.25cm}|>{\\centering}m{1.36cm}|>{\\centering}m{1.36cm}|>{\\centering}m{1.2cm}|>{\\centering}m{1.2cm}|>{\\centering}m{1.2cm}|>{\\centering}m{1.2cm}|>{\\centering}m{1.2cm}|m{1.2cm}|}\n\\hline\n\\multicolumn{1}{|c|}{}                                                           & \\multicolumn{2}{c|}{Shao et. al \\cite{shao2014scene}}                                  & \\multicolumn{2}{c|}{zanotto et. al \\cite{zanotto2012online}}                               & \\multicolumn{2}{c|}{ Yamaguchi et. al \\cite{yamaguchi2011you}}                                  & \\multicolumn{2}{c|}{Ge et. al \\cite{ge2012vision}} & \\multicolumn{2}{c|}{Solera et al. \\cite{solera2016socially}} & \\multicolumn{2}{c|}{GD-GAN} \\\\ \\cline{2-13} \n\\multicolumn{1}{|c|}{\\multirow{-2}{*}{}}                                         & $P$                                                 & $R$    & $P$                                              & $R$    & $P $                                                & $R $   & $P$          & $R$         & $P $          & $R$           & $P$            & \\hspace{.32cm} $R   $         \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \nBIWI  \\hspace*{1.2cm} $\\Delta_{GM}$ & 67.3 & 64.1 & - & -    & 84.0 & 51.2 & 89.2       & 90.9      & \\underline{97.3}        & \\textbf{97.7}        &   \\textbf{97.5}         &\\hspace{.2cm} \\textbf{97.7}            \\\\\nHotel   \\hspace*{1.2cm}  $\\Delta_{PW}$                        & 51.5                         & 90.4 & 81.0                      & 91.0  &83.7                         & \\textbf{93.9} & 88.9       & 89.3      & \\underline{89.1}        & 91.9        &  \\textbf{90.2}           &\\hspace{.2cm} \\underline{93.1}            \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \nBIWI \\hspace*{1.2cm} $\\Delta_{GM}$ & 69.3 & 68.2 & - & -    & 60.6 & 76.4 & 87.0       & 84.2      & \\underline{91.8}        & \\textbf{94.2}        &    \\textbf{92.5}          &   \\hspace{.2cm} \\textbf{94.2}          \\\\\nETH \\hspace*{1.2cm} $\\Delta_{PW}$                                                & 44.5                                              & 87.0 & 79.0                                           & 82.0 & 72.9                                              & 78.0 & 80.7       & 80.7      & \\underline{91.1}        & \\underline{83.4}        &   \\textbf{91.3}           &  \\hspace{.2cm} \\textbf{83.5}            \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \nCEB \\hspace*{1.4cm}$\\Delta_{GM}$                                                & 40.4                                              & 48.6 & -                                              & -    & 56.7                                              & 76.0 & 77.2       & 73.6      & \\textbf{81.7}        & \\textbf{82.5}        &      \\underline{81.0}         &  \\hspace{.2cm} \\underline{81.8}            \\\\\nStudent-003 \\hspace*{0.4cm}$\\Delta_{PW}$                                        & 10.6                                              & \\textbf{76.0} & 70.0                                           & 74.0 & 63.9                                              & 72.6 & 72.2       & 65.1      & \\textbf{82.3}        & \\underline{74.1}        &   \\underline{82.1}           &       \\hspace{.2cm} 63.4       \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \nVEIIG \\hspace*{1.1cm}$\\Delta_{GM}$                                              &                        -                           &     - &                -                                &  -    &             -                                      &   -   &     -       &        -   &  \\textbf{84.1}           &   \\textbf{84.1}          & \\underline{83.1}             &    \\hspace{.2cm} \\underline{79.5}          \\\\\ngall \\hspace*{1.5cm}$\\Delta_{PW}$                                               &          -                                         &  -    &               -                                 &  -    &               -                                    &  -    &      -      &   -        &     \\textbf{79.7}        &   \\textbf{77.5}          &   \\underline{77.6}           &  \\hspace{.2cm}  \\underline{73.1}           \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n \n\\begin{figure}[htb]\n  \\centering\n    \\subfigure[GVEII - Frame 2127]{\\includegraphics[width=.4\\linewidth]{figures\/groups\/002127.pdf}}\n    \\subfigure[GVEII- Frame 2320]{\\includegraphics[width=.4\\linewidth]{figures\/groups\/002320.pdf}}\n    \\subfigure[CBE - Frame 2603]{\\includegraphics[width=.4\\linewidth]{figures\/groups\/stu003_002603.pdf}}\n    \\subfigure[CBE - Frame 2910]{\\includegraphics[width=.4\\linewidth]{figures\/groups\/stu003_002910.pdf}}\n   \\caption{Qualitative results from the proposed GD-GAN methods for sequences from the CBE and GVEII datasets. Connected coloured blobs indicate groups of pedestrians.}\n  \\label{fig:group_pred}\n\\end{figure}\n\nWe selected the first 30 pedestrian trajectories from the VEIIG test set and in Fig. \\ref{fig:embedding_shift} we visualise the embedding space positions before (in blue) and after (in red) training of the proposed trajectory generator (G). Similar to \\cite{aubakirova2016interpreting} we extracted the activations using the feature extractor function $f$ and applied PCA \\cite{wold1987principal} to plot them in 2D. The respective ground truth group IDs are indicated in brackets. This helps us to gain an insight into the encoding process that $G$ utilises, which allows us to discover groups of pedestrians. Considering the examples given, it can be seen that trajectories from the same cluster become more tightly grouped. This is due to the model incorporating source positions, heading direction, trajectory similarity, when embedding trajectories, allowing us to extract pedestrian groups in an unsupervised manner. \n\n\\begin{figure}[htb]\n  \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/embedding_shift.pdf}\n   \\caption{Projections of the trajectory generator (G) hidden states before (in blue) and after (in red) training. Ground truth group IDs are in brackets. Each insert indicates the trajectory associated with the embedding. The given portion of the trajectory is in green, and the ground truth and prediction are in blue and red respectively}\n  \\label{fig:embedding_shift}\n\\end{figure}\n\n\n\\subsection{Ablation Experiment}\nTo further demonstrate the proposed group detection approach, we conducted a series of ablation experiments identifying the crucial components of the proposed methodology~\\footnote{see the supplementary material for an ablation study for the trajectory prediction}. In the same setting as the previous experiment we compare the proposed GD-GAN model against a series of counter parts as follows:  \n\n\\begin{itemize}\n\\item GD-GAN \/ GAN: removes $D$ and the model $G$ is learnt through supervised learning as in \\cite{fernando2017soft+}. \n\\item GD-GAN \/ cGAN: optimises the generic GAN objective defined in \\cite{goodfellow2014generative}.  \n\\item GD-GAN \/ $L_1$: removes sparsity regularisation and optimises Eq. \\ref{eq:gan}.\n\\item GD-GAN + hf: utilises features from $G$ as well as the handcrafted features defined in \\cite{solera2016socially} for clustering.\n\\end{itemize}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Ablation experiment evaluations}\n\\label{tab:ablation_experiment}\n\\resizebox{1.0\\textwidth}{!}{\n\\begin{tabular}{| l |>{\\centering}m{1.5cm}| >{\\centering}m{1.5cm} |>{\\centering}m{1.5cm}| >{\\centering}m{1.5cm} |>{\\centering}m{1.5cm}| >{\\centering}m{1.5cm} |>{\\centering}m{1.5cm}| >{\\centering}m{1.5cm} |>{\\centering}m{1.5cm}| m{1.5cm} |}\n\\hline\n                                          & \\multicolumn{2}{>{\\centering}m{3cm} |}{GD-GAN \/ GAN} & \\multicolumn{2}{>{\\centering}m{3cm} |}{GD-GAN \/ cGAN} & \\multicolumn{2}{>{\\centering}m{3cm} |}{GD-GAN \/ $L_1$} & \\multicolumn{2}{>{\\centering}m{3cm} |}{GD-GAN + hf} & \\multicolumn{2}{>{\\centering}m{3cm} |}{GD-GAN} \\\\ \\cline{2-11} \n\\multirow{-2}{*}{}                        & P               & R               & P                & R               & P                & R                & P               & R              & P            & \\hspace{.6cm}R            \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \nCEB \\hspace{0.98 cm} $\\Delta_{GM}$         & 73.6            & 75.1            & 76.7             & 76.2            & 77.3             & 78.0             & \\textbf{79.0}            & \\textbf{79.2}           & 78.7         & \\hspace{.5cm}\\textbf{79.2}         \\\\\nStudent-003 $\\Delta_{PW}$ & 74.1            & 52.8            & 75.5             & 60.2            & 78.1             & 65.1             & \\textbf{80.4}            & 68.0           & \\textbf{80.4}         & \\hspace{.5cm}\\textbf{68.4}         \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\nThe results of our ablation experiment are presented in Tab. \\ref{tab:ablation_experiment}. Model GD-GAN \/ GAN performs poorly due to the deficiencies in the supervised learning process. It optimises a generic mean square error loss, which is not ideal to guide the model through the learning process when modelling a complex behaviour such as human navigation. Therefore the resultant feature vectors do not capture the full context which contributes to the poor group detection accuracies. We observe an improvement in performance with GD-GAN \/ cGAN due to the GAN learning process which is further augmented and improved through GD-GAN \/ $L_1$ where the model learns a conditional behaviour depending on the neighbourhood context. $L_1$ regularisation further assists the group detection process via making the learnt feature distribution more discriminative. \n\nIn order to demonstrate the credibility of the learnt group attributes from the proposed GD-GAN model, we augment the feature vector extracted in Eq. \\ref{eq:feature_extractor} together with the features proposed in \\cite{solera2016socially} and apply subsequent process (i.e Eq. \\ref{eq:t_SNE} and \\ref{eq:DBSCAN}) to discover the groups. We utilise the public implementation \\footnote{https:\/\/github.com\/francescosolera\/group-detection} released by the authors for the feature extraction.\n\nWe do not observe a substantial improvement with the group detection performance being very similar, indicating that the proposed GD-GAN model is sufficient for modelling the social navigation structure of the crowd. \n\n\\subsection{Time efficiency}\nWe use the Keras \\cite{chollet2017keras} deep learning library for our implementation. The GD-GAN module does not require any special hardware such as GPUs to run and has 41.8K trainable parameters. We ran the test set in Sec. \\ref{sec:group_detection_eval} on a single core of an Intel Xeon E5-2680 2.50GHz CPU and the GD-GAN algorithm was able to generate 100 predicted trajectories with 30, 2 dimensional data points in each trajectory (i.e. using 15 observations to predict the next 15 data points) and complete the group detection process in 0.712 seconds.