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+{"text":"\\section{}\n\\vskip 12mm\n\n{\\bf 1. Introduction }\n\\medskip\n\nLet $S$ denote the family of functions:\n$$f(z)=z+a_2z^2+\\cdots\\eqno{(1.1)}$$\nwhich are analytic and univalent in the unit disk $E=\\{z\\colon\n|z|<1\\}$ and map $E$ onto some domains $D$. A function $f\\in S$ is said to be starlike if the domain $D$ is starlike with respect to the origin. The family of starlike functions is denoted by $S^\\ast$. It is known that a function $f\\in S$ is starlike if and only if it satisfies the geometric condition:\n$$Re\\frac{zf'(z)}{f(z)}>0.\\eqno{(1.2)}$$\n\nThe above geometric condition implies that the quantity $zf'(z)\/f(z)$ belongs to the class $P$ of analytic functions:\n$$p(z)=1+c_1z+c_2z^2+\\cdots\\eqno{(1.3)}$$\nwhich have positive real part in $E$. The family of starlike functions in the unit disk has attracted much attention in the past. The volume of work being published on this family of functions leaves no one in doubt about the importance attached to it both in the past and the present.\n\nIn this paper we determine, under certain conditions, the best possible upper bounds on some functionals defined in the coefficients space of starlike functions. These are functionals which have arisen from the study of coefficient problems of certain family of univalent functions. The study of functionals similar to those being considered in this paper is not new. For instance, bounds on $|a_n|$ and $|a_3-\\lambda a_2^2|$ can be found in many literatures (see for example \\cite{PN}-\\cite{RS}). In particular, the functional $|a_3-\\lambda a_2^2|$ is known as the Fekete-Szego functional for both real and complex values of the parameter $\\lambda$. The problem of determining the sharp bound on the Fekete-Szego functional has its origin in a conjecture of Littlewood and Parley (1932) that the true bound on the coefficients of an odd univalent function is 1, which was disproved in 1933 be Fekete and Szego via the determination of the sharp bound on the functional (see \\cite{PL}).\n\nThe functional has since continued to recieve attention of researchers in geometric function theory.\n\nIn Section 3, we consider functionals of the form $|a_4-\\gamma a_2a_3|$, $|a_4-\\gamma a_2a_3-\\eta a_2^3|$, $|a_5-\\mu a_2^2a_3|$ and $|a_5-\\xi a_2a_4-\\zeta a_2^3|$ where the parameters $\\gamma, \\eta, \\mu, \\xi, \\zeta$ are all real numbers. These functionals have been found to have applications in certain coefficient problems, which are of independent interest.\n \\medskip\n\n{\\bf 2.0 Preliminary Lemmas}\\vskip 2mm\n\nWe shall need the following well known inequalities.\\vskip 2mm\n\n{\\bf Lemma 2.1}(\\cite{PN}-\\cite{RS})\\vskip 2mm\n\n{\\em Let $p\\in P$. Then $|c_k|\\leq 2$, $k=1,2,3,\\cdots$. Equality is attained for the Moebius function\n$$L_0(z)=\\frac{1+z}{1-z}.\\eqno{(2.1)}$$}\n\n{\\bf Lemma 2.2}(\\cite{PL,SK})\\vskip 2mm\n\n{\\em Let $p\\in P$. Then $$\\left|c_2-\\frac{c_1^2}{2}\\right|\\leq 2-\\frac{|c_1|^2}{2}\\eqno{(2.2)}$$\nThe result is sharp. Equality holds for the function\n$$p(z)=\\frac{1+\\frac{1}{2}(c_1+\\varepsilon\\bar{c_1})z+\\varepsilon z^2}{1-\\frac{1}{2}(c_1-\\varepsilon\\bar{c_1})z-\\varepsilon z^2},\\;\\;|\\varepsilon|=1.\\eqno{(2.3)}$$}\n\nNote that the inequality (2.2) can be written as\n$$c_2=\\frac{1}{2}c_1^2+\\varepsilon\\left(2-\\frac{1}{2}|c_1|^2\\right),\\;\\;|\\varepsilon|\\leq 1.\\eqno{(2.4)}$$\n\n\\medskip\n\n{\\bf 3.0 Main Result}\\vskip 2mm\n\n{\\bf Theorem 3.1}\\vskip 2mm\n\n{\\em Let $f(z)$ given by (1.1) be starlike function. Then for real numbers $\\gamma, \\eta, \\mu, \\xi, \\zeta$ such that $1-\\gamma, 1-2\\mu, 1-\\xi, 1-2\\zeta$ and $1-2\\xi-2\\zeta$ are all nonnegative, we have the sharp inequalities:\n\n$$|a_4-\\gamma a_2a_3|\\leq 4-6\\gamma;\\;\\;if\\;\\;\\gamma\\leq\\frac{5}{9},$$\n\n$$|a_4-\\gamma a_2a_3-\\eta a_2^3|\\leq\n4-6\\gamma-8\\eta;\\;\\;if\\;\\;3\\gamma+4\\eta\\leq\\frac{5}{3},$$\n\n$$|a_5-\\mu a_2^2a_3|\\leq 5-12\\mu;\\;\\;if\\;\\;\\mu\\leq\n\\frac{2}{9},$$\n\n$$|a_5-\\xi a_2a_4-\\zeta a_3^2|\\leq\n5-8\\xi-9\\zeta;\\;\\;if\\;\\;5\\tau+9\\omega\\leq 2.$$}\n\n\\begin{proof} Since $f(z)$ is starlike, there exists $p\\in P$ such that\n$$zf'(z)=p(z)f(z)\\eqno{(3.1)}$$\nComparing coefficients of both sides of (3.1) using (1.1) and (1.3) we see that\n$$a_2=c_1$$\n$$2a_3=c_2+c_1^2$$\n$$6a_4=2c_3+3c_2c_1+c_1^2$$\n$$24a_5=6c_4+8c_3c_1+6c_2c_1^2+3c_2^2+c_1^4$$\n$$\\cdots\\;\\cdots\\;\\cdots\\;\\cdots\\;\\cdots$$\nso that\n$$a_4-\\gamma a_2a_3=\\frac{c_3}{3}+(1-\\gamma)\\frac{c_1}{2}\\left\\{c_2+\\frac{2(1-3\\gamma)}{3(1-\\gamma)}\\frac{c_1^2}{2}\\right\\}\\eqno{(3.2)}$$\n$$a_4-\\gamma a_2a_3-\\eta a_2^3=\\frac{c_3}{3}+(1-\\gamma)\\frac{c_1}{2}\\left\\{c_2+\\frac{2(1-3\\gamma-6\\eta)}{3(1-\\gamma)}\\frac{c_1^2}{2}\\right\\}\\eqno{(3.3)}$$\n$$a_5-\\mu a_2^2a_3=\\frac{c_4}{4}+\\frac{c_3c_1}{3}+\\frac{c_2^2}{8}+(1-2\\mu)\\frac{c_1^2}{4}\\left\\{c_2+\\frac{1-12\\mu}{3(1-2\\mu)}\\frac{c_1^2}{2}\\right\\}\\eqno{(3.4)}$$\n$$\\aligned\na_5-\\xi a_2a_4-\\zeta a_3^2\n&=\\frac{c_4}{4}+(1-\\xi)\\frac{c_3c_1}{3}+(1-2\\zeta)\\frac{c_2^2}{8}\\\\\n&+(1-2\\xi-2\\zeta)\\frac{c_1^2}{4}\\left\\{c_2+\\frac{1-4\\xi-6\\zeta}{3(1-2\\xi-2\\zeta)}\\frac{c_1^2}{2}\\right\\}\\endaligned\\eqno{(3.5)}$$\n\nRecall that the real numbers $1-\\gamma, 1-2\\mu, 1-\\xi, 1-2\\zeta$ and $1-2\\xi-2\\zeta$ are nonnegative. We eliminate $c_2$ in each of the terms in the curly brackets in (3.2) - (3.5) using the equality (2.4). For instance, we have from (3.2),\n$$c_2+\\frac{2(1-3\\gamma)}{(3(1-\\gamma)}\\frac{c_1^2}{2}=\\frac{5-9\\gamma}{3(1-\\gamma)}\\frac{c_1^2}{2}+\\varepsilon\\left(2-\\frac{|c_1|^2}{2}\\right).\\eqno{(3.6)}$$\n\nSince $2-\\frac{|c_1|^2}{2}\\geq 0$, the absolute value of (3.6) attains its maximum for $|c_1|=2$ provided $\\gamma\\leq\\frac{5}{9}$ (which is the condition given in the first inequality of the theorem). Thus (3.6) yields  \n$$\\left|c_2+\\frac{2(1-3\\gamma)}{3(1-\\gamma)}\\frac{c_1^2}{2}\\right|\\leq\\frac{2(5-9\\gamma)}{3(1-\\gamma)},\\eqno{(3.7)}$$\nso that, by triangle inequality and Lemma 2.1, (3.2) yields the first inequality of the theorem. Similar arguments and computations from (3.3) to (3.5), lead to the remaining inequalities respectively.\n\nFor each of the real numbers $\\gamma, \\eta, \\mu, \\xi$ and $\\zeta$, equality is attained in each case by the Koebe function (up to rotations) given by:\n$$k(z)=\\frac{z}{(1-z)^2}.\\eqno{(3.8)}$$\n\\end{proof}\n \n\\medskip\n\n{\\bf 4.0 Conclusion}\\vskip 2mm\n\nThe study of functionals in the theory of analytic and univalent functions is here boosted with the consideration of new ones. The functionals considered in this work are closely associated with certain coefficient problems in geometric functions theory.\n\n\\bigskip\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n   Recent ASCA, RXTE, Chandra and XMM-Newton observations\nof Seyfert~I galaxies have demonstrated the existence of the broad\niron $K_\\alpha$ line (6.4~keV) in their spectra along with a\nnumber of other weaker lines (Ne~X, Si~XIII, XIV, S~XIV-XVI,\nAr~XVII, XVIII, Ca~XIX, etc.) (see, for example,\n\\citealt{fabian1,tanaka1,nandra1,nandra2,malizia,sambruna,\nyacoob4,ogle1}).\n\n\n     For some cases when the spectral resolution is good enough,\nthe emission spectral line demonstrates the typical two-peak\nprofile with a  high \"blue\" peak and a low \"red\" peak while a long\n\"red\" wing drops gradually to the background level\n(\\citealt{tanaka1,yaqoob2}, see also \\citealt{Reynolds03} and\nreferences therein). The Doppler line width corresponds to a very\nhigh velocity of  matter.\\footnote{Note that the detected line\nshape differs essentially from the Doppler one.} E.g., the maximum\nvelocity is about $v \\approx 80000 - 100000$~km\/s for the galaxy\nMCG--6--30--15 \\citep{tanaka1,Fabian02} and $v \\approx 48000$~km\/s\nfor \\mbox{MCG--5--23--16} \\citep{krolik1}. For both galaxies line\nprofiles are known rather well. \\cite{Fabian02} analyzed results\nof long-time observations of MCG-6-30-15  using {\\it XMM-Newton}\nand {\\it BeppoSAX}. They confirmed in general the qualitative\nconclusions about the features of the Fe $K_\\alpha$ line, which\nwere discovered by ASCA satellite. \\citet{yaqoob02} discussed the\nessential importance of ASCA calibrations and the reliability of\nobtained results. \\cite{Lee02} compared data among ASCA, RXTE and\nChandra  for the MCG-6-30-15. \\cite{Iwa99,Lee99,Shih02} analyzed\nin detail the variabilities in continuum and in Fe $K_\\alpha$ line\nfor MCG-6-30-15 galaxy.\n\n     The phenomena of the broad emission lines are supposed\nto be related with accreting matter around black holes.\n\\cite{Wilms01,Ball01,Mart02} proposed physical models of accretion\ndiscs for  MCG-6-30-15 and showed their influence on the Fe\n$K_\\alpha$ line shape. \\cite{boller01} found the features of the\nspectral line near  7~keV in Seyfert galaxies with data from {\\it\nXMM-Newton} satellite. \\cite{yaqoob01a} presented results of\nChandra HETG observations of Seyfert~I galaxies. ~\\cite{Qing01}\ndiscussed a possible identification of binary massive black holes\nwith the analysis of Fe $K_\\alpha$ profiles. \\cite{Ball02} used\nthe data of X-ray observations to estimate an abundance of the\niron. \\cite{Popov01,Popov02} discussed an influence of\nmicrolensing on the distortion of spectral lines (including Fe\n$K_\\alpha$ line) that can be significant in some cases, optical\ndepth for microlensing in X-ray band was evaluated by\n\\cite{Zakharov04}. \\cite{matt02} analyzed an influence of Compton\neffect on emitted and reflected spectra of the Fe $K_\\alpha$\nprofiles. In addition, \\cite{fabian99a} presented a possible\nscenario for evolution of such supermassive black holes.\n\\cite{moral01} proposed a procedure to estimate the masses of\nsupermassive black holes.\n\n     General status of black holes was described in\na number of papers (see, e.g. \\citealt{Liang98} and references\ntherein, \\citealt{Zak00,FN01,Cherep03}). Since the matter motions\nindicate very high rotational velocities, one can assume the\n$K_\\alpha$ line emission arises in the inner regions of accretion\ndiscs at distances $\\sim (1\\div 3)~r_g$ from the black holes. Let\nus recall that the innermost stable circular orbit for\nnon-rotational black hole (which has the Schwarzschild metric) is\nlocated at the distance of $3\\,r_g$ from the black hole\nsingularity. Therefore, a rotation of black hole could be the most\nessential factor. A possibility to observe the matter motion in so\nstrong gravitational fields could give a chance not only to check\ngeneral relativity predictions and simulate physical conditions in\naccretion discs, but investigate also observational manifestations\nof such astrophysical phenomena like jets\n\\citep{romanova1,romanova2}, some instabilities like Rossby waves\n\\citep{Love99} and gravitational radiation.