\n\n\\section{Conclusions}\nIn this paper we have proposed an unsupervised learning approach for pedestrian group segmentation. We avoid the the need to handcraft sociological features by automatically learning group attributes through the proposed trajectory prediction framework. This allows us to discover a latent representation accounting for both historical and future behaviour of each pedestrian, yielding a more efficient platform for detecting their social identities. Furthermore, the unsupervised learning setting grants the approach the ability to employ the proposed framework in different surveillance settings without tedious learning of group memberships from a hand labeled dataset. Our quantitative and qualitative evaluations on multiple public benchmarks clearly emphasise the capacity of the proposed GD-GAN method to learn complex real world human navigation behaviour. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}          \n\\label{sect:intro}\n\nCurrently, a significant part of the orbital solutions found from speckle interferometric data for binaries is obtained using a small number of measurements. This may be due to the lack of observations for a particular system, as well as to large orbital periods of such objects. An additional factor is the absence or inaccurate determination of the positions of the secondaries, which is difficult to identify due to a lack of observational data. Therefore orbital parameters of objects and mass sums could be determined incorrectly. One of the solutions to this problem is the long-term monitoring of binaries, which allows for covering big part of the orbit with measurements (it depends on the orbital period). In this paper, we studied the binary HIP~53731 (HD~95175, $V_{mag}=8.85^{m}$ \\citep{bid85}). The system under study, according to \\citet{cve16}, consists of two main sequence stars of spectral types K0 and K9 with masses $\\mathfrak{M}_{A} = 0.9~ \\mathfrak{M}_{\\odot}$ and $\\mathfrak{M}_{B} = 0.48~ \\mathfrak{M}_{\\odot}$, and the total mass of the system and dynamical parallax are $\\mathfrak{M}_{tot} = 1.72~ \\pm 0.4 \\mathfrak{M}_{\\odot}$ and $\\pi_{dyn} = 29.33 \\pm 3.03$ mas. Positional parameters published earlier and obtained in this study are presented in Table \\ref{tab1}. Speckle interferometric observations and the image reduction procedure are discussed in Section \\ref{sect:Obs}, Section \\ref{sect:orbit} is dedicated to the process of orbit construction and determination of the fundamental parameters of HIP~53731, results are discussed in Section \\ref{sect:Dis}.\n\n\\section{Observations and data reduction}\n\\label{sect:Obs}\n\nSpeckle interferometric observations of HIP~53731 were carried out at the Big Telescope Alt-azimuth (BTA) of the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS) from 2007 to 2020 using a speckle interferometer \\citep{maks09} based on EMCCD detectors PhotonMAX-512B (until 2010), Andor iXon+ X-3974 (2010-2014) and Andor iXon Ultra 897 (since 2015). Speckle images were obtained under good weather conditions with seeing about 1\\arcsec-2\\arcsec. Speckle interferograms were recorded with exposure time of 20 milliseconds, the standard series consisted of 1940 (until 2010) and 2000 images. Following interference filters were used (central wavelength $\\lambda$ \/ bandpass $\\Delta\\lambda$): 550\/20, 600\/40 and 800\/100 nm. \n\nPositional parameters and magnitude differences were determined on the basis of the analysis of the power spectrum and the autocorrelation function of the speckle interferometric series described in \\citet{bali02} and \\citet{pluz05}. The reconstruction of the position of the secondary was carried out by the bispectrum method \\citep{lohm83}. The log of observations, positional parameters and the magnitude differences are presented in Table \\ref{tab1}: epoch of observations in fractions of the Besselian year; telescope; $\\lambda$\/$\\Delta\\lambda$; $\\theta$ is the position angle; $\\rho$ is the separation between the two stars; $\\Delta m$ is the magnitude difference and references. The formal errors of $\\Delta m$ corresponding to the method of model selection are presented in Table \\ref{tab1}. Wherein, an analysis of the magnitude differences from the data obtained in different epochs shows that the actual measurement accuracy is about 0.1 mag.\n\n\\begin{table*}\n\t\\begin{center}\n\t\\caption[]{Positional Parameters and Magnitude Differences.}\\label{tab1}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\\hline\\noalign{\\smallskip}\nEpoch & Telescope & $\\lambda$\/$\\Delta\\lambda$, nm & $\\theta^{\\circ}$ & $\\rho$, mas & $\\Delta m$, mag & Reference \\\\\n\t\\hline\\noalign{\\smallskip}\n1991.25 & \\textit{Hipparcos} &  & 291.0 & 287.0 &  & \\citet{hip} \\\\\n2000.1460 & 3.5-m WIYN & 648\/41 & $275.8 \\pm 1$ & $184 \\pm 3$ &  & \\citet{hor02} \\\\\n2001.2733 & BTA & 600\/30 & $268.9 \\pm 0.6$ & $177 \\pm 4$ &  & \\citet{bali06} \\\\\n2001.2733 & BTA & 750\/35 & $268.9 \\pm 0.7$ & $178 \\pm 4$ &  & \\citet{bali06} \\\\\n2002.2542 & BTA & 750\/35 & $260.0 \\pm 0.6$ & $144 \\pm 2$ &  & \\citet{bali13} \\\\\n2005.2323 & BTA & 800\/110 & $118.8 \\pm 1.1$ & $139 \\pm 3$ &  & \\citet{bali13} \\\\\n2006.3745 & BTA & 545\/30 & $107.8 \\pm 1.2$ & $180 \\pm 4$ &  & \\citet{bali13} \\\\\n2007.9019 & BTA & 600\/30 & $277.8 \\pm 0.1$ & $191 \\pm 1$ & $2.05 \\pm 0.01$ & this work \\\\\n2008.9559 & BTA & 550\/20 & $270.4 \\pm 1$ & $175 \\pm 1$ & $2.55 \\pm 0.02$ & this work \\\\\n2009.0954 & BTA & 600\/40 & $268.7 \\pm 0.1$ & $175 \\pm 1$ & $2.29 \\pm 0.01$ & this work \\\\\n2009.2645 & BTA & 600\/40 & $267.7 \\pm 0.1$ & $169 \\pm 1$ & $1.64 \\pm 0.01$ & this work \\\\\n2010.1601 & BTA & 800\/100 & $259.3 \\pm 0.1$ & $141 \\pm 1$ & $1.45 \\pm 0.01$ & this work \\\\\n2010.3416 & 2.1-m OAN & 630\/120 & $55.5 \\pm 12.9$ & $160 \\pm 30$ &  & \\citet{orl15} \\\\\n2011.1351 & BTA & 800\/100 & $240.1 \\pm 0.3$ & $83 \\pm 1$ & $1.53 \\pm 0.02$ & this work \\\\\n2011.9486 & BTA & 800\/100 & $343.9 \\pm 0.2$ & $34 \\pm 1$ & $1.51 \\pm 0.01$ & this work \\\\\n2013.3221 & BTA & 550\/20 & $296.4 \\pm 0.1$ & $150 \\pm 1$ & $2.24 \\pm 0.01$ & this work \\\\\n2014.1193 & BTA & 800\/100 & $289.5 \\pm 0.1$ & $179 \\pm 1$ & $1.46 \\pm 0.01$ & this work \\\\\n2014.9301 & BTA & 550\/20 & $283.4 \\pm 0.2$ & $188 \\pm 1$ & $2.11 \\pm 0.01$ & this work \\\\\n2015.9703 & BTA & 800\/100 & $274.8 \\pm 0.1$ & $189 \\pm 1$ & $1.55 \\pm 0.01$ & this work \\\\\n2016.1331 & 4.1-m SOAR & 788\/132 & $274.0 \\pm 0.3$ & $193.5 \\pm 0.8$ & 1.5 & \\citet{tok18} \\\\ \n2016.8847 & BTA & 800\/100 & $268.8 \\pm 0.1$ & $174 \\pm 1$ & $1.56 \\pm 0.01$ & this work \\\\\n2017.9227 & BTA & 800\/100 & $258.9 \\pm 0.1$ & $141 \\pm 1$ & $1.46 \\pm 0.01$ & this work \\\\\n2017.9227 & BTA & 800\/100 & $259.0 \\pm 0.1$ & $141 \\pm 1$ & $1.46 \\pm 0.01$ & this work \\\\\n2017.9227 & BTA & 800\/100 & $258.9 \\pm 0.1$ & $141 \\pm 1$ & $1.47 \\pm 0.01$ & this work \\\\\n2017.9227 & BTA & 800\/100 & $258.9 \\pm 0.1$ & $142 \\pm 1$ & $1.5 \\pm 0.01$ & this work \\\\\n2018.1811 & 4.1-m SOAR & 824\/170 & $255.2 \\pm 0.8$ & $134.5 \\pm 0.8$ & 1.5 & \\citet{tok19} \\\\\n2018.3214 & BTA & 800\/100 & $253.7 \\pm 0.1$ & $124 \\pm 1$ & $1.49 \\pm 0.01$ & this work \\\\\n2019.0476 & BTA & 800\/100 & $235.4 \\pm 0.1$ & $76 \\pm 1$ & $1.47 \\pm 0.02$ & this work \\\\\n2019.2744 & BTA & 550\/20 & $222.4 \\pm 0.1$ & $61 \\pm 1$ & $1.9 \\pm 0.04$ & this work \\\\\n2019.2744 & BTA & 800\/100 & $221 \\pm 0.2$ & $64 \\pm 1$ & $1.27 \\pm 0.04$ & this work \\\\\n2020.3611 & BTA & 550\/20 & $306 \\pm 0.1$ & $103 \\pm 1$ & $2.13 \\pm 0.01$ & this work \\\\\n\t\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\section{Orbit Construction}\n\\label{sect:orbit}\n\nPreliminary estimates of the orbital parameters were calculated using the Monet method \\citep{mon77}. The final orbit was constructed using the ORBIT software package \\citep{tok92}. Depending on the values of residuals and deviations from the orbital solution, the corresponding weights were selected for each measurement. Measurements by \\citet{hip}, \\citet{hor02} and \\citet{orl15} have the largest residuals (as it is shown in Figure \\ref{fig1}), consequently less weight was set to them.\n\nWhen constructing the orbit of HIP~53731, ambiguities in the positions of published measurements were found, which is probably due to the incorrect reconstruction of the position of the secondary or its absence. As a result, the position angles of the following measurements were changed by $\\pm 180^\\circ$: 2005.2323 and 2006.3745 \\citep{bali13} and 2010.3416 \\citep{orl15}. Two orbits of HIP~53731 constructed by \\citet{tok19orb} and in this study are presented in Figure \\ref{fig1}. The measurement residuals with respect to the new orbital solution are $4^\\circ$ by $\\theta$ and 18 mas by $\\rho$. However, they were overestimated due to the significant contribution of points that obviously don't match to the model solution (in Figure \\ref{fig1}, these are marked with crosses). Real estimates of residuals for $\\rho$ and $\\theta$ are 2 mas and $0.8^\\circ$, respectively.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=7.5cm, angle=0]{53731.eps}\n\t\\caption{Orbital solutions for HIP~53731. The orbit by \\citet{tok19orb} is marked with gray and the orbit constructed in this work is black. Triangles correspond to the published data; open circles - data obtained in this study; crosses - data with large residuals; the cross placed in a large circle is the first measurement for system. The arrow shows the direction of motion of the secondary. $\\Delta$ are residuals showing the angular distance between the observed and modelled value. The dashed line on the residuals plot indicates the orbital solution.}\n\t\\label{fig1}\n\t\\end{figure}\n\nTable \\ref{tab2} presents our measurements of the orbital parameters of the system and previously published ones. The columns are: the orbital period, the epoch of passing the periastron, the eccentricity of the orbit, the semimajor axis, the longitude of the ascending node, the argument of the periastron, the inclination of the orbit and the references to the publications.\n\n\\begin{table*}\n\\begin{center}\n\\caption[]{Orbital Parameters of HIP~53731.}\\label{tab2}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n  \\hline\\noalign{\\smallskip}\n$P_{orb}$, year & $T_{0}$, year & $e$ & $a$, mas & $\\Omega$, $^\\circ$ &  $\\omega$, $^\\circ$ & $i$, $^\\circ$ & Reference \\\\\n\\hline\\noalign{\\smallskip}\n16.244 & 2003.412 & 0.150 & 202.6 & 95.3 & 256.6 & 115.8 & \\citet{cve16} \\\\ \n$\\pm 0.682$ & $\\pm 0.682$ & $\\pm 0.041$ & $\\pm 10.9$ & $\\pm 2.8$ & $\\pm 15.8$ & $\\pm 1.6$ & \\\\ \n\\hline\\noalign{\\smallskip}\n15.83 & 2000.92 & 0.095 & 214 & 95.1 & 192.2 & 105.8 & \\citet{tok19orb} \\\\\n\\hline\\noalign{\\smallskip}\n7.83 & 2004.00 & 0.886 & 123.6 & 151 & 61 & 140.1 & this \\\\\n$\\pm 0.01$& $\\pm 0.01$ & $\\pm 0.004$ & $\\pm 1.8$ & $\\pm 3$ & $\\pm 3$ & $\\pm 1.4$ & work \\\\\n  \\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*} \n\nA comparison of orbital solutions shows, that the orbit of HIP~53731 constructed in this work fits the observational data much better, than the orbits by \\citet{cve16} and by \\citet{tok19orb}, obtained from a small amount of observational data. Also, mass sum, absolute magnitudes of components, their spectral types and masses were determined using two independent methods and are presented in Table \\ref{tab3}. Both \\textit{Hipparcos} \\citep{hippi} and \\textit{Gaia} \\citep{gapi} parallaxes and orbital parameters were used in the first method. This method allows for calculation of mass sum via the \\textit{Kepler's} law:\n\n\\begin{equation}\n\\sum \\mathfrak{M}=\\frac{(a\/\\pi)^{3}}{P_{orb}^{2}},\n\\label{eq1}\n\\end{equation}\n\nand the uncertainty is calculated using\n\n\\begin{equation}\n\\begin{gathered}\n\\sigma(\\mathfrak{M})=\\\\\n\\sqrt{\\frac{9(\\sigma_{\\pi})^2}{\\pi^2}+\\frac{9(\\sigma_{a})^2}{a^2}+\\frac{4(\\sigma_{P_{orb}})^2}{P_{orb}^2}}*\\mathfrak{M}.\n\\end{gathered}\n\\label{eq2}\n\\end{equation}\n\nThe second method allows for obtaining of the masses of components via the Pogson's relation. The magnitude of the object in the V band \\citep{bid85} and the average magnitude difference ($\\Delta m_{550} = 2.19^{m} \\pm 0.10^{m}$ from Table \\ref{tab1}) together with parallaxes ($\\pi_{\\textit{Hip}} = 26.35 \\pm 1.29$ mas and $\\pi_{\\textit{Gaia}} = 31.0803 \\pm 0.6137$ mas) were used. The work by \\citet{pec13} was applied to match the calculated absolute magnitudes of the components with spectral types and masses.\n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\caption[]{Comparison of Fundamental Parameters.}\\label{tab3}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\\noalign{\\smallskip}\n\t& Parallax & $M_{V,A}$, m & $Sp_{A}$ & $\\mathfrak{M}_{A}$, $\\mathfrak{M}_{\\odot}$ & $M_{V,B}$, m &  $Sp_{B}$ & $\\mathfrak{M}_{A}$, $\\mathfrak{M}_{\\odot}$ & $\\sum \\mathfrak{M}$, $\\mathfrak{M}_{\\odot}$ \\\\\n\t\\hline\\noalign{\\smallskip}\n\\citet{cve16} & \\textit{Hipparcos} & $5.99 \\pm 0.12$ & K0 & 0.90 & $8.57 \\pm 0.77$ & K9 & 0.48 & $1.72 \\pm 0.40$ \\\\ \n\t\\hline\\noalign{\\smallskip}\nThis & \\textit{Hipparcos} & $6.11 \\pm 0.10$ & K2 & 0.78 & $8.30 \\pm 0.14$ & K7 & 0.63 & $1.68 \\pm 0.26$ \\\\ \\cline{2-9}\nwork & \\textit{Gaia} & $6.47 \\pm 0.10$ & K3 & 0.75 & $8.66 \\pm 0.14$ & K9 & 0.56 & $1.03 \\pm 0.07$ \\\\\n  \\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\section{Discussion}\n\\label{sect:Dis}\nAn analysis of speckle interferometric data obtained at the 6-m telescope of the SAO RAS from 2007 to 2020 made it possible to halve the previously known value of the orbital period of HIP~53731. It should be noted that the residuals of positional parameters are small, which indicates the high-precision of the orbit. This fact indicates a high accuracy of new orbital parameters and the justification for long-term monitoring of such objects carried out in the group of high-resolution methods in astronomy of the SAO RAS. As a result, the orbital solutions by \\citet{cve16} and by \\citet{tok19orb} are very different from one, presented in this work, because they were obtained using small number of measurements, some of which have $\\pm 180^\\circ$ ambiguities. \n\nThe mass sum of the HIP~53731 components was determined with an accuracy of 15\\% (using the \\textit{Hipparcos} parallax) and 8\\% (using the \\textit{Gaia} parallax). The masses of the components obtained by the second method in this study are consistent with the mass sums calculated by the first method. The values obtained using \\textit{Hipparcos} parallax agree better with each other. The masses obtained using \\textit{Gaia} parallax in this work are less consistent with each other. The reason is probably \\textit{Gaia} parallax, so we are looking forward to the new data release of this mission. The proximity of the new parameters to the previous values is explained by the fact that \\citet{cve16} used the magnitude difference from the \\textit{Hipparcos} catalog together with the table of star parameters from the book by \\citet{gra05}, as well as erroneous values of both the orbital period and the semimajor axis. \n\nThe classification of the obtained orbital solutions was carried out using the qualitative grade of \\citet{wor83}. The orbit of HIP~53731 is ''good'' (Grade 2) - the observations correspond to different phases and cover more than half of the orbital period, which allows for fitting of orbit accurately enough.\n\n\\begin{acknowledgements}\nThe reported study was funded by RFBR, project number 20-32-70120. The work was performed as part of the government contract of the SAO RAS approved by the Ministry of Science and Higher Education of the Russian Federation. This work has made use of data from the European Space Agency (ESA) mission \\textit{Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the \\textit{Gaia} Data Processing and Analysis Consortium (DPAC, \\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the \\textit{Gaia} Multilateral Agreement. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.  \n\\end{acknowledgements}\n\n\n\\bibliographystyle{raa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAll turbulent flows are characterized by spatially and temporally chaotic evolutions on a wide range of scales and frequencies \\cite{Frisch95}.  As a result,  direct numerical simulations (DNS) approaches are still not practical to study many turbulent flows occurring\nin nature and in engineering applications.  The control parameter is given by the Reynolds number, $Re_L = UL\/\\nu$ a dimensionless measure of the relative importance of advective and viscous terms in the Navier-Stokes equations (NSE),\nwhere\n$U$ denotes\nthe rms velocity fluctuations at the energy injection scale, $L$.  It is possible to estimate  that in   homogeneous and isotropic turbulent flows the number of active degrees of freedom grows as $ Re_L^{9\/4}$ \\ml{\\cite{Landau59}}, leading to extremely demanding numerical resources already for moderate turbulent intensities.\\\\  \nTo overcome the problem, numerical tools based on a modeling of small-scale turbulent fluctuations are often introduced, and called  large eddy simulations (LES). \nThis technique is based on filtering out the small-scale interactions\nand replacing them with subgrid-scale (SGS)\nmodels~\\cite{Piomelli99,meneveau2000,pope2001}.\\\\ \nThe demand for LES is\nincreasing for magnetohydrodynamic (MHD) problems, too, as e.g. in  \nheliophysical and astrophysical applications~\\cite{miesch2015} \n\\ml{and in the context of liquid metal MHD. Furthermore, the LES technique \nis a useful alternative to spectral approaches in theoretical analyses of \ninterscale energy transfer \\cite{Aluie17}, in particular with a view towards\napplications in wall-bounded (i.e. liquid metal) flows}.\n\\ml{In MHD-LES,} the small-scale nonlinear magnetic interactions and the  \nvelocity\/magnetic correlations have to be replaced with SGS models, too.\nThis introduces additional complexity to the MHD-LES method \\cite{Kessar16,Aluie17}, leading to \ndifferent modeling approaches\n\\cite{Zhou91,Theobald94,Agullo01,Mueller02a,muller2002,Knaepen04,haugen2006,\nBaerenzung08,Balarac10,Chernyshov10}. \n\\lb{As in LES of nonconducting fluids, the success of a given  model is usually assessed in terms of reproducing mean profiles of large scale quantities. }\n\\mb{However, it is more and more clear that SGS velocity fluctuations are characterized by extreme events with magnitudes comparable to that of the large-scale velocity root mean squares. Departure from Gaussian distribution becomes larger and larger by decreasing the scales where velocity and\/or magnetic fluctuations are evaluated, a phenomenon known as {\\it intermittency} \\cite{Frisch95,Biskamp03,Verma04}. As a result, due to their statistical relevance and intensity, extreme events cannot be neglected when modeling SGS dynamics \\cite{cerutti1998intermittency,Buzzicotti18a}. Intermittency and anomalous scaling have attracted the attention of several studies on  MHD turbulence \\cite{Politano95,Servidio09,Mininni09,Sahoo11,Yoshimatsu11,Imazio13,Meyrand15,Yang17}, with particular interest in high Reynolds number astrophysical applications, e.g. solar wind \\cite{Veltri99,Salem09,Wan12,Matthaeus15}. }\nThe development of SGS models which are sophisticated enough to capture extreme events, \nand therefore provide a more faithful representation of turbulent dynamics, requires\na detailed analysis of  SGS quantities. {\\em A-priori} studies of \nDNS data \nprovide a first test-bed from where to extract the necessary information. The aim is to  \n analyse the SGS correlations of the original fields\nand understand what the key features are that must be modeled.  To our knowledge, there are very few {\\em a-priori} studies \nfor the MHD-LES formulation \\cite{Balarac10,Kessar16,Grete16}\\ml{\\cite{Grete15}}, all of which  concerning  statistically stationary nonlinear dynamos and without any focus on\nintermittency.  \nThe aim of this paper is to analyse the SGS properties of a MHD turbulent flow\nat different temporal instants during the evolution of a small-scale dynamo\nsuch as to be able to assess both regimes, when the magnetic field is passively\nadvected by, or  actively reacting on, the velocity field. In particular, we\nperform a systematic analysis  of the different components of the SGS total\nenergy transfer. We first split it in two  sub-channels, involving velocity or\nmagnetic temporal dynamics only and we analyse the mutual scale-by-scale energy\nexchanges. Second, we further decompose \nthe kinetic SGS energy component into two contributions, one  \ncoming from the advection and one from the Lorentz force.\\\\\nFurthermore, we also \\ml{apply a}\nformulation of the filtered fields,\nbased on an exact projection on a finite number of Fourier modes \n(P-LES) \\cite{Buzzicotti18a} that disentangles the signal due to the  \ncoupling between resolved and unresolved scales from that\ndue to interactions between resolved fields only. \\\\\n     \nThe main results of this study are: \\\\\n\\ml{ (i) The SGS energy transfer shows some degree of intermittency in all\nevolutionary stages of the dynamo. Its component coming from the Lorentz force\nbecomes successively more intermittent while that originating from\nhydrodynamics shows decreased intermittency. \\\\ (ii) In terms of guidance for\nLES modelling, we find that dissipative models should be well suited for the\nSGS stresses connected with the Lorentz force, while not being suitable for\nthose coming from purely inertial dynamics.  }\n\nThis paper is organized as follows: We begin in section \\ref{sec:dataset} \nwith a description of\nthe DNS dataset.  In section \\ref{sec:theory}, we introduce the P-LES formulation for MHD.\nSection \\ref{sec:apriori} presents the results from the {\\em a-priori} analysis of the\nstatistical properties of the SGS energy transfers. We summarize and discuss\nour results in section \\ref{sec:conclusions}.\n\n\\section{Description of the dataset}\n\\label{sec:dataset}\n\n\\begin{table}[t\n  \\centering\n  \\begin{tabular}{ccccccccc}\n    \\ml{$M$} & $\\Re_L$ & ${\\rm Re}_\\lambda$\t& $\\varepsilon$ & $U$ & $L$ & $\\nu$ & $T$  & $k_{max}\\eta_u$\t\\\\\n    \\hline\n    512 & 889 & 164\t& 0.14\t& 0.61\t& 1.0\t& 0.0007 & 1.7 & 1.3\t\\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:dataset}\n  \\caption{Description of the statistically stationary hydrodynamic simulation used as\n          an initial condition for the velocity fields in the dynamo runs.\n   \\ml{$M$} denotes the number of grid points in each Cartesian coordinate,\n  \n   \\ml{$\\Re_L$ the integral-scale Reynolds number, ${\\rm Re}_\\lambda$ the Reynolds number with respect to the Taylor microscale}, \n   $\\varepsilon$ the total\n   dissipation rate, $U$ the rms velocity, $L$ the integral length scale of the\n   turbulence, $\\nu$ the kinematic viscosity,\n  \n   $T=L\/U$ the large-eddy turnover time, \\ml{$k_{\\rm max}$ the \\ml{largest}\n    resolved wave number and $\\eta_u$ the Kolmogorov microscale.} All observables are time averaged. \n}\n\\end{table}\n\n\\noindent\nThe data for the \\textit{a priori} study is generated through DNSs of \nthe three-dimensional incompressible MHD equations\n\\begin{align}\n  \\p{t}\\vec{u} + \\left(\\vec{u} \\cdot \\nabla\\right)\\vec{u} &= -\\nabla p + \\left(\\nabla \\times \\vec{b}\\right) \\times \\vec{b} + \\nu \\Delta \\vec{u} + \\vec{f}, \\label{eq:mom}\\\\\n  \\p{t}\\vec{b} &= \\nabla \\times \\left(\\vec{u} \\times \\vec{b}\\right) + \\eta \\Delta \\vec{b}, \\label{eq:ind}\\\\\n  \\nabla \\cdot \\vec{u} & = 0, \\ \\\n  \\nabla \\cdot \\vec{b}  = 0, \\label{eq:sol}\n\\end{align}\nwhere $\\vec{u}$ is the velocity field, $\\vec{b}$ the magnetic field in Alfv\\'en units, \n$p$ the pressure {divided by the density}, $\\nu$ the kinematic viscosity, \n$\\eta$ the magnetic resistivity, and \n$\\vec{f}$ an external mechanical force which is solenoidal at all times. \nThe density has been set to unity for convenience. \n\nEquations \\eqref{eq:mom}-\\eqref{eq:sol} are solved numerically on the periodic domain \n$V = [0,2\\pi]^3$ using the \npseudospectral method \\cite{Orszag69} with full dealiasing by the $2\/3$rds rule \\cite{Orszag71}.\nAn ensemble of 10 runs is generated, where the initial velocity field configurations \nare obtained from a statistically stationary hydrodynamic \nDNS on $512^3$ grid points by sampling in intervals of one large-eddy turnover time\n\\ml{$T=L\/U$, where $U$ is the rms velocity and $L$ the integral scale of the \nturbulence.\nThe mechanical force $\\vec{f}$ is a Gaussian-distributed and delta-in-time correlated random process acting at wavenumbers $1 \\leq k \\leq 2.5$ with a flat spectrum and without injection of kinetic helicity.  \nThe magnetic seed fields are randomly generated with a Gaussian distribution and concentrated at wavenumber $k_{\\rm s}=40$.}\nDetails of the stationary hydrodynamic simulation are summarized in table I.\n\n\\begin{table*}[t]\n\t\\begin{tabular}{cccccccccccccc}\n\t\t& $\\Re_L$ & ${\\rm Re}_\\lambda$ & \\ml{$\\rm Pm$} & $\\varepsilon_u$ & $\\varepsilon_b$ & $U$ & $L$ & $B$ & $L_b$ & \\ml{$\\eta$} & $k_{\\rm max} \\eta_u$ & $k_{\\rm max} \\eta_b$ & $t_S\/T$  \\\\\n\t\t\\hline\n\t\t(I) & 811 & 161 & 1 & 0.099 & $2.6\\cdot 10^{-3}$ & 0.59 & 0.97 & 0.020 & 0.092 & 0.0007 & 1.3  & 3.2  & 8.8 \\\\\n\t\t(II) & 851 & 208 & 1 & 0.057 & 0.056 & 0.58 & 1.0 & 0.13 & 0.15 & 0.0007 & 1.5  & 1.5  & 17.6 \\\\\n\t\t(III) & 870 & 211 & 1 & 0.032 & 0.076 & 0.51 & 1.2 & 0.25 & 0.29 & 0.0007 & 1.7  & 1.4 & 32.3 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{tab:stage_properties}\n\t\\caption{\n         Summary of the dynamo simulations during kinematic (I), nonlinear (II) and saturated stages (III).\n         $\\Re_L$ denotes the integral-scale Reynolds number, ${\\rm Re}_\\lambda$ the Reynolds number with respect to the \n         Taylor microscale, \\ml{$\\rm Pm$ the magnetic Prandtl number}, $\\varepsilon_u$\n         the kinetic dissipation rate, $\\varepsilon_b$ the magnetic dissipation rate, $U$\n         the rms velocity, $L$ the integral length scale of the turbulence, $B$ the rms of the magnetic\n         field, $L_b$ the magnetic integral length scale, \\ml{$\\eta$ the resistivity}, \n         $k_{\\rm max}$ the \\ml{largest}\n         resolved wavenumber, $\\eta_u$ and $\\eta_b$ are the kinetic and magnetic Kolmogorov microscales,\n         respectively, and $t_S$ is the sampling time of each evolutionary stage of the dynamo.\n         All observables are ensemble-averaged over an ensemble of 10 simulations.\n         }\n\\end{table*}\n\n\\begin{figure}[t\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG1a.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG1b.pdf}\n  \\caption{\n   Panel (a): \n   Time evolution of the kinetic energy $E_u$, the magnetic energy \n   $E_b$ and the total energy $E_u+E_b$ with time measured in units of \n   large-eddy turnover time $T$ (see table 1). \n   The inset shows the evolution $E_b$ on a\n   linear-logarithmic scale to highlight its initial exponential growth \n   phase. The different stages of dynamo evolution are indicated\n   by arrows: (I): kinematic stage, (II) nonlinear stage, (III) saturated stage.\n   Panel (b): Kinetic energy spectra $E_u(k)$ (dashed) and magnetic energy spectra \n   $E_b(k)$ (solid) measured at $t\/T =8.8$ in stage (I), at $t\/T =17.6$ \n   in stage (II) and at $t\/T = 32.3$ in stage (III).}\n  \\label{fig:En_evo}\n\\end{figure}\n\nThe time evolution of the kinetic and magnetic energies\nper unit volume\n\\begin{align}\nE_u(t) &= \\frac{1}{2} \\left \\langle |\\vec{u}(\\vec{x},t)|^2 \\right \\rangle_{{V,N}} \n        \\equiv \\left \\langle \\frac{1}{2|V|}\\int d\\vec{x} \\ |\\vec{u}(\\vec{x},t)|^2 \\right \\rangle_{{N}} \\ , \\\\\nE_b(t) &= \\frac{1}{2} \\left \\langle |\\vec{b}(\\vec{x},t)|^2 \\right \\rangle_{{V,N}} \n        \\equiv \\left \\langle \\frac{1}{2|V|}\\int d\\vec{x} \\ |\\vec{b}(\\vec{x},t)|^2 \\right \\rangle_{{N}} \\ , \n\\end{align}\n{where the subscript $N$ denotes an ensemble average over $N=10$ realisations,}\nand the total energy for the ensemble are shown in Fig.~\\ref{fig:En_evo}(a).\nFrom the time evolution of $E_b$, which is also shown on a \nlinear-logarithmic scale in the inset, it can be seen that\nthe simulation can be divided in three stages. \nFirst, during the kinematic stage (I), the magnetic field \ngrows exponentially. During that stage the Lorentz force {in Eq.~\\eqref{eq:mom}} is negligible and \nthe evolution equations are linear in the magnetic field. The exponential growth phase \nends once the Lorentz force \nis large enough such that the back-reaction of the magnetic field on the flow needs to \nbe taken into account. \nThis is the nonlinear, unsteady, stage (II) of the evolution, during which \n$E_b$ continues to increase sub-exponentially \\ml{\\cite{Haugen04,Schekochihin04}}.  \nFinally, $E_b$ is approaching a statistically stationary\nstate. \\ml{That is,} it enters the saturated stage (III)\n\\ml{which at unity magnetic Prandtl number and sufficiently large $\\Re$ \nis characterized by the ratios of the dissipation rates\n$\\varepsilon_b\/(\\varepsilon_u + \\varepsilon_b) \\simeq 0.7$ and energies $E_b\/(E_u+E_b) \\simeq 0.25$ \\cite{Haugen03,Brandenburg14,Linkmann17,McKay17}. Our data in stage (III)\nis consistent with these ratios, as can be seen from the values listed in \ntable \\ref{tab:stage_properties}, where a summary of the dynamo runs in the kinematic (I), nonlinear (II) and saturated (III) stages is provided.} \nThe SGS energy transfers will be studied during stages (I)-(III), with each stage\nanalysed separately. \n\n\nThe kinetic and the magnetic energy spectra \n\\begin{align}\nE_u(k,t) & = \\frac{1}{2} \\left \\langle \\int_{|\\vec{k}|=k} |\\fvec{u}_{\\vec{k}}(t)|^2  \\ d\\vec{k} \\right \\rangle_N \\ , \\\\ \nE_b(k,t) & = \\frac{1}{2} \\left \\langle \\int_{|\\vec{k}|=k} |\\fvec{b}_{\\vec{k}}(t)|^2 \\ d\\vec{k} \\right \\rangle_N \\ ,  \n\\end{align}\nare shown in Fig.~\\ref{fig:En_evo}(b) for different instances in time corresponding to \nstages (I)-(III) as specified in table \\ref{tab:stage_properties}.\nThe kinetic energy spectrum is dominated by the forcing in the interval $1\\leq k\\leq2.5$. \nDuring the kinematic stage, an inertial subrange with Kolmogorov scaling can be identified, \nas indicated in the figure by the straight solid line. During stages (II) and (III) we observe a \nsteepening of $E_u(k)$ at successively smaller wavenumbers.   \nThe magnetic energy spectrum grows self-similarly during {stage (I)}\nwhich is typical for a small-scale dynamo~\\cite{Schekochihin04, Mininni05a, Brandenburg05}.\nIn the saturated stage (III), the magnetic energy exceeds the kinetic energy\nat the small scales while the large scales remain essentially hydrodynamic and \nforcing-dominated. A crossover-wavenumber $k^*$ can be identified where $E_u(k^*) = E_b(k^*)$, \nin the present dataset $k^* = 9$.\n\\ml{Since the peak of the saturated magnetic energy spectrum depends on the forcing scale \\cite{Brandenburg05}, the equipartition scale we measure will also not be universal, that is, it should depend on the forcing.  \n}\n\n\\section{P-LES formulation for MHD}\n\\label{sec:theory}\nThe governing equations\nare derived by applying a filtering operation to the MHD equations \\cite{Zhou91,Kessar16,Yang16a,Aluie17}, {with}\nthe filtered {component}\n\\ml{$\\overline h$ of a function $h$} \ndefined as\n\\begin{align}\n \n \n \n  \\overline{h}(\\vec{x},t) \\equiv \n\t\\int_{V}d\\vec{y}\\, G\\left(\\vec{x} - \\vec{y}\\right) h(\\vec{y},t) = \n  \\sum\\limits_{\\vec{k} \\in \\mathbb{Z}^3} \\hat{G}(\\vec{k}) \\hat{h} (\\vec{k},t) e^{i\\vec{k}\\vec{x}},\n  \\label{eq:filter}\n\\end{align}\nwhere $G$ is the filter function and $\\hat{\\cdot }$ denotes the Fourier transform.  Applying\nthis filtering operation to Eqs.