\n\n     Observations and theoretical interpretations of broad\nX-ray lines (particularly, the iron $K_\\alpha$ line) in AGNs are\nactively discussed in a number of papers\n\\citep{yaqoob1,wanders,sulentic1,sulentic2,paul,Bia02,Tur02,\nLev02a}. The results of numerical simulations are also presented\nin the framework of different physical assumptions on the origin\nof the broad emissive iron $K_\\alpha$ line in the nuclei of\nSeyfert galaxies\n\\citep{Matt92a,Matt92b,bromley,pariev2,pariev1,cui,bromley2,pariev3,\nma02a,Ma02,karas01}. The results of Fe $K_\\alpha$ line\nobservations and their possible interpretation are summarized by\n\\citet{fabian2}.\n\nAccording to the standard interpretation these $K_\\alpha$ lines\nare formed due to the  cold thin and optically thick accretion\ndisk illumination by hot clouds \\citep{Fabian89,Laor91}, however\nanother geometry for regions of hot and cold clouds located near\nblack holes is not excluded. For example,\n\\cite{Hartnoll00,Hartnoll01,Hartnoll02,Blackman02} considered a\nmore complicated structure of accretion disks including warps,\nclumps and spirals. \\cite{Karas00} investigated a possibility to\nexplain Fe $K\\alpha$ line with the model that the innermost part\nof a disk is disrupted owing to disk instabilities and forms cold\nclouds which move not exactly in the equatorial plane, but they\nform a layer (or shell) covering a significant part of sky from\nthe point of view of central X-ray source (actually, that is a\ndetailed analysis of ideas suggested by \\cite{Collin96}). Other\nfeatures of such a model were discussed by \\cite{Malzac01}. An\ninfluence of warps on X-ray emission line shapes were investigated\nrecently by \\cite{Cadez03} analyzing photon geodesics in the\nSchwarzschild black hole metric.\n\n     Broad spectral lines are considered to be formed by\nradiation  emitted in the vicinity of black holes.  If there are\nstrong magnetic fields near black holes these lines are split by\nthe field into several components. Such lines have been found in\nmicroquasars, GRBs and other similar objects\n\\citep{Balu99,grein99,mira00,Lazz01,Mart02a,Mira02a,\nMiller02,zaman02}.\n\n     To obtain an estimation of the magnetic field we simulate\nthe formation of the line profile for different values of magnetic\nfield in the framework of the simple model of  non-flat accretion\nflows assuming that emitting particles move along orbits with\nconstant radial coordinates, but not exactly in the equatorial\nplane. Earlier, \\cite{ZKLR02} analyzed an influence of magnetic\nfield on a distortion of $K_\\alpha$ line considering equatorial\ncircular motion of emitting region of the Fe $K_\\alpha$ line\nradiation.\\footnote{Recently, \\cite{Loeb03} has analyzed Zeeman\nsplitting for X-ray absorbtion lines in the X-ray spectrum of the\nbursting neutron star EXO 0748-676.} Here we will use the simple\nmodel of a non-flat accretion flow \\citep{ma02a,Ma02}\n to analyze\nthe non-equatorial plane motion of particles emitting X-band\nphotons. Actually, we will use a generalization of the previous\nannulus model described earlier by \\cite{zak_rep1}.\n As a result we find the minimal $H$ value of magnetic field at which the\ndistortion of the line profile becomes significant. Here we do not\nuse an approach, which is based on numerical simulations of\ntrajectories of the photons emitted by annuli moving along a\ncircular geodesics near black hole, described earlier by\n\\cite{zakharov6,zakharov1,zakharov5,zak_rep1}. In this paper  we\ngeneralize previous considerations for the the simple model of\nnon-flat accretion flow.\n\n\\section{Magnetic fields in accretion discs}\n\n     Magnetic fields play a key role in dynamics of accretion\ndiscs and jet formation. \\cite{Bis74,Bis76} considered a scenario\nto generate super strong magnetic fields near black holes.\nAccording to their results magnetic fields near the marginally\nstable orbit could be about $H \\sim 10^{10} - 10^{11}$~G. However,\nif we use a model of the Poynting -- Robertson magnetic field\ngeneration then only small magnetic fields are generated\n\\citep{BKLB02}.\n \\citet{Kard95,Kard00,Kard01a,Kard01} has shown that\nthe strength of the magnetic fields near super massive black holes\ncan reach the values of $H_{max} \\approx 2.3\\cdot 10^{10}\nM_9^{-1}$~G due to the virial theorem\\footnote{Recall that\nequipartition value of magnetic field is only $\\sim 10^4$~G.}, and\nconsidered a generation of synchrotron radiation, acceleration of\n$e^{+\/-}$ pairs and cosmic rays in magnetospheres of super massive\nblack holes  at such high fields. It is the magnetic field that\nplays a key role in these models. Below, based on the analysis of\niron $K_\\alpha$ line profile in the presence of a strong magnetic\nfield, we describe how to detect the field itself or at least\nobtain an upper limit of the magnetic field.\n\nOne of the basic problems in understanding the physics of quasars\nand microquasars is the \"central engine\" in these systems, in\nparticular, a physical mechanism to accelerate charged particles\nand generate energetic electromagnetic radiation near black holes.\nThe construction of such \"central engine\" without magnetic fields\ncould  hardly ever be possible. On the other hand, magnetic fields\nmake it possible to extract energy from rotational black holes via\nPenrose process and Blandford -- Znajek mechanism, as it was shown\nin MHD simulations by \\cite{Koide02,Koide02a}. The Blandford --\nZnajek process could provide huge energy release in AGNs (for\nexample, for MCG-6-30-15) and microquasars when the magnetic field\nis strong enough \\citep{Wilms01}.\n\n    Physical aspects of generation and evolution\nof magnetic fields were considered in a set\nof reviews (e.g. \\cite{Ass87,Giov01}).\nA number of papers conclude that in the vicinity of\nthe marginally stable orbit the magnetic fields could be\nhigh enough \\citep{Bis74,Bis76,Krolik99}.\n     \\cite{Agol99} considered the influence of magnetic\nfields on the accretion rate near the marginally stable orbit and\nhence on the disc structure. They found the appropriate changes of\nthe emitting spectrum and solitary spectral lines. \\cite{vietri98}\ninvestigated the instabilities of accretion discs in the case when\nthe magnetic fields play an important role.\n\\cite{Li02a,Li02b,WLM03} analyzed an influence of magnetic field\non accretion disk structure and its emissivity through the\nmagnetic coupling of a rotating black hole with its surrounding\naccretion disk.\n\nMagnetic field could play a key role in Fe $K\\alpha$ line\nemission, since coronae around  accretion disks could be magnetic\nreservoirs of energy to provide a high energy radiation\n\\citep{Merloni01} or magnetic flares could help to understand an\norigin of narrow Fe $K\\alpha$ lines and their temporal dependences\n\\citep{Nayakshin01,Nayakshin02}. \\cite{Collin03} calculated X-ray\nspectrum for the flare model  and pointed out some signatures of\nthe model to distinguish it from the well-known lamppost model\nwhere it is assumed that an X-ray source illuminates the inner\npart of accretion disk in a relatively steady way.\n\n\\section{Influence of a magnetic field on the distortion\n         of the iron $K_\\alpha$ line profile}\n\n    The magnetic pressure at the inner edges of the accretion\ndiscs and its correspondence with the black hole spin parameter\n$a$ in the framework of disc accretion models is discussed\nby~\\cite{krolik01}. However, the numerical value of magnetic field\nis determined there from a model-dependent procedure, in which a\nnumber of parameters cannot be found explicitly from observations.\n\n    Here we consider the influence of  magnetic field on\nthe iron $K_\\alpha$ line profile \\footnote{We can also consider\nX-ray lines of other elements emitted by the area of accretion\ndisc close to the marginally stable orbit; further\nwe talk only\nabout iron $K_\\alpha$ line for brevity.} and show how one can\ndetermine the value of the magnetic field strength or at least\nan upper limit.\n\n    The profile of a monochromatic line\n\\citep{zak_rep1,zak_rep2,zak_rept} depends on the angular momentum\nof a black hole, the inclination angle of observer, the value of\nthe radial coordinate if the emitting region represents an\ninfinitesimal ring (or two radial coordinates for outer and inner\nbounds of a wide disc). The influence of accretion disc model on\nthe profile of spectral line was discussed by\n\\cite{zak_rep3,Zakharov_Repin_Ch03}.\n\n    We assume that the emitting region is located in\nthe area of a strong quasi-static magnetic field. This field\ncauses line splitting due to the standard Zeeman effect. There are\nthree characteristic frequencies of the split line that arise in\nthe emission \\citep{Blokh63,Dir58,Mess99}. The energy of central\ncomponent $E_0$ remains unchanged, whereas two extra components\nare shifted by $\\pm \\dfrac{eH}{2mc}=\\pm \\mu_B H$, where\n$\\mu_B=\\dfrac{e \\hbar}{2m_{\\rm e}c}=9.273\\cdot 10^{-21}$~erg\/G is\nthe Bohr magneton. Therefore, in the presence of a magnetic field\nwe have three energy levels: $E_0-\\mu_B H,~ E_0$ and $E_0+\\mu_B\nH$. For the iron $K_\\alpha$ line they are as follows:\n$E_0=6.4\\left(1 - \\dfrac{0.58}{6.4} \\cdot \\dfrac{H}{10^{11}\\,{\\rm\nG}} \\right) $ keV, $E_0=6.4$~keV and $E_0=6.4\\left(1  +\n\\dfrac{0.58}{6.4} \\cdot \\dfrac{H}{10^{11}\\,{\\rm G}}\\right) $ keV.\n\n\\cite{Loeb03} pointed out that for a strong field, there is also a\nnet blueshift of the centroid of the transition line component\nwhich is quadratic in $B$. For hydrogen-like ions\n\\cite{Jenkins39,Schiff39,Preston70} give\n\n\\begin{eqnarray}\n(\\Delta E) _{\\rm shift} \\sim \\frac{e^2\na_0^2}{8Z^2m_ec^2}n^4(1+M_l)B^2=\n \\nonumber \\\\\n = 9.2\\times 10^{-4}\n{\\rm eV}\n\\left(\\frac{Z}{26}\\right)^{-2}n^4(1+M_l^2)\\left(\\frac{B}{10^9 {\\rm\nG}}\\right),\n\\end{eqnarray}\nwhere $n$ and $M_L$ are the principal and orbital quantum numbers\nof the upper state, $a_0$ is the Bohr radius, and $Z$ is the\nnuclear charge (=26 for Fe).\n\n   Let us discuss how the line profile changes when photons\nare emitted in the co-moving frame with energy $E_0 (1+\\epsilon)$,\nbut not with $E_0$. In that case the line profile can be obtained\nfrom the original one by $1+\\epsilon$ times stretching along the\nenergy axis, the component with $E_0 (1-\\epsilon)$ energy should\nbe $(1-\\epsilon)$ times stretched, respectively. The intensities\nof different Zeeman components are approximately equal\n\\citep{Fock76}, each of which depends on the direction of the\nquantum escape with respect to the direction of the magnetic field\n\\citep{BLP89}. However, we neglect this weak dependence\n(undoubtedly, the dependence can be counted and, as a result, some\ndetails in the spectrum profile can be slightly changed, but the\nqualitative picture, which we discuss, remains unchanged). As a\nconsequence, the composite line profile can be found by summation\nthe initial line with energy $E_0$ and two other profiles,\nobtained by stretching this line along the $x$-axis in\n$(1+\\epsilon)$ and $(1-\\epsilon)$ times correspondingly.\n\n     Another indicator of the Zeeman effect is a significant\ninduction of the polarization of X-ray emission: the extra\nlines possess a circular polarization (right and left,\nrespectively, when they are observed along the field direction)\nwhereas a linear polarization arises if the magnetic field is\nperpendicular to the line of sight.\\footnote{Note that another\npossible polarization mechanisms in $\\alpha$-disc were discussed\nby~\\citet{Saz02}.} Despite of the fact that\nthe measurements of polarization of X-ray emission have not\nbeen carried out yet, such experiments can\nbe realized in the nearest future \\citep{Cos01}.\n\n     With increase of the magnetic field\nthe peak profile structure becomes apparent and can be distinctly\nrevealed, however, the field $H \\sim 2\\cdot 10^{11}$~G is rather\nstrong, so the classical linear expression for the Zeeman\nsplitting\n\\begin{equation}\n     \\epsilon=\\frac{ \\mu_B H}{E_0}\n      \\label{eq15a}\n\\end{equation}\nshould be modified. Nevertheless, we use Eq.(\\ref{eq15a}) for any\nvalue of the magnetic field, assuming that the qualitative picture\nof peak splitting remains unchanged, whereas for $H = 2\\cdot\n10^{11}$~G the exact maximum positions may appear slightly\ndifferent. If the Zeeman energy splitting $\\Delta E$ is of the\norder of $E$, the line splitting due to magnetic fields is\ndescribed in a more complicated way. The discussion of this\nphenomenon is not a point of this paper. Our aim is to pay\nattention to the qualitative features of this effect.\n\nThus, besides magnetic field, the line profiles depend on the\naccretion model as well as on the structure of emitting regions.\nProblems of such kind may become actual with much better accuracy\nof observational data in comparison with their current state.\n\n\\section{Non-flat accretion flows and  iron K$\\alpha$ line shapes}\n\nThe relativistic generalization of Liouville's theorem was used to\ncalculate the spectral flux by many authors (e.g.\n\\citealt{thorne67,ames,gerlach}). Just after \\cite{thorne74}\n  finished the analysis of the time-averaged\nstructure of a thin, equatorial disk of material accreting onto a\nblack hole, \\cite{cunningham} used Liouville's theorem to give a\nprediction about the X-ray continuum from the disk.\n\nSimulations of iron line profiles started from the paper by\n\\cite{Fabian89}. These calculations are based on assumptions about\ngeometrically thin, optically thick disks.\n These results were generalized by \\cite{Laor91}\n to a Kerr black hole case. Many authors\n(e.g. \\citealt{bromley,dabrowski}) also used the thin-disk model\nwith Cunningham's approach: The solid angle was evaluated as a\nfunction of both the emission radius $r$ and the frequency shift\n$g$; The propagation of the line radiation was considered to emit\nfrom the thin disk in the range from the innermost radius $r_{in}$\nto the outermost radius $r_{out}$. Thus, the flux carried by a\nbundle of photons from a whole disk needs the integration over\n$r$. In this case, a single value of $\\theta=90^\\circ$ was fixed\nin advance to simulate the thin-disk. Another approach to simulate\nline profiles for flat accretion disk was proposed by\n\\cite{zakh91,zakharov1,zakharov5,zak_rep1,Zak02pr,zak_rep2,\nzak_rept,Zak_rep02_xeus,Zak_rep02_Gamma,Zak_Rep03_Lom,Zak_rep03_aa,Zak_rep03_azh,Zak03_Sak,Zakharov_Repin_Pom04,zak_rep3}\nwhich is based on qualitative analysis developed in previous\npapers by \\cite{zakh86,zakh89}.\n\nIn fact, different from the continuum, iron $K\\alpha$ line is\nbelieved to originate via fluorescence in the very inner part of a\ndisc (e.g. \\citealt{tanaka1,iwasawa}), within which particles\nexist in spherical orbits between the minimum and maximum\nlatitudes about the equatorial plane of the central black hole\n\\citep{wilkins,ma00} in the form of a hot torus \\citep{chen} or a\nshell (or a layer) formed by cold clouds which could be\nilluminated by hot clouds \\citep{Karas00,Malzac01}. Therefore, the\nmechanism of the Fe line emissions should be re-considered with\nthe non-disk formulation, which is connected with several\nparameters in both sets of coordinates, such as the spin of the\nblack hole, particles' constants of motion, photons's impact\nparameters, the thickness and the radial position of the shell,\nthe polar angle of the shell, etc. In our work, as done by\nprevious authors with the thin-disk model\n\\citep{gerlach,Laor91,dabrowski}, we make following assumptions:\n(1) The emitting \"shell\" is geometrically thin; i.e., at radius\n$r$ its thickness $\\delta r$ is always much less than $r$. This\npermits us to treat particles as a thin-shelled ensemble at $r$\nsurrounding a BH. (2) The emission is isotropic and the emitted\nfluorescent Fe K$\\alpha$ line from particles can be described by a\n$\\delta$-function in frequency, which gives each emitted photon an\nenergy of 6.4~keV. That is, particles are monochromatic. (3)\nPhotons emitted by shell particles are homogeneous and are free to\nreach the observer.\n\nFor a given $a$, there are sets of three constants of motion at\none radius $r$. That is, a thin shell means a collection of sets\nof three constants of motion, $q$, $p$, and $\\varepsilon^{2}$.\nConsidering the monotonous relations between $p$ and $q$ (or\n$\\varepsilon^{2}$), the number of the sets can be simply\nrepresented by $q$. Therefore, different from the fact that the\ncontinuum is an integration over $r$ in disk models, the line flux\nshould be a superimposition of all possible individual emissions\nwith every $q$ at $r$.\n\nWith the thin-disk model, previous authors (e.g.,\n\\citealt{Laor91}) considered the emitted intensity $I_{e}(r,\\\n\\nu)$ as a created parameter $I_{e}(r,\\\n\\nu)=\\delta(\\nu-\\nu_{e})J(r)$, in which $J(r)$ is defined as \"the\nline-emissivity law\" with different artificial forms versus the\nradius of emission $r$ only; $\\nu_{e}$ is the rest frequency of\nthe emission. Fortunately, a more realistic form of $J$ was\ndeduced by \\cite{george}, in which $J$ is expressed as a function\nof $r$, $\\nu$ and $\\theta$. However, Liouville's equation contains\nnot only the invariant photon four-momentum, but the four-velocity\n\\& four-coordinate components of particles as well.  Let $I$ (ergs\ns$^{-1}$ cm$^{-2}$ sr$^{-1}$ eV$^{-1}$) be the specific intensity\nand $N$ (cm$^{-3}$ dyn$^{-3}$ s$^{-3}$) the photon distribution\nfunction. The relationship between $I$ and $N$ is\n(\\cite{thorne67}): $I=\\dfrac{1}{2}\\times2hN\\nu_e^3$, where $h$ is\nPlanck constant, $\\nu_e=\\dfrac{6.4\\mathrm{~keV}}{h}$ is the\nK$\\alpha$ frequency of an iron atom with 4-coordinate $x^{\\mu}$\nand 4-velocity $u^{\\mu}$. Coefficient ${1}\/{2}$ means only half of\nphotons emitted outwards and 2 indicates that there exists both\nstates of the photon quantum per phase-space. The expression of\n$N$ is (\\cite{gerlach})\n\\begin{equation}\nN(x^{\\mu},\\ u^{\\mu},\\ s^{\\mu})=C(x^{\\mu})\\ \\cdot\\ \\delta\n(u^{\\mu}s_{\\mu}-h\\nu_e)\n\\end{equation}\nwhere $C(x^{\\mu})$ (cm$^{-3}$) is photon's number density.\nAccording to the last assumption, $C=1$.\n\nThe dimensionless relative flux versus the shift\n$\\dfrac{\\nu}{\\nu_e}$ depends on three parameters: the black hole\nspin $a$ , the radial position $r$ of emitters, and the\ninclination angle ${\\rm Obs}$ of the observer. The flux expression\nis (c.f., e.g., \\cite{Laor91,bromley})\n\\begin{equation}\nF=\\frac{r_0^2}{h\\nu_e}\\ F_{\\rm line}=\\frac{r_0^2}{h\\nu_e}\\\n\\sum_q\\int d\\nu d\\Omega\\cdot I\\\n\\cdot\\left(\\frac{\\nu}{\\nu_e}\\right)^3\\cdot\\mathrm{\\cos}\\alpha\n\\end{equation}\nin which the integration over the element of the solid angle\n$d\\Omega$ covers the image of the spherical ring in the observer's\nsky plane; the solid angle is expressed by impact parameters\n$\\alpha$, $\\beta$ of the observer which are related to the\nconstants of motion; $E=h\\nu$; $\\dfrac{\\nu}{\\nu_e}$ is the general\nrelativistic frequency shift;\nthe sum $\\sum$ is to all values of $q$, which reflects that the\nflux is contributed by all photons emitted from all particles (see\n\\citealt{Ma02} for details).\n\nThis model could be interpreted as a simplified version of some\ndistribution of clouds in the \"quasi-spherical\" accretion model\ndeveloped earlier by \\cite{Celotti92,Collin96}  to evaluate\noptical\/UV\/soft X-ray emission of AGNs. We use such a distribution\nof clouds to calculate the Fe $K\\alpha$ line shapes in presence of\na strong magnetic field which could play a significant role in\nsuch models \\citep{Celotti92}.\n\nIn numerical simulations of photon geodesics we used their\nanalytical analysis to reduce numerical errors.\n    A qualitative analysis of the geodesic equations\nshowed that types of photon motion can drastically vary with small\nchanges of chosen geodesic parameters \\citep{zakh86,zakh89}. In\nour approach we use results of this analysis and numerical\ncalculations of photon geodesics \\citep{zakh91}.\n\n\\section{Simulation results}\n\nWe have calculated spectral line shapes for different parameters\nof the model. Below we briefly describe results of these\ncalculations.