~\\eqref{eq:mom} and \\eqref{eq:ind}, we obtain\nthe filtered momentum and induction equations\ngiven here in tensor notation\n\\begin{align}\n\n  \\p{t}\\overline{u}_i &= - \\p{j}\\left(\\overline{\\overline{u}_i\\overline{u}_j} - \\overline{\\overline{b}_i\\overline{b}_j} + \\tij{I} - \\tij{M} + \\overline{p}\\delta_{ij}\\right) + \\nu \\p{jj} \\overline{u}_i + {\\overline{f}_i}, \n  \\label{eq:momentum_PLES}\\\\\n \n\t\\p{t}\\overline{b}_i &= - \\p{j}\\left(\\overline{\\overline{b}_i\\overline{u}_j} - \\overline{\\overline{u}_i\\overline{b}_j} + \\ml{\\tij{b}} \\right) + \\eta \\p{jj} \\overline{b}_i,\n  \\label{eq:induction_PLES}\n\\end{align}\n{where a summation over repeated indices is implied, and}\nwith \n\\begin{align}\n\n\n\n\n  \\tij{I} =& \\ \\overline{u_i u_j} - \\overline{\\overline{u}_i \\overline{u}_j}, \\label{eq:SGS_tensor_V} \\\\\n  \\tij{M} =& \\ \\overline{b_i b_j} - \\overline{\\overline{b}_i \\overline{b}_j}, \\label{eq:SGS_tensor_M} \\\\\n  \\tij{b} =& \\ \\overline{b_i u_j} - \\overline{\\overline{b}_i \\overline{u}_j} \n\t           - (\\overline{u_i b_j} - \\overline{\\overline{u}_i \\overline{b}_j}), \\label{eq:SGS_tensor_b} \n\\end{align}\nwhere $\\tij{I}$ is the inertial SGS tensor, $\\tij{M}$ the Maxwell SGS tensor,\n\\ml{and $\\tij{b}$ the SGS tensor originating from the electromotive force \nin Eq.~\\eqref{eq:ind}.}\n\\ml{\nIt consists of two SGS-stresses which} \nare related to each other by transposition. \nThey are associated with different dynamics, \nthat is with advection, $(\\vec{u}\\cdot\\nabla)\\vec{b}$, \\ml{in case of $\\overline{b_i u_j} - \\overline{\\overline{b}_i \\overline{u}_j}$} \nor dynamo action through magnetic field-line stretching, $(\\vec{b}\\cdot\\nabla)\\vec{u}$, \n\\ml{in case of $\\overline{u_i b_j} - \\overline{\\overline{u}_i \\overline{b}_j}$}. \n\\ml{However, as} they have a common physical origin, \nthe electric field $\\vec{E} = \\vec{u} \\times \\vec{b}$, \n\\ml{we do not consider them separately}. \n\n{Equations \\eqref{eq:momentum_PLES}-\\eqref{eq:induction_PLES}} are obtained by\nusing \\ml{solenoidality}\nof both fields, the linearity of the filtering operator\nand including the terms which can be written as a gradient into the pressure gradient.\n\\ml{As usual, the\nequations are not closed in terms of the resolved fields only, due to the fact\nthat the SGS stress tensors depend on the product of two unresolved fields.}\n\\ml{Equations ~\\eqref{eq:momentum_PLES}-\\eqref{eq:induction_PLES}\ndiffer from those usually given in the MHD literature on LES \\cite{Zhou91,Kessar16,Aluie17}\nthrough the additional filtering of products of two resolved fields. \nIn conjunction with a projector filter,\nthe latter ensures that after introducing SGS models, \nEqs.~\\eqref{eq:momentum_PLES} and \\eqref{eq:induction_PLES} \ncan be evolved on a finite computational grid \\cite{Sagaut06, Buzzicotti18a}, \nwhich can seen by supposing that $G$ in Eq.~\\eqref{eq:filter}\nis a Galerkin projector on a finite number of \nFourier modes \\cite{Buzzicotti18a}.  \nIn what follows, we consider $G$ to be a projector and \nEqs.~\\eqref{eq:momentum_PLES}-\\eqref{eq:SGS_tensor_b}\nare referred to as the P-LES formulation.\n}\n\n\nThe P-LES formulation has the further advantage that, unlike in the \\ml{usual} \nLES formulation, the SGS energy transfers based on the P-SGS tensors \n\\ml{defined in Eqs.~\\eqref{eq:SGS_tensor_V}-\\eqref{eq:SGS_tensor_b}} \ndo not contain couplings between the resolved fields \\cite{Buzzicotti18a}. \nThe latter is very important for the evaluation of backscatter in {\\em a priori} analyses of SGS energy\ntransfers, since residual couplings between the resolved fields can be wrongly interpreted \nas backscatter events. We will come back to this point in Secs.~\\ref{sec:energy_transfers} \nand \\ref{sec:energy_transfers-tot}.\n\n\\blue{\nFinally, we point out that care must be taken in {\\em a-posteriori} \nstudies of MHD LES concerning $\\overline{p}$ since it contains the magnetic \nSGS pressure term, which is not closed in terms of the resolved magnetic field. \nAs such, a closure of Eq.~\\eqref{eq:SGS_tensor_M} would lead to two models \nfor the magnetic pressure term: an explicit one coming from the choice of model\nand an implicit one from the solution of the Poisson equation. \nHowever, the magnetic pressure term does not affect the global energy transfers\nand is thus not of direct relevance to the present {\\em a-priori} study.\n}\n\n\n \n \n\n\n\\subsection{The resolved-scale energy transfer}\n\\label{sec:energy_transfers}\nNeglecting viscous, Joule dissipation {and  forcing} terms, \nthe {P-LES} kinetic and magnetic energy evolution equations read\n\\begin{align}\n  \\p{t}\\frac{1}{2}\\overline{u}_i\\overline{u}_i + \\p{j}A^{u}_j &= -\\Pi^{u} + (\\p{j}\\overline{u}_i) (\\overline{\\overline{u}_i \\overline{u}_j}) - (\\p{j}\\overline{u}_i) (\\overline{\\overline{b}_i \\overline{b}_j}), \\label{eq:kin_en}\\\\\n  \\p{t}\\frac{1}{2}\\overline{b}_i\\overline{b}_i + \\p{j}A^{b}_j &= -\\Pi^{b} + (\\p{j}\\overline{b}_i) (\\overline{\\overline{b}_i \\overline{u}_j}) - (\\p{j}\\overline{b}_i) (\\overline{\\overline{u}_i \\overline{b}_j}), \\label{eq:mag_en}\n\\end{align}\nwhere $A_j^{u} = \\overline{u}_i(\\overline{\\overline{u}_i \\overline{u}_j} - \\overline{\\overline{b}_i \\overline{b}_j} +\n\\overline{p} \\delta_{ij} + \\tij{I} - \\tij{M})$ and  $A_j^{b} =\n\\overline{b}_i(\\overline{\\overline{b}_i \\overline{u}_j} - \\overline{\\overline{u}_i \\overline{b}_j} +\\ml{\\tij{b}})$\n{result in flux terms}\nthat redistribute the energies in space and vanish under\nspatial averaging: \n{$\\langle \\p{j}A_j^{u}\\rangle_V = \\langle \\p{j}A_j^{b}\\rangle_V = 0$ }.  \nThe {P-SGS}\nenergy transfers $\\Pi^{u}$ and $\\Pi^{b}$ are defined as\n\\begin{align}\n  \\Pi^{u} &= \\Pi^{I} - \\Pi^{M} = (\\p{j} \\overline{u}_i) \\tij{I} -  (\\p{j} \\overline{u}_i) \\tij{M},\\\\\n  \\Pi^{b} &= \\ml{(\\p{j} \\overline{b}_i) \\tij{b}},\n\\end{align}\nwhere $\\Pi^{I} = -(\\p{j}\\overline{u}_i)\\tij{I}$ is the inertial SGS energy transfer,\n$\\Pi^{M} = -(\\p{j}\\overline{u}_i)\\tij{M}$ the Maxwell SGS energy transfer\n\\ml{and $\\Pi^{b}$ the SGS energy transfer associated with \nthe electromotive force.}\nEquations~\\eqref{eq:kin_en} and \\eqref{eq:mag_en} contain four extra terms:\n$(\\p{j}\\overline{u}_i) (\\overline{\\overline{u}_i \\overline{u}_j})$, $(\\p{j}\\overline{u}_i) (\\overline{\\overline{b}_i\n\\overline{b}_j})$, $ (\\p{j}\\overline{b}_i) (\\overline{\\overline{b}_i \\overline{u}_j})$ and $(\\p{j}\\overline{b}_i)\n(\\overline{\\overline{u}_i \\overline{b}_j})$. \n\\ml{Using $\\nabla \\cdot \\overline{\\vec{u}} =0$,\n$\\nabla \\cdot \\overline{\\vec{b}} =0$ \nand the projector property $\\hat{G}^2 = \\hat{G}$}, it can be shown that \n\\begin{align}\n& \\avg{(\\p{j}\\overline{u}_i)(\\overline{\\overline{u}_i \\overline{u}_j})}_V = 0 \\ , \\\\  \n&\\avg{(\\p{j}\\overline{b}_i) (\\overline{\\overline{b}_i \\overline{u}_j})}_V = 0 \\ ,\n\\end{align}\nand \n\\begin{equation}\n\\avg{(\\p{j}\\overline{u}_i) (\\overline{\\overline{b}_i \\overline{b}_j})}_V = - \\avg{(\\p{j}\\overline{b}_i) (\\overline{\\overline{u}_i \\overline{b}_j})}_V \\ , \n\\end{equation}\n{hence} they do not contribute to the global {total energy} balance.\nFurthermore, it is easy to verify that out of the four terms \nonly $(\\p{j}\\overline{u}_i) (\\overline{\\overline{b}_i\\overline{b}_j})$ is Galilean invariant. \nGalilean invariance is important to prevent the occurrence\nof unphysical fluctuations in the measured SGS energy transfer \n\\cite{aluie2009I,aluie2009II,Buzzicotti18a}.  \nThis problem can be solved by adding and subtracting energy transfers \noriginating from the Leonard stress components for each SGS \ntensor~\\cite{leonard1975,Zhou91} in Eqs.~\\eqref{eq:kin_en} and \n\\eqref{eq:mag_en}.\n{The Leonard stresses are defined as}\n\\begin{align}\n \n \n \n \n  \\tij{I,L} & = \\overline{\\overline{u}_i \\overline{u}_j} - \\overline{u}_i \\overline{u}_j,\\\\\n  \\tij{M,L} & = \\overline{\\overline{b}_i \\overline{b}_j} - \\overline{b}_i \\overline{b}_j,\\\\\n  \\tij{b,L} & = \\ml{\\overline{\\overline{b}_i \\overline{u}_j} - \\overline{b}_i \\overline{u}_j -(\\overline{\\overline{u}_i \\overline{b}_j} - \\overline{u}_i \\overline{b}_j)}, \n\\end{align}\n{which give rise to the following energy transfer terms}\n\\begin{align}\n  \\label{eq:Leo_u}\n\t\\Pi^{u,L} & = \\Pi^{I,L} - \\Pi^{M,L} = (\\p{j} \\overline{u}_i) \\tij{I,L} -  (\\p{j} \\overline{u}_i) \\tij{M,L},\\\\\n  \\label{eq:Leo_b}\n \n\t\\Pi^{b,L} & = \\ml{(\\p{j} \\overline{b}_i) \\tij{b,L}} .\n\\end{align}\n{Including the Leonard terms in Eqs.~\\eqref{eq:kin_en} and \\eqref{eq:mag_en}} results in \n\\begin{align}\n  \\label{eq:evol_Eu}\n  \\p{t}\\frac{1}{2}\\left(\\overline{u}_j \\overline{u}_j\\right) & + \\p{j}\\left(A^{u}_j + \\overline{u}_i\\tij{u,L}\\right) \\nonumber \\\\\n                                           &= - \\Pi^{u} - \\Pi^{u,L} - (\\p{j}\\overline{u}_i) (\\overline{b}_i \\overline{b}_j),\\\\\n  \\label{eq:evol_Eb}\n  \\p{t}\\frac{1}{2}\\left(\\overline{b}_j \\overline{b}_j\\right) &+ \\p{j}\\left(A^{b}_j + \\overline{u}_i\\tij{b,L}\\right) \\nonumber \\\\\n                                           &= - \\Pi^{b} - \\Pi^{b,L} + (\\p{j}\\overline{u}_i) (\\overline{b}_i \\overline{b}_j).\n\\end{align}\nNow all terms in the resolved energy evolution equations are Galilean invariant.\\\\\n\n\\noindent\nIt is important to remark that the Leonard SGS transfers vanish under spatial averaging,\ni.e. they do not alter the global balances. {Furthermore, they couple only the \nresolved fields, hence they cannot be associated with transfers between resolved and \nSGS quantities. Therefore the LES formulation differs from the P-LES formulation\nin a fundamental way: All SGS-tensors in the LES formulation are the sum of the \nrespective P-SGS and Leonard tensors, e.~g. $\\tij{I,\\rm LES} = \\tij{I} + \\tij{I,L}$, \nand the corresponding SGS energy transfers of the LES formulation contain \nthe contribution from the Leonard stresses. That is, the SGS energy transfers\nin the LES formulation have contributions from interactions\nbetween the resolved fields \\cite{Buzzicotti18a}. We will come back to this\npoint in the context of backscatter in Sec.~\\ref{sec:energy_transfers-tot} and in the Appendix.}\n\n\\noindent\n{Finally, the term $(\\p{j}\\overline{u}_i) (\\overline{b}_i \\overline{b}_j)$ occurs in Eqs.~\\eqref{eq:evol_Eu} and \\eqref{eq:evol_Eb}} with opposite sign. \nSince it\nis closed in terms of the resolved fields and exchanges kinetic and magnetic energy,\n{$(\\p{j}\\overline{u}_i) (\\overline{b}_i \\overline{b}_j)$} has been named \n{\\em resolved-scale conversion term}~\\cite{Aluie17}. \n\\ml{It is positive if kinetic energy is converted to magnetic energy and \nnegative vice versa.}\nWith $\\Pi^{u} = \\Pi^{I} - \\Pi^{M}$ and $\\Pi^{b}$\nwe now have key benchmark quantities to study the properties of the\ndifferent SGS energy transfers. \nFurthermore, as the total energy is conserved in the absence of forcing and dissipation, \nthe total SGS energy transfer is also a quantity of interest.\nWe define the resolved total energy transfer \\ml{$\\Pi$  through the resolved-scale total energy balance}\n\\begin{align}\n  \\p{t}\\frac{1}{2}\\left(\\overline{u}_i \\overline{u}_i\\right) + \\p{t}\\frac{1}{2}\\left(\\overline{b}_i \\overline{b}_i\\right) &+  \\p{j}\\left(A_j + \\overline{u}_i\\tij{L}\\right) \\nonumber \\\\ \n     &= - \\Pi - \\Pi^L\n  \\label{eq:total_ples}\n\\end{align}\nwhere $A_j = A^u_j + A^b_j$, $\\tij{L} = \\tij{u,L} + \\tij{b,L}$, $\\Pi = \\Pi^{u} + \\Pi^{b}$ and $\\Pi^L = \\Pi^{u,L} + \\Pi^{b,L}$.