\n\n\\begin{figure*}[!tbh]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{fig1.EPS}\n\\end{center}\n  \\vspace{-7mm}\n  \\caption{Spectral line shapes  for $a=0.998$, $r=25$, the inclination angle of\n  observer (Obs$=90^\\circ$),\n           for different magnetic fields $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G.}\n  \\label{figure2}\n\\end{figure*}\n\nFig. \\ref{figure2} shows a series of spectral line profiles for\n$a=0.998$,\n $r=25$ and the inclination angle of observer\nObs$=90^\\circ$ (it means that an observer is located in the\nequatorial plane)\n    for magnetic fields $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G respectively.\n           Corresponding parameter values are indicated in the\n           left top angle of each panel. One could see from these\n           panels that magnetic field  $H= 5.5 \\times 10^9$G does\n           not distort significantly the spectral line profiles,\n           but  $1.1 \\times 10^{10}$G gives a significant\n           broadening the peaks of the profile and as a result the shortest red\n           peak may be not distinguishable from observational\n           point of view. The spectral line profile for $H=5.5 \\times  10^{10}$G\ndemonstrates a significant difference from the spectral line\nprofile with respectively \"low\" magnetic field like in first panel\n$H=1.1 \\times 10^9$ G, because there is an evident three peaked\nstructure of the spectral line profile for  $H=5.5 \\times\n10^{10}$G.\n\n\\begin{figure*}[!tbh]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{fig2.EPS}\n\\end{center}\n  \\vspace{-7mm}\n  \\caption{Spectral line shapes  for $a=0.998$, $r=40$, Obs$=50^\\circ$,\n           for different magnetic fields $H=1.1 \\times 10^9$G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$G, $5.5 \\times  10^{10}$G.\n           }\n  \\label{figure3}\n\\end{figure*}\nFig. \\ref{figure3} shows a series of spectral line profiles for\n$a=0.998$,\n $r=40$ and\n  Obs$=50^\\circ$ and for magnetic fields  $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G, respectively.\nIn this case\nthe magnetic field $H= 5.5 \\times 10^9$G also does not distort\nsignificantly the spectral line profiles,\n           but  $H=1.1 \\times 10^{10}$G gives a significant\n           broadening the profile and as a result the shortest extra blue\n peak is disappeared and the initial structure of the the spectral line profile\nwith  high and low blue peaks and a red peak is changed to a\ntwo-peaked structure with a red peak and a blue peak.  As in Fig.\n\\ref{figure2},  the spectral line profile for $5.5 \\times\n10^{10}$G demonstrates a significant difference from the spectral\nline profile with respectively \"low\" magnetic field like in the\nfirst panel $H=1.1 \\times 10^9$ G. This result is general for our\ncalculations and applicable for all cases considered below.\n\n\\begin{figure*}[!tbh]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{fig3.EPS}\n\\end{center}\n  \\vspace{-7mm}\n  \\caption{Spectral line shapes  for $a=0.998$, $r=14$,\n  Obs$=30^\\circ$,\n           for different magnetic fields $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G.\n           }\n  \\label{figure3a}\n\\end{figure*}\n\nFig. \\ref{figure3a} shows a series of spectral line profiles for\n$a=0.998$,\n $r=14$\nand  Obs$=30^\\circ$ for magnetic fields  $H=1.1 \\times 10^9$ G,\n$5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G,\n           respectively.\n           In this case the blue peak is higher than the red one\n(such types of peaks were calculated also by \\cite{Cadez03} for\nthe warped disks).\n           As for previous case (Fig. \\ref{figure2}) magnetic\nfield $H= 5.5 \\times 10^9$G does not distort significantly the\nspectral line profiles,\n           but  $H=1.1 \\times 10^{10}$G gives a significant\n           broadening the peaks of the profile and as a result\nsub-peaked structure between blue and red peaks is disappeared.\n\n\\begin{figure*}[!tbh]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{fig4.EPS}\n\\end{center}\n  \\vspace{-7mm}\n  \\caption{Spectral line shapes  for $a=0.3$, $r=5$,\n  Obs$=50^\\circ$,\n           for different magnetic fields $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G.}\n  \\label{figure4}\n\\end{figure*}\n\nThe Zeeman split of one peaked spectral line profiles is presented\nin Fig. \\ref{figure4}-\\ref{figure6}. Similar spectral line\nprofiles (without Zeeman splitting) were calculated in the\nframework of a warped disk model by \\cite{Cadez03} and for a cloud\nmodel by \\cite{Karas00}. Fig. \\ref{figure4} shows that for\n$a=0.3$, $r=5$ and Obs$=50^\\circ$, there is only one single peak,\nbut there is\n  no bump around this peak for low magnetic fields.\nFig. \\ref{figure5} shows that for $a=0.998$, $r=5$ and\nObs$=30^\\circ$, there is also only one single peak, but there is\n  a broad bump around this peak for low magnetic fields (evidences\n  for such a kind of bump was found by \\cite{Wilms01} using data\n  of XMM-EPIC observations of MCG-6-30-15\\footnote{\\cite{Wilms01} suggested that there\n  are magnetic fields ($h \\sim 10^4$\\,G) in the Seyfert galaxy MCG-6-30-15\n  and even there is magnetic extraction of energy because of Blandford -- Znajek\n  effect, but   of course, these magnetic fields are too low to lead to a\n  significant changes of the iron $K_\\alpha$ line due to Zeeman splitting.}).\nFig. \\ref{figure6} shows that for $a=0.75$, $r=5$ and\n  Obs$=50^\\circ$,  there is also only one single peak and there is\n  a bump around this peak for low magnetic fields.\nFigs. \\ref{figure5},\\ref{figure6} demonstrate two cases where\nZeeman splitting gives significant changes of spectral line\nprofiles, in which evident single peak structure for low magnetic\nfields is changed into a three-peaked structure for high magnetic\nfields $H \\sim 5.5 \\times 10^{10}$G.\n\n\\begin{figure*}[!tbh]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{fig5.EPS}\n\\end{center}\n  \\vspace{-7mm}\n  \\caption{Spectral line shapes  for $a=0.998$, $r=5$, Obs$=30^\\circ$\n           for different magnetic fields $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G.}\n  \\label{figure5}\n\\end{figure*}\n\n\\begin{figure*}[!tbh]\n   \\begin{center}\n\\includegraphics[width=0.98\\textwidth]{fig6.EPS}\n\\end{center}\n  \\caption{Spectral line shapes  for $a=0.75$, $r=5$,\n  Obs$=50^\\circ$,\n           for different  magnetic fields $H=1.1 \\times 10^9$ G, $5.5 \\times\n           10^9$G, $1.1 \\times 10^{10}$ G, $5.5 \\times  10^{10}$G.}\n  \\label{figure6}\n\\end{figure*}\n\nSummarizing these results of calculations for considered examples,\nwe note that magnetic fields $H \\sim 5 \\times 10^{10}$G produce\nsignificant changes of spectral line profiles, but $H \\sim\n10^{10}$ G could be responsible for essential broadening the\nprofile peaks. Therefore, in principle there is a possibility to\nmeasure magnetic fields about $5 \\times 10^{10}$G that could be\ngenerated near Galactic Black Hole Candidates and probably near\nblack hole horizons in some AGNs.\n\n\\section{Summary and discussions}\n\nIt is known that Fe $K_\\alpha$ lines are found not only in AGNs\nand microquasars but also in X-ray afterglows of gamma-ray bursts\n(GRBs) \\citep{Lazz01}. A theoretical model for GRBs was suggested\nrecently by \\cite{Van_Putten01,Van_Putten03}. In the framework of\nthis model a magnetized torus (shell) around rapidly rotating\nblack hole could be formed after black hole-neutron star\ncoalescence. In this case magnetic fields could be even much\nhigher than $10^{11}$~G. Therefore, an influence of magnetic\nfields on spectral line profiles can be very significant and we\nmust take into account Zeeman splitting.\n\n Results of 3D\nmagnetohydrodynamical (MHD) simulations  demontrated that there\nare non-equatorial and non-axisymmetric density patterns and some\nconfigurations like tori or shells could be formed\n\\citep{Mineshige02}. Moreover, an analysis of instabilities of\naccretion flows showed that warps, tilts and caustic surfaces\ncould arise not only in the equatorial plane \\citep{Illarionov01}.\nObservations also gave some indirect evidences for more\ncomplicated accretion flows (than the standard thin accretion\nflow) because there are some signatures for a precession and a\nnutation  (for example, there is a significant precession of the\naccretion disk for the SS433 binary system\n\\citep{Cher02}).\\footnote{\\cite{Shak72} predicted that if the\nplane of an accretion disk is tilted relative to the orbital plane\nof a binary system, the disk can precess.}\n\n    It is evident that duplication (triplication)\nof a blue peak could be caused not only by the influence of a\nmagnetic field (the Zeeman effect), but by a number of other\nfactors. For example, the line profile can have multiple peaks\nwhen the emitting region represents multiple shells with different\nradial coordinates (it is easy to conclude that two emitting rings\nwith finite widths separated by a gap, would yield a similar\neffect). Actually, such an explanation was proposed by\n\\cite{Tur02} to fit the Fe $K\\alpha$ shape in NGC 3516.\n Despite\nof the fact that a multiple blue peak can be generated by many\ncauses (including the Zeeman effect as one of possible\nexplanation), the absence of the multiple peak can lead to a\nestimation of an upper limit of the magnetic field.\n\n    It is known that neutron stars (pulsars) could have\nhuge magnetic fields. So, it means that the effect discussed above\ncould appear in binary neutron star systems and in single neutron\nstars as well \\citep{Loeb03}. The quantitative description of such\nsystems, however, needs more detailed computations.\n\nSimilar to considerations presented in paper by \\cite{ZKLR02},\nanalyzing Fe $K\\alpha$ shapes for MCG-6-30-15 galaxy and using\nASCA data \\citep{tanaka1} one could evaluate a magnetic field for\nthis case; namely a magnetic field should be less than $5 \\times\n10^{10}$~G and the estimate is independent on a character of\naccretion flow. So, we could use the estimate for non-flat\naccretion flows for MCG-6-30-15 Seyfert galaxy and generalize\nconclusions by \\cite{ZKLR02} for more general cases of accretion\nflows.\n\nAs an extended work of the first paper by Zakharov et al. (2003),\nthe estimates of magnetic field may seem not very precise. But one\ncould mention that the rough estimates are caused by a noisy\nobservational data since as a matter of fact, present spaceborne\ninstrumentations vary greatly in their sensitivities and\nresolutions and precisions of measurements are not very high to\nhave good estimates. Moreover, it would be difficult to reveal a\ncompromisable list quantitatively of possible limiting effects\nrestricted by different detections. However, \\cite{Zakharov_Ma04}\nare trying to focus on  specific ASCA observations of PKS 0637-752\nin the range 1.3-24.8 keV \\citep{Yaqoob98} and provide an\nestimation of the minimum magnetic field detectable with the\nsatellite taking into account  a new X-ray emission hypothesis\n\\citep{Varshni99}, other than a fluorescent assumption. The\nanalysis is based on the laboratory measurements and\nidentification of iron line experiments in Lawrence Livermore\nNational Laboratory \\citep{Brown02}.\n\n\n    With further increase of observational facilities it\nmay become possible to improve the above estimation. The\nConstellation-X launch suggested in the coming decade seems to\nincrease the precision of X-ray spectroscopy as many as\napproximately 100 times with respect to the present day\nmeasurements \\citep{weaver1}. Therefore, there is a possibility in\nprinciple that the upper limit of the magnetic field can also\ngreatly improved in the case when the emission of the X-ray line\narises in a sufficiently narrow region.\n\n\\section{Acknowledgements}\nAuthors are grateful to J.-X.~Wang for fruitful discussions.\nAuthors thank an anonymous referee for very useful remarks.\n\n This work was supported by the National Natural Science Foundation\nof China, No.:10233050.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCollision models, first studied in the seminal paper by Rau~\\cite{Rau1963}, have seen a revival of interest in recent years~\\cite{Ziman2002,Scarani2002,Englert2002}. \nThey replace the complex system-bath dynamics by a series of sequential collisions between a system of interest and a continuous stream of small units, called ancillas. \nThis not only makes the dynamics simpler, but also more controllable. \nFor example, collisional models have proven to be crucial in developing the basic laws of thermodynamics in the quantum regime~\\cite{DeChiara2018,Landi2020a,Barra2015,Strasberg2016}, or to further our understanding of non-Markovianity~\\cite{Campbell2019a,Man2018,Lorenzo2017,Donvil2021,Mascarenhas2017,McCloskey2014,Campbell2018b,Rybar2012a,Cilluffo,Ciccarello2013b,Bernardes2017,Taranto2018a,Ciccarello2013a,Filippov2017,Kretschmer2016,Jin2018,Ciccarello2013,Bernardes2014,Cakmak2017,Daryanoosh2018,DeChiara2020}. \nFor a recent review, see~\\cite{Ciccarello2020}.\n\nA particularly nice feature of these models is that they allow for a clean implementation of autonomous processes: \nAncillas arrive, undergo some physical process, and then leave. \nDifferent implementations can  be used to perform different tasks, which are gauged by the changes in the ancilla's state.\nMoreover, the process is allowed to continue indefinitely, as long as new ancillas continue to arrive.\nIndeed, there have already been several proposals which employ collision models in e.g. quantum heat engines~\\cite{Quan2007,Allahverdyan2010,Uzdin2014,Campisi2014,Campisi2015,Denzler2019,Pezzutto2019,Mohammady2019,Molitor2020} or quantum thermometers~\\cite{Seah2019,Shu2020,Alves2021}.\n\nIn this paper we discuss the implementation of an autonomous collision model engine aimed at charging quantum batteries.\nBattery charging in the quantum domain is an active field of study~\\cite{Skrzypczyk2013,Campaioli2017,Teo2017,Andolina2019,Mitchison2021}. \nThe present framework aims to produce a model in which this charging occurs autonomously, for an arbitrary number of charging units, and in a way which works for arbitrary initial battery states.  \n\n\nThe input of the engine is a stream of ancillas, drawn randomly from some ensemble of states.\nThe thermodynamic ``usefulness'' of each ancilla will be characterized by its ergotropy~\\cite{Allahverdyan2004}, which quantifies the maximum amount of work that can be extracted from it by means of a unitary interaction. \nThe goal of the engine is then to increase the average ergotropy of the outgoing ancillas.\nThis is accomplished by using information extracted from measurements in the system, as depicted in Fig.~\\ref{fig:drawing} (the ancillas are never measured). \nThis setup was inspired by Ref.~\\cite{Francica2016a}, which studied the ergotropy that could be extracted from quantum correlations between a system and a single ancilla. \nAnd it is  opposite in spirit to, e.g., continuously monitored systems~\\cite{Wiseman2009,Jacobs2014}, where one uses measurements in the ancillas to learn something about the system~\\cite{Gross2017a,Rossi2020,Landi2021a,Belenchia2019}; here we use instead information about the system to learn about the ancillas. \n\nThe measurement outcomes are used to classify the ancillas as having either high or low ergotropy, which we model  using Bayesian decision theory~\\cite{Duda2000}.\nThis therefore implements a Maxwell demon~\\cite{Maxwell1888}, which autonomously decides what to do with each ancilla. \nHigh ergotropy ancillas (defined according to some threshold) are allowed to leave, while low ergotropy ones are flagged for further processing. \nThat is, they are redirected to go through another quantum channel, aimed at increasing their ergotropy further (Fig.~\\ref{fig:drawing}). \nIn our case, we will model this in terms of an additional unitary pulse, but more general quantum channels can also be used. \n\nThe system in this case plays the role of a memory. \nAs is well known, the process of acquiring information can in principle be done without any energetic cost. \nHowever, there is a fundamental cost in erasing the information~\\cite{Bennett1973,Plenio2001}, given by Landauer's principle~\\cite{Landauer1961}.\nWe model this by assuming that the system is coupled to a cold heat bath that acts for a finite time, in between collisions. \nAs we show, this is crucial for the engine to operate autonomously.\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{drawing.pdf}\n    \\caption{Autonomous collision model for enhancing the ergotropy in an ensemble of ancillas. \n    A stream of ancillas, drawn from random states, interact with a system $S$. Measurements in $S$ are then used to distinguish whether the ancillas have low or high ergotropy. \n    This information is used by a (space invader) demon, operating under the paradigm of Bayesian Decision Theory, to decide whether or not the ancillas should be further processed or not, with the goal of increasing their ergotropy even further.\n    }\n    \\label{fig:drawing}\n\\end{figure*}\n\n\\section{Basic model}\n\nWe consider a stream of ancillas, each prepared in a state $|\\psi_A\\rangle$ drawn from an ensemble of $d$ possible states $\\{|\\psi_i\\rangle\\}$ (not necessarily orthogonal), with probability $q_i$.\nOften, in the collision model literature, one assumes that the ancillas are in mixed states. \nThis is a natural choice if one is interested in the steady-state properties of the system. \nBut here, for the task at hand, it is much more natural to assume that the ancillas are in pure states. \nNotwithstanding, all results below also hold for ensembles of mixed states.\nThe notations $\\psi_A = |\\psi_A\\rangle \\langle \\psi_A|$ will be used whenever the ancilla state is pure. \n\nThe thermodynamic utility of each ancilla can be quantified by its ergotropy~\\cite{Allahverdyan2004} which, for a generic ancilla state $\\rho_A$, is defined as\n\\begin{equation}\\label{ergotropy}\n    \\mathcal{W}(\\rho_A) = \\mathrm{tr}(\\rho_A H_A) - \\min\\limits_{V} \\mathrm{tr}(\\rho_A V^\\dagger H_A V),\n\\end{equation}\nwhere $H_A$ is the ancilla Hamiltonian and the minimization is over all unitaries $V$. When the state is pure, this reduces to the more intuitive result \n\\begin{equation}\\label{ergotropy2}\n    \\mathcal{W}(\\psi_A) = \\langle \\psi_A | H_A | \\psi_A\\rangle - E_{\\rm gs}^A,\n\\end{equation}\nwhere $E_{gs}^A$ is the ground state of $H_A$.\n\nThe stream of ancillas are first put to interact with a system $S$, one at a time, for a fixed duration $\\tau_{SA}$, according to some Hamiltonian $H_{SA}$.\nIf the system is in $\\rho_S$ and the ancilla is in $\\psi_A$, this produces the map\n\\begin{equation}\n    \\rho_{SA|\\psi_A} = e^{-i H_{SA} \\tau_{SA}} (\\rho_S \\otimes \\psi_A) e^{i H_{SA} \\tau_{SA}}.\n\\end{equation}\nImmediately afterwards, the system is measured, which we  describe by a set of Kraus operators $\\{M_x\\}$, with $m$ possible outcomes, $x = 1, \\ldots, m$, and satisfying $\\sum_x M_x^\\dagger M_x = 1$.\nThe probability of outcome $x$, conditioned on the initial ancilla state, is \n\\begin{equation}\\label{likelihood}\n    P(x|\\psi_A) = \\mathrm{tr}\\Big\\{ (M_x^\\dagger M_x \\otimes I_A )\\rho_{SA|\\psi_A}\\Big\\},\n\\end{equation}\nwhere $I_A$ is the identity acting on the ancilla.\nMoreover, if outcome $x$ is observed, the reduced state of $SA$ should be updated to \n\\begin{equation}\\label{conditional_joint_state}\n    \\rho_{SA|x,\\psi_A} = (M_x \\otimes I_A) \\rho_{SA|\\psi_A} (M_x^\\dagger \\otimes I_A). \n\\end{equation}\nFrom this, the reduced states of system and ancilla, $\\rho_{S|x,\\psi_A}$ and $\\rho_{A|x,\\psi_A}$, can be obtained by taking the partial trace.\n\nIn between collisions, the state of the system is allowed to relax in contact with a heat bath, which we describe by a Lindblad master equation acting for a fixed time $\\tau_{SE}$. \nIt is assumed for simplicity that $\\tau_{SA} \\ll \\tau_{SE}$ so that, during the system-ancilla interaction, the system is approximately uncoupled from the bath.\n\nBased on the outcome $x$, a demon tries to classify whether an ancilla has a high or a low ergotropy $\\mathcal{W}$ (according to some model-specific threshold).\nThe former can leave the process, while the latter are redirected for additional processing, aimed at increasing their ergotropy further. \nWe describe this in terms of a unitary pulse $\\mathcal{O}$, so that the final state of the ancilla will be \n\\begin{equation}\\label{cases}\n    \\rho_A' = \\begin{cases}\n    \\rho_{A|x,\\psi_A} & \\text{high ergotropy in } \\psi_A,\n    \\\\[0.2cm]\n    \\mathcal{O} \\rho_{A|x,\\psi_A} \\mathcal{O}^\\dagger, & \\text{low ergotropy in } \\psi_A.\n    \\end{cases}\n\\end{equation}\nThe meaning of low or high ergotropy is model specific, and will be discussed further below.