\nFigure~\\ref{fig:diagram} gives a schematic overview of the different \nSGS energy transfers.\n\\begin{figure}[htp]\n\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{FIG2.pdf}\n\t\\caption{\n        \n         A schematic representation of the energy transfer between the \n         resolved-scale energies and the SGS energy. The exchange between\n         magnetic and kinetic energies at the resolved scales is carried\n         by the resolved-scale conversion term \n         $(\\p{j}\\overline{u}_i)\\overline{b}_i\\overline{b}_j$. {According to Eqs.~\\eqref{eq:evol_Eu} and \\eqref{eq:evol_Eb},} the exchange \n         of energy between resolved scales and SGS follows different channels, \n         $\\Pi^{b}$ couples the resolved-scale magnetic energy to the SGS and combines \n         the physical processes of advection of magnetic energy and magnetic field \n         line stretching, while $\\Pi^{u}$ transfers kinetic energy between resolved scales and SGS. \n         The latter itself has two components, an inertial channel $\\Pi^{I}$ which is due to \n         vortex-stretching and advection and a magnetic channel $\\Pi^{M}$ originating from \n         the Lorentz force. During the kinematic stage of the dynamo, $\\Pi^{M}$ is negligible\n         compared to $\\Pi^{I}$.\n        \n        }\n\t\\label{fig:diagram}\n\\end{figure}\n\n\\section{\\textit{A priori} analysis of the SGS energy transfers}\n\\label{sec:apriori}\n\nThe \\textit{a priori} analysis of the statistical properties of the SGS energy\ntransfers is carried out using a sharp spectral cut-off filter, \n\\ml{which is \ndefined through its action on a generic \\ml{function $h$}\n\\begin{align}\n \n \n  \\overline{h}(\\vec{x},t) \\equiv \n  \\sum\\limits_{|\\vec{k}| < k_c} \\hat{h} (\\vec{k},t) e^{i\\vec{k}\\vec{x}},\n  \\label{eq:galerkin_filter}\n\\end{align}\nwhere $k_c$ is the cut-off wavenumber, which corresponds to \nthe configuration-space filter width $\\Delta = \\pi\/k_c$.}\n\\mb{Although sharp projectors produce\nGibbs oscillations in \\ml{configuration}\nspace \\cite{ray2011resonance} resulting in\nSGS stress tensors \\cite{vreman1994realizability}\n\\ml{that are not positive-definite}, they have\nthe advantage to create a clear distinction between resolved and unresolved\nscales and to allow all terms in the equations\n\\eqref{eq:momentum_PLES}-\\eqref{eq:induction_PLES} to evolve on the same\nFourier subspace for all times.  Moreover, for nonconducting flows, a good\nagreement between the statistics of the SGS energy transfer obtained from a\nsharp cutoff and Gaussian filter was found \\cite{Buzzicotti18a}, suggesting\nthat \n\\ml{effects specific to Galerkin projection have}\nonly a subleading effect at the level of the energy evolution equations. \n} \\\\\n\nIn what follows, we study the mean ({here, mean refers to the combined spatial and ensemble average})  \n{P-}SGS energy transfers\nand their spatial fluctuations for different $k_c$.  \n\\mb{The fluctuations are investigated through the probability density functions (pdfs)\nof the respective P-SGS energy transfers.\n\\ml{I}n order to quantify the departure from Gaussianity at \ndifferent scales, it is customary to evaluate the flatness $F_x$ of the standardized pdfs, \n$$\nF_x(k_c) = \\langle x^4 \\rangle\/ \\langle x^2 \\rangle^2 \\sim k_c^\\zeta, \n$$\nas a function of the cutoff wavenumber, where $x$ represents the different contributions of the P-SGS energy transfer.}\nSince the Leonard stresses do not provide information relevant to modelling, \nwe summarize results specific to the Leonard stresses in the Appendix, \nwhich is referenced in the text where necessary.\n\n\\noindent\nWe begin with\n$\\Pi$,\nand subsequently increase the level of detail by first splitting \n$\\Pi$ into\n$\\Pi^{u}$ and $\\Pi^{b}$, \n{followed by the decomposition of} \n$\\Pi^{u}$ into\n$\\Pi^{I}$ and $\\Pi^{M}$.  \nNote that $\\Pi^{b}$ is not decomposed any further, because\nthe stress tensors \\ml{associated with the advection and field-line stretching terms \nin the induction equation}\noriginate both from the electric field and are related to each other \nby transposition, as discussed in Sec.~\\ref{sec:theory}. As such, a single LES\nmodel term should be used in the induction equation. \n\n\n\\subsection{The total SGS energy transfer} \n\\label{sec:energy_transfers-tot}\n\n\\begin{figure}[t\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG3.pdf}\n  \\caption{\n\tThe mean\n    total SGS energy transfer $\\langle \\Pi\\rangle_{V,N} $ normalized\n\twith the total dissipation $\\varepsilon$ versus the cutoff wavenumber $k_c$ at the\n\tkinematic (I), nonlinear (II) and stationary (III) stages.}\n  \\label{fig:TOT_mean}\n\\end{figure}\n\nFigure \\ref{fig:TOT_mean} presents $\\avg{\\Pi}_{V,N}$ as function of $k_c$ \nat three different instants during the time evolution which are representative\nof the three stages (I)-(III).\n\\mb{Since $\\Pi$ is obtained using a spectral cut-off projector, its mean value equals the total energy flux in Fourier space across the cut-off wavenumber $k_c = \\pi\/\\Delta$, see  \\cite{eyink2005locality,Buzzicotti18a}.}\nAs can be seen from Fig.~\\ref{fig:TOT_mean}, \n$\\avg{\\Pi}_{V,N} \\geqslant 0$, which is representative of a mean total energy transfer \nfrom large scales to small scales. Furthermore, we find that $\\avg{\\Pi}_{V,N}$\ndoes not change significantly during the different evolutionary stages of the \ndynamo, which implies that the exchange of kinetic and magnetic energy proceeds\nin a way that leaves the total scale-by-scale transfer unaffected. We will \ncome back to this point in further detail when assessing the decomposed\nSGS energy transfers.  \n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{FIG4a.pdf}\n  \\includegraphics[width=0.45\\textwidth]{FIG4b.pdf}\n  \\includegraphics[width=0.45\\textwidth]{FIG4c.pdf}\n  \\caption{Total SGS energy transfer $\\Pi$ during the kinematic stage (I), non-linear stage (II) and the stationary stage (III).\n\tPanel (a): pdfs of $\\Pi$ at $k_c=20$.\n\tPanel (b): The flatness of $\\Pi$ versus the\n\tcutoff wavenumber $k_c$. \\mb{The inset shows a zoom of the flatness in the inertial range of scales, with error bars estimated from the different configurations. The black line represents the fit of the flatness scaling exponent in the range $2 \\leqslant k_c \\leqslant 30$.\n\tPanel (c): pdfs of $\\Pi$ at $k_c=8$, $k_c=20$ and $k_c=70$ during stage (II).}}\n  \\label{fig:TOT}\n\\end{figure*}\n\nSince $\\avg{\\Pi}_{V,N} \\geqslant 0$, it can be expected that the pdf of\n$\\Pi$ is positively skewed such that events leading to a forward\ntransfer of total energy across the filter scale are more likely than\nbackscatter events. This is indeed the case as shown by the \\ml{standardized} pdf of $\\Pi$ \nin Fig.~\\ref{fig:TOT}(a) at $k_c=20$ for stages (I)-(III).  \nApart from more pronounced tails occurring in stage (II),\nthe \\ml{standardized} pdfs are remarkably similar.  \n{However,} while the pdf of $\\Pi$ is positive skewed \nat all stages, the pdf of $\\Pi^L$ is symmetric\n(see Fig.~\\ref{fig:TOT_LEO} in Appendix\n). Hence, by measuring  \n$\\Pi = \\Pi + \\Pi^L$ as the total SGS energy transfer, the \nresidual transfer amongst the resolved scales carried by the Leonard component \ncould lead to the conclusion\n{of backscatter events being more frequent}\nthan they actually are. \n\n\\mb{In Fig.\\ref{fig:TOT}(b) we show the flatness of $\\Pi$, ($F_{_\\Pi}$), as a\nfunction of the cut-off wavenumber $k_c$. From this analysis we can see that\nthe flatness\nshows a similar power-law behavior\n\\ml{in the inertial range $2\\le k_c\\le 30$}\nduring all stages in the evolution. The\nflatness scaling exponent $\\zeta = 0.55 \\pm 0.05, $ see inset of\nFig.~\\ref{fig:TOT}(b), has been measured by a least-squares fit and its error has\nbeen estimated by \\ml{varying the fitting interval within the inertial range $2 \\le k_c \\le 30$.}\nA small temporal variability of the flatness is observed only in the dissipative range where it is also found to increase exponentially suggesting strong deviation from Gaussianity at all\ntimes. From both the pdfs and flatness analysis it follows that the statistical\nproperties of $\\Pi$ are conserved during the temporal evolution. \\ml{Fig.~\\ref{fig:TOT}(c) presents} the pdfs of $\\Pi$ at a fixed time during the non-linear stage (II) at three different cut-off wavenumbers \\ml{$k_c=8$, $k_c=20$ and $k_c=70$. As can be seen from Figs.~\\ref{fig:En_evo}(b) and (\\ref{fig:TOT_mean}), $k_c=8$ and $k_c=20$ correspond to the beginning \nand the end of the inertial range, respectively, while $k_c=70$ lies in the dissipative range.}\nFrom the comparison of the three standardized pdfs in Fig.\\ref{fig:TOT}(c) we can\nclearly observe the presence of intermittency in the statistics of $\\Pi$ \\ml{through an increasing} departure from Gaussianity at successively smaller scales.}\n\\mb{The same information can be extracted by the power-law behavior of the flatness over the inertial range of scales which also shows the intermittent properties of the SGS energy transfer in MHD turbulence. It is interesting to note that the value of $\\zeta$ measured from the data is in agreement with the prediction of the She-Leveque model \\cite{she1994universal}. Indeed, from the scaling estimate, \n$$\\langle |\\Pi|^n \\rangle = O\\left( \\Delta^{\\zeta_{3n}-n}\\right),$$\n\\cite{eyink1996multifractal,Buzzicotti18a} and from the She-Leveque values of the exponents for n=2: $\\zeta_6 \\sim 1.77$ and n=4: $\\zeta_{12} \\sim 1.94$ (note that there is a typo in the value of $\\zeta_{12}$ reported in ref.~\\cite{Buzzicotti18a}), we obtain for the flatness the She-Leveque prediction $\\zeta_{_{SL}} \\sim 0.6$.}\n\n\n\\subsection{Kinetic and magnetic SGS energy transfers}\n\nAs discussed in Sec.~\\ref{sec:theory}, $\\Pi$ can be further decomposed\ninto\n$\\Pi^{u}$ and\n$\\Pi^{b}$.  Furthermore, the resolved-scale conversion term,\n$(\\p{j}\\overline{u}_i)\\overline{b}_i\\overline{b}_j$, {in Eqs.~\\eqref{eq:evol_Eu} and \\eqref{eq:evol_Eb}}, \n{which cancels out in Eq.~\\eqref{eq:total_ples} for the total resolved-scale energy, \nmust also be measured.}\n{It contains information on the scale-dependence of the conversion of kinetic \nto magnetic energy, an assessment of which is essential in order to provide guidance\nfor SGS models of MHD dynamos.}   \n\\\\\nThe averages $\\avg{\\Pi^{u}}_{V,N}$, $\\avg{\\Pi^{b}}_{V,N}$ and\n$\\langle (\\p{j}\\overline{u}_i)\\overline{b}_i\\overline{b}_j\\rangle_{V,N}$ are shown in Fig.~\\ref{fig:VTOTBTOT_mean}(a-c),\nrespectively.  We first notice that $\\avg{\\Pi^{u}}_{V,N}$ gets depleted towards\nstage (III) while $\\avg{\\Pi^{b}}_{V,N}$ increases.  