\nThe ultimate goal of the engine is thus to produce an ensemble with average ergotropy higher than that of the initial ensemble $\\{q_i, |\\psi_i\\rangle\\}$:\n\\begin{equation}\n    \\mathcal{W}_{\\rm raw} = \\sum\\limits_i q_i \\mathcal{W}(\\psi_i),\n\\end{equation}\nwhere the subscript ``raw'' will always refer to the ancillas before entering the engine.\n\n\\section{Bayesian risk analysis}\n\nBefore discussing an actual implementation, we must first discuss the type of rationale that will be used by the demon in deciding whether the ergotropy is high or low. \nWe do this using the concept of  Bayesian risk analysis, as a general tool for implementing the decision process. \n\nThere are $d$ possible preparations $\\psi_i$, and $m$ possible outcomes $x$, each pair associated to a certain quantum state $\\rho_{A|x,\\psi_i}$ [Eq.~\\eqref{conditional_joint_state}].\nIt is assumed that the demon knows the possible set of states $\\{\\psi_i\\}$, but does not know the current ancilla state, nor the probabilities $q_i$ with which they were sampled (the latter restriction could be lifted without significantly altering the problem). \nAt each collision, all the demon knows is therefore the outcome $x$.\nBased on this, it may take one of a set of $a$ actions $\\alpha_k$, $k = 1,\\ldots, a$. \nGenerally speaking, we could associate each action with a quantum channel $\\mathcal{E}_k$, which will process the quantum state of the ancilla further. \nFor example, in the case of Eq.~\\eqref{cases}, action $\\alpha_1$ stands for ``do nothing,'' while $\\alpha_2$ stands for the unitary channel $\\mathcal{O}\\bullet \\mathcal{O}$. \nBut, more generally, all kinds of channels can in principle be used. \n\nIn Bayesian risk analysis, we quantify each action by a certain gain, described by a  non-negative function $\\lambda(\\alpha_k |x, \\psi_i)$ determining how much is gained from using action $\\alpha_k$ when the outcome is $x$ and the state is $\\psi_i$ (one could equivalently frame the problem in terms of risks, instead of gains). \nThis set of function determines the type of strategy the demon will use, and different functions will lead to different engine performances. \nAn example could be the\nergotropy~\\eqref{ergotropy} of \n$\\mathcal{E}_k(\\rho_{A_x,\\psi_i})$; that is, \n\\begin{equation}\n    \\lambda(\\alpha_k |x, \\psi_i) = \\mathcal{W}\\Big(\\mathcal{E}_k(\\rho_{A_x,\\psi_i})\\Big)\n\\end{equation}\nHowever, as we will show below, in specific models simpler functions often be employed.\n\nFor each outcome $x$, the demon's decision will then be to choose the action $\\alpha_k$ which maximizes the Bayesian gain\n\\begin{equation}\\label{gain}\n    G(\\alpha_k | x) = \\sum\\limits_i \\lambda(\\alpha_k | x, \\psi_i) P(\\psi_i|x),\n\\end{equation}\nwhere $P(\\psi_i | x)$ is the probability that the initial state was $|\\psi_i\\rangle$ given that the outcome in the system was $x$. \nAccording to Bayes's rule, this is further given by \n\\begin{equation}\\label{bayes_theorem}\n    P(\\psi_i|x) = \\frac{P(x|\\psi_i) P(\\psi_i)}{\\sum\\limits_i P(x|\\psi_i) P(\\psi_i)},\n\\end{equation}\nwhere $P(x|\\psi_i)$ is the likelihood function, given in Eq.~\\eqref{likelihood}, and \n$P(\\psi_i)$ is the prior probability the demon associates to the ancilla being in $|\\psi_i\\rangle$.\n\nIf the demon does not know in advance how the ancillas are  sampled, the priors $P(\\psi_i)$ will in general differ from the $q_i$. In fact, in the beginning of the process, a natural choice of prior would be $P(\\psi_i) = 1\/d$.\nAfter each collision, however, the demon updates $P(\\psi_i)$ to the posterior $P(\\psi_i|x)$, which can then be used as the prior for the next step.\nUnder mild conditions, it is  expected that in the steady state this should converge to the true sampling probabilities $q_i$. \n\nWe also mention that, in general, the state of the system is constantly changing. \nAs a consequence, when the above procedure is used sequentially, it may cause $P(\\psi_i|x)$ at the $n$-th step to depend on the outcomes of all past collisions, thus making the process highly non-Markovian. \nIn fact, even in the limiting case of projective measurements,  $P(\\psi_i|x)$ would still depend on the previous outcome.\nThis is directly associated with Benett's exorcism of Maxwell's demon~\\cite{Bennett1973}: while there is no minimum cost to acquire information, there is always a fundamental heat cost for erasing it (see also~\\cite{Plenio2001}). \nIf the engine is to operate autonomously, the memory (which is in this case the system) must be reset at each step. \nIn practice, the demon may continue to employ the same gain function~\\eqref{gain}, which would happen when it is unaware of whether the system has been fully reset or not.\nThe only problem is that this may cause it to make wrong decisions. \nThe better is the memory reset, the more accurate is the demon's decision.\n\n\n\\section{Qubit-qubit model}\n\nWe now consider a concrete implementation of this approach, where we assume that the system and ancillas are all made of qubits. \nThe ancilla Hamiltonian is taken to be $H_A = -\\omega \\sigma_z^A\/2$, where $\\sigma_z$ is a Pauli matrix. \nThe ground-state is thus the computational basis state $|0\\rangle$; i.e., $\\sigma_z|0\\rangle = |0\\rangle$. \nThe ergotropy~\\eqref{ergotropy} is then bounded between $\\mathcal{W} \\in [0,\\omega]$, with the maximum being for the excited state $|1\\rangle$.\n\n\nThe system-ancilla interaction is taken as\n\\begin{equation}\\label{example_HSA}\n    H_{SA} = g \\sigma_y^S \\otimes \\sigma_z^A. \n\\end{equation}\nThis is a typical pointer-basis type of measurement~\\cite{Zurek1981}, with information on the ancilla's population being directly encoded in the system, while at the same time causing the coherence's to dephase. \nThe ergotropy~\\eqref{ergotropy} has  contributions from both the populations and  coherences~\\cite{Francica2020}. \nThe interaction with the system will keep the former intact, but disturb the latter (measurement backaction). \nThe goal, therefore, is to see if one can increase the ergotropy from the populations while, at the same time, not excessively harm that from the coherences.\n\nThe system is  measured after each step in the eigenbasis $|\\pm\\rangle = (|0\\rangle \\pm |1\\rangle)\/\\sqrt{2}$ of the $\\sigma_x$ operator. \nTo understand why this is a good measurement strategy, suppose that the system is initially prepared in $\\rho_S = |0\\rangle\\langle 0|$, while the ancilla is in  $|\\psi_A \\rangle = \\cos(\\theta\/2)|0\\rangle + e^{i \\phi} \\sin(\\theta\/2)|1\\rangle$. \nThen Eq.~\\eqref{likelihood} will produce the likelihoods\n\\begin{equation}\\label{example_likelihood}\n    P(x| \\psi_A) = \\frac{1}{2} \\Big[1 +x \\sin(2g \\tau_{SA}) \\cos\\theta\\Big].\n\\end{equation}\nFor $\\theta \\in [0,\\pi\/2]$ (northern hemisphere in Bloch's sphere), the outcome $x = +1$ is more likely, while for $\\theta \\in [\\pi\/2,\\pi]$ (southern hemisphere) it is actually $x=-1$. \nBut the ergotropy is directly related to the position in Bloch's sphere, being low in the former and high in the latter.\nThis means that if $x=+1$ is observed, it is more likely that the ancilla has a low ergotropy. \nA very simple Bayesian strategy is thus to take the gain of no action ($\\alpha_1$) as $\\lambda(\\alpha_1|x, \\psi_i) = 1$ when $x=-1$, and zero otherwise; and similarly  $\\lambda(\\alpha_2|x, \\psi_i) = 1$ when $x = 1$, and zero otherwise. \n\nWhen the ancilla is flagged, it is more likely to be in the northern hemisphere. In this case, we can then apply an additional unitary pulse  $\\mathcal{O} = \\sigma_x^A$, which flips the ancilla's state to the southern hemisphere. \nNote that if the ergotropy is already high, this will generally spoil it. \nThat is to say, whenever the demon makes a mistake, it will actually be degrading the ancilla's ergotropy. \nBut since correct decisions are more likely, it will on average increase it. \n\n\nFinally, between measurements the system is taken to interact with a zero temperature heat bath for a time $\\tau_{SE}$, described by the master equation \n\\begin{equation}\\label{QME}\n    \\frac{d\\rho_S}{dt} = -i[H_S, \\rho_S] + \\gamma  D[\\sigma_+^S]\\rho_S,\n\\end{equation}\nwhere $\\gamma$ is the coupling strength and $D[L]\\rho = L \\rho L^\\dagger - \\frac{1}{2}\\{L^\\dagger L, \\rho\\}$. \nMoreover, we assume $H_S = -\\omega_S\\sigma_z^S\/2$, with $\\omega_S$ not necessarily resonant with the ancilla frequency $\\omega$. \n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{example.pdf}\n    \\caption{Battery charging in a qubit-qubit collisional model. \n    (a) Histogram of the ergotropies obtained from randomly sampled ancilla states (red), as compared with the final ergotropies after they are passed through the engine. The data was sampled from   $N = 10^4$ simulations, with the system-ancilla interaction strength  fixed at $g\\tau_{SA} = \\pi\/8$.\n    (b) Average ergotropy as a function of $g\\tau_{SA}$. Raw values (which are independent of $g\\tau_{SA}$) and processed values are shown in the same color code as in (a). The points marked as ``engine'' refer to the ergotropy when all ancillas are passed through the engine, irrespective of the outcome $x$.\n    }\n    \\label{fig:example}\n\\end{figure*}\n\n\\section{Results}\n\nIn what follows, the ancillas are all uniformly sampled from generic states $|\\psi_i\\rangle$ within the Bloch sphere, using the appropriate Haar measure.\nWe start by assuming that $\\gamma\\tau_{SE}$ is sufficiently large so that, after each step, the state of the system is fully reset back to $\\rho_S = |0\\rangle\\langle 0|$.\nIllustrative results are shown in Fig.~\\ref{fig:example}. \nThe histogram in Fig.~\\ref{fig:example}(a) compares the raw ergotropy with that obtained at the output of the engine, for fixed $g\\tau_{SA} = \\pi\/8$. \nAs is evident, the engine charges the ancillas, leading to a final ensemble with clearly larger ergotropy. \n\nIn Fig.~\\ref{fig:example}(b) we show the average ergotropy as a function of $g\\tau_{SA}$, where it is evident that stronger interactions lead to monotonic improvements in the charging process.\nThis is expected since higher $g\\tau_{SA}$ imply more information is available to the demon to make the decision.\nWe also show, for comparison, the ergotropy which would be obtained if all ancillas were to be processed by the engine, irrespective of the measurement outcomes (labeled ``engine''). \nIn this case the interaction with the system causes an overall degradation of $\\mathcal{W}$. \nThis happens because the interaction~\\eqref{example_HSA} dephases the ancillas. \nHence, the coherent part of the ergotropy tends to be lost (while the population part is unaffected). \n\nNext we investigate what happens when the state of the ancilla is not fully reset after each step. \nDue to the projective nature of the measurement, after each collision the system will either be in $|+\\rangle$ or in $|-\\rangle$. \nThe state, after a time $\\gamma\\tau_{SE}$, under the action of Eq.~\\eqref{QME}, will thus be \n\\begin{equation}\n    \\rho_{S|\\pm}(t) = \\begin{pmatrix}\n    1 - e^{-\\gamma\\tau_{SE}}\/2 & \\pm e^{-\\gamma\\tau_{SE}\/2 + i \\omega_S \\tau_{SE}}\/2\n    \\\\[0.2cm]\n    \\pm e^{-\\gamma\\tau_{SE}\/2- i \\omega_S \\tau_{SE}}\/2 & e^{-\\gamma\\tau_{SE}}\/2\n    \\end{pmatrix},\n\\end{equation}\nwhich are thus taken as the initial states of the next collision.\nResults for the average ergotropy are shown in Fig.~\\ref{fig:example2}. \nAs can be seen, when $\\gamma\\tau_{SE}$ is finite, the ergotropy is gradually reduced. \nThis happens because when the system is not properly erased, it affects the demon's ability to make proper decisions. \nIn fact, if $\\gamma\\tau_{SE}$ is very small, one can even obtain an average ergotropy which is worse than that of a fully random ensemble. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{example2.pdf}\n    \\caption{The curve marked ``finite reset'' depicts the dependence  of the average ergotropy on the system relaxation time $\\gamma\\tau_{SE}$. \n    The data was sampled from   $N = 10^4$ simulations, with the system-ancilla interaction strength  fixed at $g\\tau_{SA} = \\pi\/8$.\n    The other two curves, marked  ``raw'' and ``processed,''  are shown for comparison, and are similar to those from Fig.~\\ref{fig:example}(b).\n    }\n    \\label{fig:example2}\n\\end{figure}\n\n\\section{\\label{sec:energetics}Energetics}\n\nWe now discuss in further detail the energetics of the problem. \nWe divide the problem in 3 steps: interaction, measurement and conditional unitary pulse.\nFor simplicity, we focus on full system resets ($\\gamma\\tau_{SE} \\to \\infty$).\nThe interaction~\\eqref{example_HSA} does not affect the energy of the ancillas since $[H_{SA}, H_A] = 0$. \nBut it does affect the energy of the system. \nThe net change in energy of system plus ancilla, in one collision, assuming the ancilla is in $\\psi_A$, is thus given by \n\\begin{equation}\n    \\Delta E_{\\rm col} = \\mathrm{tr}\\Big\\{(\\rho_{S|\\psi_A} - \\rho_S) H_S\\Big\\}.\n\\end{equation}\nThis change reflects the inherent work cost associated to the interaction $H_{SA}$, known as on\/off work~\\cite{DeChiara2018,Landi2021a}.\nNotice, however, that this will depend on the Hamiltonian in the system, which has a generic gap $\\omega_S$ (not necessarily resonant with the ancilla's gap $\\omega$). \nThe on\/off work can thus be made arbitrarily small by choosing $\\omega_S$ to be small. \nThis  means that it is well possible to operate the engine in a regime where the energy cost of the collision is negligible. \n\nNext we turn to the effects of the measurement. \nWe assume that the ancilla's initial state has the generic form \n$|\\psi_A \\rangle = \\cos(\\theta\/2)|0\\rangle + e^{i \\phi} \\sin(\\theta\/2)|1\\rangle$. \nThe average energy of the ancillas after the measurement, given outcomes $x = \\pm1$, will then be \n\\begin{equation}\n    E_{A|x} = -\\frac{\\omega}{2} \\frac{\\cos\\theta + x \\sin(2g\\tau_{SA})}{1 + x \\cos\\theta \\sin(2g\\tau_{SA})}.\n\\end{equation}\nAveraging this over the probabilities~\\eqref{example_likelihood} recovers the initial average energy $\\langle \\psi_A | H_A |\\psi_A\\rangle$.\nThus, up to this point, no work is performed in the ancillas (on average). \n\nThe actual work comes from the controlled unitary pulse, which is applied only when $x = + 1$. \nThis causes the energy of the ancillas to change to \n\\begin{equation}\n    \\tilde{E}_{A|+1} = \\mathrm{tr}\\Big\\{\\sigma_x \\rho_{A|+1} \\sigma_x H_A\\Big\\} = -E_{A|+1}.\n\\end{equation}\nThe net work is therefore \n\\begin{equation}\n    \\mathbb{W}_+ = \\tilde{E}_{A|+1} - E_{A|+1} = \\omega \\frac{\\cos\\theta +  \\sin(2g\\tau_{SA})}{1 +  \\cos\\theta \\sin(2g\\tau_{SA})},\n\\end{equation}\nwhich $\\mathbb{W}_- = 0$ when $x = -1$.\nThe average work is thus\n\\begin{equation}\n    \\mathbb{W} = P(+1|\\psi_A) \\mathbb{W}_+ + P(-1|\\psi_A) \\mathbb{W}_- = \\frac{\\omega}{2} (\\cos\\theta + \\sin(2g\\tau_{SA})).\n\\end{equation}\nNotice how work is still performed even if the system and ancilla do not interact ($g\\tau_{SA} = 0$).\nThis happens because, even though they don't interact, we assume that the system is nonetheless still measured, thus yielding equally likely outcomes $x = \\pm 1$.\nThat is to say, half of the time the pulse is applied. \n\nWe now analyze this from the perspective of the ergotropy. \nThe initial ergotropy is $\\mathcal{W}_0 = \\omega \\sin^2(\\theta\/2)$. \nAfter the measurements (but before the pulse), the ergotropies conditioned on each outcome are \n\\begin{equation}\n    \\mathcal{W}_x = \\mathcal{W}(\\rho_{A|x,\\psi_A}) = \\omega \\sin^2(\\theta\/2) \\frac{1-x \\sin(2g\\tau_{SA})}{1+ x \\cos\\theta \\sin(2g\\tau_{SA})}.\n\\end{equation}\nSince the measurement does not perform any work, on average, we simply have $\\sum_x P(x|\\psi_A) \\mathcal{W}_x = \\mathcal{W}_0 = \\omega \\sin^2(\\theta\/2)$, as it must be.\n\nWhen the pulse is performed, however, the ergotropy changes to \n\\begin{equation}\n    \\tilde{\\mathcal{W}}_+ = \\omega \\cos^2(\\theta\/2) \\frac{1+ \\sin(2g\\tau_{SA})}{1+ \\cos\\theta \\sin(2g\\tau_{SA})}.\n\\end{equation}\nThe net change in ergotropy is, of course, the work injected, \n\\begin{equation}\n    \\tilde{\\mathcal{W}}_+ - \\mathcal{W}_+ = \\mathbb{W}_+.\n\\end{equation}\nThe final average  ergotropy is then \n\\begin{IEEEeqnarray}{rCl}\n\\mathcal{W}_{\\rm processed} &=& P(+1|\\psi_A) \\tilde{\\mathcal{W}}_+ + P(-1|\\psi_A) \\mathcal{W}_- \n\\nonumber\\\\[0.2cm]\n&=& \\frac{\\omega}{2} \\Big[ 1 + \\sin(2g\\tau_{SA})\\Big].\n\\end{IEEEeqnarray}\nIf $g\\tau_{SA}$, this reduces to $\\omega\/2$, which is half the maximum value it may have. \nThus, if the machine is applied under no information about the ancillas whatsoever, it would result in an average ergotropy of $\\omega\/2$. \nAnd if $g\\tau_{SA} = \\pi\/$, the average ergotropy achieves its maximum value $\\omega$. \nThis therefore fully accounts for the behavior observed in  Fig.~\\ref{fig:example}.\n\n\n\\section{Discussion}\n\nIn this paper we put forth the idea of an autonomous engine, which processes random incoming ancillas with the goal of increasing their ergotropy.\nThere are endless possible variations of such an engine that one might construct.\nThe goal of the present proposal was to build a minimal engine, where the basic effects could be made evident. \nIn particular, they are the following. \nFirst, the idea that, in reality, ancillas are usually sampled from an ensemble of pure states. \nCollision models often assume that the ancillas arrive in mixed states $\\rho_A$, which could be viewed as the ensemble average. \nBut for the present purposes, it is much more realistic to assume that in each collision, the state of the ancilla is pure, but not necessarily known.\nIn fact, for the example in Fig.~\\ref{fig:example} the ensemble average would be simply the identity $\\rho_A = I_A\/2$. \n\nThe second relevant aspect of this construction is the need for the state of the system to be properly reset after each step, as it plays the role of a memory. \nIf this is not done, the ability of the demon in making a decision based on the measurement outcomes is severely degraded, as Fig.~\\ref{fig:example2} illustrates very clearly. \n\nFinally, the third relevant point is the energetic balance of the problem. \nThis has long been a major advantage of collisional models, as it enables for precise accounting of all possible energy sources and sinks. \nThe analysis in Sec.~\\ref{sec:energetics} showed how this can be used to pinpoint, at the level of each possible measurement outcome, whether or not work is being performed, and how this affects the ergotropy at each step. \n\n\n\n\\section*{Acknowledgments}\n\nThe author acknowledges the financial support of the S\\~ao Paulo Funding Agency FAPESP (Grant No.~2019\/14072-0.), and the Brazilian funding agency CNPq (Grant No. INCT-IQ 246569\/2014-0). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider the wave equation on a Riemannian manifold $X:$\n$$\n\\Box u=0\\text{ on } \\RR\\times X\n$$\nwhere $\\Box=D_t^2-\\Lap_g,$\n$$\n\\Lap_g=\\sum \\frac{1}{\\sqrt{g}} D_j g^{jk} \\sqrt{g} D_k\n$$\nand $D_j\\equiv i^{-1} \\pa_{x_j}$.\n\nIf $X$ happens to be an odd dimensional Euclidean space, then \\emph{Huygens' Principle} applies, i.e.,\nthe solution\n$$\n\\cos t\\sqrt{\\Lap} \\delta_q\n$$\nwhich has initial data a delta-function (and initial derivative zero)\nis supported exactly on sphere of radius $\\abs{t}.$ In even space\ndimensions, or on a general odd dimensional manifold, this principle\nis well known to fail, but quite a nice proxy for it persists: we in\ngeneral have\n$$\n\\singsupp u(t) \\subset \\big\\{p:\\ \\text{there exists a geodesic of length } \\abs{t} \\text{ with endpoints }p,q\\big\\}.\n$$\n(Recall that the singular support of a distribution is the set of points near which is it not locally a smooth function.)\nA more precise result yet is the refinement of this statement to deal\nwith the \\emph{wavefront set} of the distribution $u;$ $\\WF u$ is a\nconic closed subset of $T^*X$ such that $\\pi \\WF u=\\singsupp u.$   H\\\"ormander's\nrather general theorem \\cite{Hormander9} on propagation of\nsingularities tells us in this special case that for a solution $u$ of\nthe wave equation, $\\WF u$ is invariant\nunder the (forward and backward) geodesic flow on $T^*X.$  Thus the\ninitial wavefront given by (the lift to the light cone of)\n$N^* \\{q\\}$ then spreads into the conormal bundle of expanding\ndistance spheres.\n\nGeneralizing this result to manifolds with boundary (with Dirichlet or\nNeumann boundary conditions) turns out be a rather complicated story.\nChazarain \\cite{Ch:73} showed that singularities striking the boundary\ntransversely simply reflect according to the usual law of geometric\noptics (conservation of energy and tangential momentum, hence ``angle\nof incidence equals angle of reflection'') for the reflection of\nbicharacteristics.  