From a comparison of the large\nincrease of {$\\langle (\\p{j}\\overline{u}_i)\\overline{b}_i\\overline{b}_j\\rangle_{V,N}$}\nrelative to the smaller decrease of\n$\\avg{\\Pi^{u}}_{V,N}$ during stages (I)-(III), it follows that the growth of the\nmagnetic field is due to direct interactions between $\\overline{\\vec{u}}$ and\n$\\overline{\\vec{b}}$. \n\\ml{The data presented in Fig.~\\ref{fig:VTOTBTOT_mean}(a,b)}\nalso show that both the kinetic and magnetic\nSGS energy transfers are forward.\n\\ml{From Fig.~\\ref{fig:VTOTBTOT_mean}(c) it can be seen}\nthat\n\\blue{ $\\langle (\\p{j}\\overline{u}_i)\\overline{b}_i\\overline{b}_j\\rangle_{V,N}$ \nhas an inflection point that saturates at $k^* \\approx 20$. Since the \nlarge-scale conversion term is the running integral in $k$ \nof the energy transfer at $k$, \nan inflection point in $\\langle (\\p{j}\\overline{u}_i)\\overline{b}_i\\overline{b}_j\\rangle_{V,N}$ at $k^*$ \nimplies an extremum in the energy conversion at $k^*$, corresponding \nto a saturation length scale for the conversion of kinetic to magnetic energy.}\nThe existence of a\nsaturation length scale implies the breaking of inertial self-similarity and\nputs a natural constraint on any LES for MHD. Either we use an extremely\nresolved model with $k_c\\gg k^*$, and we fully resolve the dynamics leading to\nthe non-linear dynamo saturation, or we use $k_c \\sim k^*$ and a very\nsophisticated SGS model must be used. Certainly one cannot further push and use\n$k_c \\ll k^*$, or a fully ad-hoc magnetic field growth must be supplied.  \n\\ml{An in-depth investigation of the statistical properties of \n    the resolved-scale conversion term would provide guidance \n    for cases where very coarse grids require the aforementioned \n    ad-hoc magnetic forcing term. \n}\nA quantitative assessment of this issue \\ml{also} \nrequires {\\em a posteriori} analyses and would \nconstitute a useful contribution to MHD LES.\n\n\n\\begin{figure}[htbp\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG5a.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG5b.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG5c.pdf}\n  \\caption{\n\tThe mean\n    P-SGS energy transfers $\\langle \\Pi^{u}\\rangle_{V,N}$ (panel (a)),\n\n    and $\\langle \\Pi^{b}\\rangle_{V,N}$ (panel (b)), and the mean\n\tof resolved-scale conversion term\n\t$\\langle (\\p{j}\\overline{u_i})\\overline{b}_i\\overline{b}_j\\rangle_{V,N}$ (panel (c)), \n    normalized with the total energy dissipation rate $\\varepsilon$ \n    versus the cutoff wavenumber $k_c$ during the kinematic\n\t(I), non-linear (II) and stationary (III) stages.}\n  \\label{fig:VTOTBTOT_mean}\n\\end{figure}\n\n\n\nFigure~\\ref{fig:VTOTBTOT}(a,b) presents the standardized pdfs of\n$\\Pi^{u}$ and $\\Pi^{b}$ at $k_c=20$. {We note that the pdfs of $\\Pi^{b}$ are only\nshown for stages (II) and (III), as $\\Pi^{b}$ is negligible in stage  (I), because the system\nis dominated by magnetic field amplification which occurs through the term $(\\p{j}\\overline{u_i})\\overline{b}_i\\overline{b}_j$.}\n{Although $\\langle \\Pi^{u}\\rangle_{V,N}$ and $\\langle \\Pi^{b}\\rangle_{V,N}$\nare positive, that is, kinetic and magnetic energies are transferred downscale on average, \nthe pdfs of $\\Pi^{u}$ and $\\Pi^{b}$ develop negative tails. The latter is particularly pronounced \nfor $\\Pi^{b}$ in stage (III), as shown in Fig.~\\ref{fig:VTOTBTOT}(b).}\n{That is} backscatter events in the \nmagnetic SGS energy transfer cannot be neglected {for a fully nonlinear dynamo}. The latter implies that \ndissipative {approaches}\nsuch as the Smagorinsky {closure} \\cite{Smagorinsky63} are\nhardly optimal to model the SGS stresses in the induction equation.\nThe flatness of $\\Pi^{u}$ and $\\Pi^{b}$ as a function of $k_c$ is shown \nFigs.~\\ref{fig:VTOTBTOT}(c,d). \nThere {appears to be} a slight indication of increased  \nintermittency in stage (III) compared to stages (I) and (II) for both $\\Pi^{u}$ and $\\Pi^{b}$\n{since the flatness becomes more scale-dependent in the inertial range. \nAs can be seen from the \nfigures, $\\Pi^b$ appears to be less intermittent than $\\Pi^u$.\nHowever, the latter statements on intermittency require further assessment \nusing higher-resolved datasets with a more extended inertial range}. \n\n\\begin{figure}[htbp\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG6a.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG6b.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG6c.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG6d.pdf}\n  \\caption{\n        Kinetic and magnetic SGS energy transfers $\\Pi^{u}$ and $\\Pi^{b}$ during the\n        kinematic stage (I), non-linear stage (II) and the stationary stage (III):\n\tpdfs of $\\Pi^{u}$ (a) and $\\Pi^{b}$ (b) at $k_c=20$;\n\tflatness of $\\Pi^{u}$ (c) and $\\Pi^{b}$ (d) against the\n\tcutoff wavenumber $k_c$.}\n  \\label{fig:VTOTBTOT}\n\\end{figure}\n\n\\begin{figure*}[htbp]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{FIG7a.png}\n \\includegraphics[width=0.9\\textwidth]{FIG7b.pdf}\n \\caption{(Colour online)\n   {Top:} Two-dimensional visualisations of\n   $\\Pi^{u}$ (left) and $\\Pi^{b}$ (right)  at $k_c=20$ \n   in stage (III). \n   Positive values correspond to forward energy transfer while negative values\n   indicate backscatter. \n   {Bottom: Corresponding joint pdf of $\\Pi^{u}$ and $\\Pi^{b}$.}\n }\n \\label{fig:Visu_Piub}\n\\end{figure*}\n\nVisualisations of $\\Pi^{u}$ and $\\Pi^{b}$ obtained during stage (III) \nare presented in {the top panels of} Fig.~\\ref{fig:Visu_Piub}. \nA striking feature is the localized \\ml{elongated}\nnature \nof intense forward-transfer events in $\\Pi^{u}$. Similar\nstructures are \nare also visible in $\\Pi^{b}$\n, and the colour-mapping suggests an \\blue{inverse relation}\nbetween $\\Pi^{u}$ and $\\Pi^{b}$, \\blue{where large values of $\\Pi^{u}$ are correlated with small values of $\\Pi^{b}$\nand vice versa}.\nThe correlation between $\\Pi^{u}$ and $\\Pi^{b}$ is quantified\nthrough their joint pdf shown in the bottom panel of Fig.~\\ref{fig:Visu_Piub}. \n\\ml{The data in the figure show a tendency towards higher probabilities along the axes where either \n$\\Pi^{u} = 0$ or $\\Pi^{b} = 0$, which suggest a mild \\blue{inverse proportionality}\nbetween the two. \n}\nAs will be seen later, the intense forward-transfer events in $\\Pi^{u}$ originate\nfrom the {P-SGS} Maxwell stresses in\n{Eq.~\\eqref{eq:momentum_PLES}}.\n\n\\subsection{Inertial and Maxwell SGS energy transfers}\nThe term {$\\Pi^{u}$ in Eq.~\\eqref{eq:momentum_PLES}} \nis now further decomposed into\n$\\Pi^{I}$ and\n$\\Pi^{M}$, as introduced in Sec.~\\ref{sec:theory}. \nFigure~\\ref{fig:VELMAG_mean} presents $\\avg{\\Pi^{I}}_{V,N}$ and $\\avg{-\\Pi^{M}}_{V,N}$\nas functions of $k_c$ {where the sign convention for $\\Pi^{M}$ reflects\nthe sign with which it occurs in Eq.~\\eqref{eq:momentum_PLES}}. During the kinematic stage (I), $\\avg{\\Pi^{M}}_{V,N}$ is \nnegligible and the total SGS energy transfer is carried by $\\avg{\\Pi^{I}}_{V,N}$.  \nAs expected $\\langle \\Pi^{I}\\rangle_{V,N}$ gets depleted towards stage (III) while \n$\\langle -\\Pi^{M} \\rangle_{V,N}$ increases. \nBoth $\\avg{\\Pi^{I}}_{V,N}$ and $\\avg{-\\Pi^{M}}_{V,N}$ are positive, that is, \nthe resolved-scale kinetic energy is transferred from large to small scales through inertial transfer \nas well as through the Maxwell component.\n\n\n\\begin{figure}[htbp\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG8a.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG8b.pdf}\n  \\caption{\n\tThe mean components\n    $\\avg{\\Pi^{I}}_{V,N}$ (a)\n\tand\n    $\\avg{\\Pi^{M}}_{V,N}$ (b) versus the cutoff\n\twavenumber $k_c$ at the kinematic stage (I), non-linear stage (II) and the\n\tstationary stage (III).}\n  \\label{fig:VELMAG_mean}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG9a.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG9b.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG9c.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG9d.pdf}\n  \\caption{\n       \n        {Fluctuations of}\n        $\\Pi^{I}$ and $\\Pi^{M}$ during\n        kinematic (I), non-linear (II) and stationary (III) stages:\n\tpdfs of $\\Pi^{I}$ (a) and $\\Pi^{M}$ (b) at $k_c=20$;\n\tflatness of $\\Pi^{I}$ (c) and $\\Pi^{M}$ (d) against the\n\tcutoff wavenumber $k_c$.\n        }\n  \\label{fig:VELMAG}\n\\end{figure}\n\nFigures~\\ref{fig:VELMAG} (a,b) show the standardized pdfs of \n{$\\Pi^{I}$ and $\\Pi^{M}$, respectively, where we note that \nthe pdf of $\\Pi^{M}$ is only shown in stages (II) and (III) as it \nis negligible in stage (I)}. \nDuring stages (II) and (III) the pdf of $\\Pi^{I}$ changes significantly\ncompared to its shape during stage (I), where the inertial dynamics\nare approximately unaffected by the magnetic field. The most striking \nfeature here is the development of wide tails and a much more symmetric\nshape. That is, the inertial SGS energy transfer fluctuates very differently\nin presence of a fluctuating magnetic field as in the nonconducting case: \nFirst, the wide tails indicate that extreme events are more likely than in \nthe nonconducting case. Second, the symmetric shape implies that backscatter\nevents in the inertial SGS energy transfer become significant. \nIn contrast, as can be seen from Fig.~\\ref{fig:VELMAG} (b), \nthe pdf of $\\Pi^{M}$ has a clear positive skewness. That is, \nbackscatter events are much less important than for all other\nSGS energy transfer components and the contributions from the SGS \nMaxwell stresses should be well approximated by a dissipative model.\n\nMeasurements of the pdfs of ${\\Pi^{I} + \\Pi^{I,L}}$ and $\\Pi^{M} + \\Pi^{M,L}$ during the saturated\nstage of a small-scale dynamo have been reported recently \\cite{Kessar16}.  \nBy comparison of Figs.~\\ref{fig:VELMAG} (a,b) with the left panel of Fig.~7 in \nRef.~\\cite{Kessar16}, one observes that the shape of the pdfs measured in Ref.~\\cite{Kessar16}\nis quite different from the results found here {for $\\Pi^{I}$ and $\\Pi^{M}$} . More precisely, \nthe pdf of $\\Pi^{I} + \\Pi^{I,L}$ in Ref.~\\cite{Kessar16} \nlacks the wide tails seen {for $\\Pi^{I}$} here,\nand the pdf of $\\Pi^{M} + \\Pi^{M,L}$\nis much more symmetric than that presented \n{for $\\Pi^{M}$} in Fig.~\\ref{fig:VELMAG}(b). \n{There are two reasons for latter difference. First,  \nthe Leonard component is included in the measurement of the SGS\nenergy transfer in Ref.~\\cite{Kessar16} while it is not included here. \nSecond, the Reynolds numbers and filter widths also differ. \nIn Ref.~\\cite{Kessar16} the the pdfs were measured at ${\\rm Re}_\\lambda =75$ at a filter scale coresponding to \n$k_c=64$. For comparison, in our dataset ${\\rm Re}_\\lambda =211$, and the pdfs in Figs.~\\ref{fig:VELMAG} (a,b) \nare measured at $k_c=20$. Even in our simulations, it can be seen from \nthe energy spectra (Fig.~\\ref{fig:En_evo}(b)) and the mean SGS energy transfer \n(Fig.~\\ref{fig:TOT_mean}) that the dynamics at $k_c=64$ is significantly affected \nby viscous and Joule dissipation. This will be even more so for lower ${\\rm Re}_\\lambda$.\nIn order to provide a like-for-like comparison, we measured of the pdfs of \n$\\Pi^M$, $\\Pi^{M,L}$ and $\\Pi^{M} + \\Pi^{M,L}$ for $k_c = 80$, which for our data at ${\\rm Re}_\\lambda =211$ \nis comparable to $k_c=64$ for ${\\rm Re}_\\lambda =75$.\n}\nAs can be seen in Fig.~\\ref{fig:MAG_LEO} in the Appendix,\n{the pdf of $\\Pi^{M,L}$ in the viscous range is sizeable and symmetric, such that}\nthe inclusion of {$\\Pi^{M,L}$}\nin the measurement of the Maxwell SGS transfer\nmasks the distinctive positive skewness of its PDF.\n\n\nFigures~\\ref{fig:VELMAG}(c,d) present the flatness of $\\Pi^{I}$ and $\\Pi^{M}$\nas functions of $k_c$. For $\\Pi^{I}$, the development of \n{strongly non-Gaussian statistics}\nis also reflected in the flatness, which has higher values in stage (III) compared \nto stages (I) and (II). Furthermore, the flatness has a much weaker scale-dependence \nduring stage (III) as shown in Fig.~\\ref{fig:VELMAG}(c). This indicates a depletion of intermittency \nof the velocity field in presence of a saturated dynamo. \n{Indeed, a} comparison of the $p^{\\rm th}$-order \nscaling exponents $\\zeta_p$ of the velocity-field structure functions \nfor hydrodynamic turbulence \\cite{Gotoh02} and for a saturated MHD dynamo \\cite{Haugen04} \nreveals differences in $\\zeta_p$ for $p \\geqslant 5$. According to these results, \nthe velocity field is less intermittent in presence of a saturated dynamo, as\nobserved here. \nSince $(\\Pi^{I})^p$ is related to the $3p^{\\rm th}$-order velocity-field \nstructure function \\cite{eyink2} the scaling properties of high-order structure functions determine\nthe behavior of the flatness of $\\Pi^{I}$. Therefore differences concerning intermittency between MHD and \nhydrodynamic turbulence are more clearly visible in measurements of the flatness \nof $\\Pi^{I}$ compared to direct measurements of $\\zeta_p$. \nHowever, a quantitative assessment of the scaling properties of the flatness \nof $\\Pi^{I}$ requires a further extended scaling range.    \nIn contrast to the results for $\\Pi^{I}$, the flatness of $\\Pi^{M}$ shown in \nFig.~\\ref{fig:VELMAG}(d) retains its scale-dependence after \ndynamo saturation. \nAs can be seen from the figure, the flatness of $\\Pi^{M}$ has a much stronger scale \ndependence compared to $\\Pi$. \n\\ml{The stronger intermittent signal in $\\Pi^{M}$ may be related to the \nfact that the saturated magnetic field is much more intermittent than the velocity \nfield that maintains it, as shown by measurements of scaling exponents of inertial and magnetic\nstructure functions obtained from DNSs of stationary small-scale dynamos \\cite{Haugen04}.}\nAs in the present data, no mean magnetic field was present in the data analysed in \nRef.~\\cite{Haugen04}. \n\n{As shown in Fig.~\\ref{fig:VELMAG_mean}(a), the mean inertial interscale energy\ntransfer is weakened in presence of a saturated dynamo. This partly occurs\nthrough cancellations of forwards and inverse transfers since backscatter\nevents in $\\Pi^{I}$ now occur more frequently as already discussed.\nAdditionally, an overall depletion of the fluctuations of $\\Pi^{I}$ occurs, as\ncan be seen from  the comparison of the pdfs of $\\Pi^{I}$ and $\\Pi^{M}$ and\n$\\Pi^{u}$ presented in Fig.~\\ref{fig:VELMAG-comp}.}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{FIG10a.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG10b.pdf}\n  \\includegraphics[width=0.5\\textwidth]{FIG10c.pdf}\n  \\caption{\n       \n        {Fluctuations of}\n        $\\Pi^{I}$ and $\\Pi^{M}$ during\n        kinematic (I), non-linear (II) and stationary (III) stages at $k_c=20$.\n        }\n  \\label{fig:VELMAG-comp}\n\\end{figure}\n\n\\begin{figure*}[htbp]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{FIG11a.png}\n \\includegraphics[width=0.9\\textwidth]{FIG11b.pdf}\n \\caption{(Colour online)\n   Top: Two-dimensional visualisations of the inertial and Maxwell SGS energy \n   transfers $\\Pi^{I}$ (left) and $-\\Pi^{M}$ (right)  at $k_c=20$ \n   in stage (III). \n   Positive values correspond to forward energy transfer while negative values\n   incicate backscatter.\n   {Bottom: Corresponding joint pdf of $\\Pi^{I}$ and $-\\Pi^{M}$.}\n }\n \\label{fig:Visu_PiVM}\n\\end{figure*}\n\nThe clear forward transfer of energy {in stage (III)} associated with the Maxwell stress is also\nvisible in the 2D visualisations of $\\Pi^{I}$ and $\\Pi^{M}$ presented \nin {the top panels of} Fig.~\\ref{fig:Visu_PiVM}. Unlike $\\Pi^{I}$, $\\Pi^{M}$ shows very intense and \nlocalized regions of forward transfer.\n{As discussed earlier, the pdf of $\\Pi^{I}$ becomes quite symmetric in stage (III), indicating that\npositive and negative fluctuations of $\\Pi^{I}$ occur with similar probabilities. \nThis is also visible in the visualisations, where we see regions of forward and inverse transfer which are of comparable intensity. The \nfluctuations of $\\Pi^{I}$ also appear to be  much weaker than those of $\\Pi^{M}$. \nFinally, we find that $\\Pi^{I}$ and $\\Pi^{M}$ \\blue{have a relation of weak inverse proportionality}\nas can be seen from their joint \npdf presented in the bottom panel of Fig.~\\ref{fig:Visu_PiVM}. \nThe latter suggests that the transfer of kinetic energy between resolved scales and SGS is more \nlikely to occur {\\em separately} through $\\Pi^{I}$ or $\\Pi^{M}$ rather than simultaneously through \nboth.}  \n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper, we investigated the different components of the SGS energy transfer \nthrough three stages of dynamo evolution considering mean and fluctuating properties. \nWe decomposed the total SGS energy transfer in the components corresponding to\neither the momentum or the induction equation, thus separating kinetic from \nmagnetic SGS energy transfer. The kinetic SGS energy transfer was then further\nsplit into an inertial component and a component originating from the Lorentz\nforce. By also distinguishing between the actual SGS energy transfers and \nresidual contributions from interactions amongst the resolved scales, we got \nclear measurements of the fluctuating individual SGS energy transfers.\n\nConcerning the velocity field, important differences are present between the\nstatistical properties of the inertial SGS energy transfer in presence of a\nsaturated dynamo and in the nonconducting case.  First, the kinetic energy\ncascade is depleted in the saturated dynamo regime{, see Figs.~\\ref{fig:VELMAG_mean}(a) and \n\\ref{fig:VELMAG-comp}}.  Second, we find that the\npdf of the inertial SGS energy transfer becomes more symmetric and less\nGaussian than in the non-conducting case with wider tails suggesting more\nextreme events also in terms of backscatter{, see Figs.~\\ref{fig:VELMAG}(a) and \n\\ref{fig:VELMAG-comp}}.  Third, we found quantitative evidence \nthat the flatness of the\ninertial SGS energy transfer has a weaker scale dependence, which\n\\ml{suggests} that the velocity field \\ml{may be}\nless intermittent in presence of a saturated small-scale\ndynamo than in the nonconducting case{, see Fig.~\\ref{fig:VELMAG}(c)}.  \nThis latter case deserves a more quantitative investigation \nby increasing the statistics and by extension of the involved scales.\nConcerning the magnetic field, we find\nthat the pdf of the magnetic energy transfer is pretty symmetric in both the\nnonlinear and the saturated dynamo regimes{, see Fig.~\\ref{fig:VTOTBTOT}(b)}.  \nIn contrast, the SGS energy\ntransfer originating from the Maxwell stress in the momentum equation is\nclearly skewed towards positive values{, see Figs.~\\ref{fig:VELMAG}(b) and \n\\ref{fig:VELMAG-comp}}.\n\n\\ml{In terms of fundamental results on interscale energy transfer in MHD turbulence, \nthe filtering technique is a useful alternative to spectral approaches.  \nAccording to analyses of shell-to-shell transfers, magnetic and velocity-field modes couple \nat disparate wave number shells \\cite{Verma04,Alexakis05a,Mininni05a,Alexakis07}, \nleading to nonlocal contributions to the conversion of kinetic to magnetic energy in Fourier space. \n\\blue{\nAs can be seen from Eqs.~\\eqref{eq:evol_Eu} and \\eqref{eq:evol_Eb}, \nthe conversion of resolved-scale kinetic to\nmagnetic energy involves resolved scales only. Although not assessed\nhere, the energy conversion term for the SGS energies is\nalso closed in terms of the SGS \\cite{Aluie17}. That is, the conversion terms do\nnot couple the resolved scales with the SGS. In summary, the\n}\t\nfiltering technique shows that energy conversion across the filter scale does \nnot occur \\cite{Aluie17}.\nThe degree of locality of energy cascades is certainly affected by the presence of large-scale fields, such as in \nrotating turbulence, two-dimensional flows or in the presence of magnetic and kinetic helicity \\cite{Alexakis18}, \nrequiring further analysis of the effect of SGS closures on higher-order statistics \\cite{Linkmann18}. \nHence, separate {\\em a-priori} studies are required in order to provide guidance for LES modeling in such cases,\nas e.g for large-scale dynamos \\cite{Kessar16}.  \n}\n\nIn terms of guidance for LES modelling, the symmetry  of the\nmagnetic SGS energy transfer pdf implies that backscatter events are important,\nwhich calls applications of dissipative models for the stresses in the\ninduction equation into question. For the momentum equation, a similar situation\noccurs for the inertial SGS energy transfer in the saturated stage of the dynamo.\nAs a result, while a dissipative model for the inertial stresses may be suitable \nduring the kinematic stage, a more sophisticated approach is required to adequately \ncapture the increased backscatter in the nonlinear and saturated stages.   \nOn the other hand, dissipative models would be well suited for the Maxwell stress in both \nnonlinear and saturated stages.  \n{Finally, we find that the \\blue{correlation between the}\nindividual SGS energy transfers appears to be \\blue{of inverse proportionality}\nin the saturated stage. \nThis holds for $\\Pi^u$ and $\\Pi^b$ and also for $\\Pi^I$ and $\\Pi^M$. That is, the energy transfers in the different channels \nappear to occur separately, which should be taken into account in the design of more sophisticated LES models for MHD.}\nHowever, measurements of the correlations between the different SGS energy transfers {at higher Reynolds numbers} \nneed to be carried out in order to better quantify the effect.\n\n\n\n\\section*{Acknowledgements}\nThe research leading to these results has received funding from the European Union's Seventh\nFramework Programme (FP7\/2007-2013) under grant agreement No. 339032.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}