The difficulties arise, however, in the treatment\nof geodesics tangent to the boundary: in \\cite{Melrose-Sjostrand1} and\n\\cite{Melrose-Sjostrand2} Melrose--Sj\\\"ostrand showed that, at\nthese ``glancing points,'' singularities may only propagate along certain\ngeneralized bicharacteristics.  By parametrix constructions of Melrose\n\\cite{Melrose14} and Taylor \\cite{Taylor1}, these $\\CI$ singularities\ndo \\emph{not} propagate along concave boundaries (e.g.\\ they do not\n``stick'' to the exterior of a convex obstacle).  Note that this last\nresult ceases to be true in the analytic, rather than smooth,\ncategory.\n\nA simple summary of some of the fundamental results in the subject is provided by Figure~\\ref{figure:fundamental}.\n\\begin{figure}[bth]\\label{figure:fundamental}\n\\includegraphics[scale=1.3,angle=-90]{obstacle3.pdf}\\caption{Singularities\n  of the fundamental solution of the wave equation exterior to a convex obstacle.}\\end{figure}\nThis figure shows the singularities of the fundamental solution the\nwave equation in the exterior of a convex obstacle in the plane.\nThere is (part of) a circular front of directly propagated\nsingularities as well as a curved front of singularities reflected off\nthe obstacle in accordance with Snell's law.  Most crucially,\n\\emph{there are no singularities behind the obstacle} in the ``shadow\nregion,'' as a consequence of the parametrix construction of Melrose and Taylor.\n\nBy contrast, it has been known since the late 19th century (starting\nwith work of Sommerfeld \\cite{Sommerfeld1}) that if the obstacle has a\nsharp corner, singularities \\emph{do} propagate, i.e., \\emph{diffract,}\ninto the shadow region behind the obstacle.  Figure~\\ref{figure:wedge}\nshows the fundemental solution of the wave equation in the exterior of\na wedge; we can easily see a circular wave of singularities emanating\nfrom the tip of the wedge and giving rise to singularities in the\nshadow region.\n\\begin{figure}\n\\includegraphics[scale=0.7]{wedge-edited.pdf}\\caption{Singularities\n  of the fundamental solution of the wave equation exterior to a\n  wedge. \\label{figure:wedge}}\\end{figure}\n\nAs alluded to above, general boundaries present special difficulties\nof their own, so in order to study the diffraction phenomenon in a\nsimple setting, we now mostly set aside this class of manifolds, and focus on\n\\emph{manifolds with conic singularities} where wave equation\nsolutions will exhibit diffraction, but the geometry of geodesics is\nrelatively manageable.\n\n\\section{Conic geometry}\n\nWe define a \\emph{conic manifold}\nto be a manifold $X$ (of dimension $n$) with boundary $Y=\\pa X,$ and a Riemannian metric\non $X^\\circ$ such that in terms of some boundary defining function $x$\nwe have in a collar neighborhood of $Y,$\n$$\ng=dx^2 +x^2 h\n$$\nwhere $h$ is a smooth symmetric 2-cotensor such that $h|_{Y}$ is a\nmetric on $Y.$ Note in particular that $g$ degenerates at $\\pa X$ so\nas not to be a metric uniformly up to the boundary.  \n\nThe upshot is that while $X$ looks like a manifold with boundary from\nthe point of view of $\\CI$ structure, it is metrically a manifold with\nconic singularities: from the point of view of metric geometry, if we\nwrite the connected components of the boundary as \n$$\nY=\\bigsqcup Y_i\n$$\nthen each boundary component $Y_i$ should be viewed as a \\emph{cone\n  point}.  (See Figure~\\ref{figure:conicgeometry}.)\n\n\\begin{figure}[bth]\n\\includegraphics[scale=.2,angle=-90]{conicmfld.pdf}\n\\includegraphics[scale=.2,angle=-90]{conicmfld2.pdf}\n\\caption{Smooth structure, and Riemannian picture of $X$ \\label{figure:conicgeometry}}\n\\end{figure}\n\nThe conic manifold as defined here should thus be viewed as a manifold\nwith conic singularities \\emph{already equipped} with the blow-up that\nhas desingularized it to a smooth manifold with boundary.  Here the\ncost of having a smooth manifold is of course having a degenerate metric.\n\nA very special case of a conic manifold is that of a surface\nobtained by gluing together two copies of the interior (or exterior)\nof a polygonal planar domain along their common edges.  This gives a\nflat surface with cone points where the polygon had vertices.  The\nstudy of the wave equation on the original domains with Dirichlet\/Neumann\nconditions is equivalent to the study of odd\/even solutions of the wave equation on the\ndoubled manifold---see Hillairet \\cite{Hillairet:2005}.\n\nThe behavior of geodesics on conic manifolds is of considerable\ninterest near the cone point.  The crucial observation is that it is\nin fact quite hard to aim a geodesic so as to hit the cone point: most\nwill pass nearby and miss.  Indeed, starting out near the cone point,\nthere is a unique direction to aim in, in order to reach a nearby cone point.\n\\begin{proposition}[Melrose--Wunsch \\cite{Melrose-Wunsch1}]\nEvery $y \\in Y=\\pa X$ is the endpoint of a unique geodesic; these\ngeodesics foliate a collar neighborhood of $Y$:\n\\end{proposition}\nThis is equivalent to a normal-form statement for the metric: we can\nfind coordinates so that $h=h(x,y,dy)$ has no $dx$ components, and\nthus the curves $x=x_0 \\pm t, y=y_0$ are unit-speed geodesics.\n\nA crucial point in trying to make sense of propagation of singularites\nis to make a reasonable definition of the continuation of a geodesic\nthat reaches a cone point.  There are two reasonable candidates for\nthis definition, one more restrictive than the other, and both play a\nrole here:\n\n\\begin{definition} We define geodesics passing through $\\bigsqcup Y_j \\equiv \\pa X$ as follows:\n\\begin{itemize}\n\\item A \\emph{diffractive geodesic} is a geodesic which, upon reaching\n  the boundary component $Y_i$ along a geodesic ending at a point\n  $y\\in Y_i,$ immediately then leaves the boundary from some point $y'\n  \\in Y_i.$\n\\item A \\emph{geometric geodesic} is a geodesic which, upon reaching\n  the boundary component $Y_i$ along a geodesic ending at a point\n  $y\\in Y_i,$ immediately then leaves the boundary from some point $y'\n  \\in Y_i$ such that $y,y'$ are endpoints of a geodesic \\emph{in $Y_i$}\n  (w.r.t.\\ the metric $h|_{Y_i})$ of length $\\pi.$\n\\item A \\emph{strictly diffractive} geodesic is one which is\n  diffractive but not geometric.\n\\end{itemize}\n\\end{definition}\n\nA more intuitive definition of geometric geodesics is as follows: they\nare the geodesics that are \\emph{locally approximable} by families of geodesics in\n$X^\\circ$.  We refer the reader to \\cite{Melrose-Wunsch1} for more\ndetail on these definitions.\n\n\\section{Propagation of singularities on conic manifolds}\n\nConsider now solutions to the wave equation on a manifold with conic\nsingularities.  We always employ the \\emph{Friedrichs extension} of\nthe Laplacian acting on $\\mathcal{C}_c^\\infty(X^\\circ).$\n(This stipulation is important only in dimension two, where $\\Lap$ is not\nessentially self-adjoint.)\n\nWe now can (roughly) state the following:\n\\begin{theorem}[Melrose--Wunsch \\cite{Melrose-Wunsch1}]\nSingularities for solutions to the wave equation propagate along diffractive geodesics; strictly\ndiffractive geodesics generically propagate \\emph{weaker}\nsingularities than geometric geodesics.\n\\end{theorem}\nThe genericity condition is that the incident singularities not be\nprecisely \\emph{focused} on the cone tip and applies, e.g., to Cauchy data\nthat are conormal with respect to a manifold that is at most simply\ntangent to the hypersurfaces at constant distance from a cone tip.  In\nthis case---and in particular for the fundamental solution---we\nfind that \\emph{the diffracted wave for the fundamental solution is\n  $(n-1)\/2-\\epsilon$ derivatives smoother than the main wavefront,}\nwhere $n$ is the dimension of $X.$\n\nWe remark that this result has been subsequently generalized to cover\nthe cases of manifolds with incomplete edge singularities \\cite{MVW1},\nas well as manifolds with corners \\cite{Va:04}, \\cite{MeVaWu:13}.\n\nThe rest of this paper is essentially applications and extensions of this result in various contexts.\n\n\\section{Local energy decay on conic manifolds with Euclidean ends}\\label{section:BW}\n\nConsider now a noncompact $n$-manifold $X$ with ends that are\nEuclidean.  We will consider solutions to the wave equation\n$$\n\\Box u=0\n$$\non $X$ with compactly supported Cauchy data in the energy space.\n\nIf $X$ is a smooth manifold, it has long been known that the decay of\nlocal energy can be obstructed by the trapping of geodesics; recall\nthat a geodesic is said to be forward- or backward-\\emph{trapped} if\nit remains in a compact set as $t \\to \\pm \\infty.$ Classic work of\nLax--Philips \\cite{Lax-Phillips1} and Morawetz \\cite{Morawetz:Decay}\nshows that, for odd $n,$ absence of trapping implies exponential local\nenergy decay; on the other hand, results starting with Ralston\n\\cite{Ralston:Localized} show that trapping of rays implies that\nexponential local energy decay cannot hold.  The usual line of\nreasoning in obtaining such estimates involves obtained bounds on the\n\\emph{cutoff resolvent}\n$$\n\\chi (\\Lap-\\lambda^2)^{-1} \\chi,\\quad \\chi \\in \\mathcal{C}_c^\\infty.\n$$\nIt is well known that in odd dimensions this operator can be\nmeromorphically continued from $\\Im \\lambda>0$ to $\\CC,$ and its poles\nare known as \\emph{resonances}.  Exponential local energy decay is\nthen obtained by showing that no resonances lie in some \\emph{strip}\n$\\Im \\lambda>-\\nu,$ $\\nu>0$ (and that the resolvent has an upper bound\nwith polynomial growth in this strip).\n\nThe situation with conic manifolds is thus interesting for the\nfollowing reason: as soon as we have more than one cone point (or,\nindeed,\\footnote{The author is grateful to Yves Colin de Verdi\\`ere\n  for pointing out this possibility. In practice, it seems hard to create an\n  interesting example of a non-simply connected manifold where the\n  \\emph{only} trapping is a strictly diffractive geodesic of this form.  On the other hand one may probably add a complex absorbing\n  potential to the problem to destroy other trapping and create\n  non-simply connected examples.} at least one\ncone point if the manifold is non-simply connected) there must be trapped\ndiffractive geodesics: we can simply continue traversing geodesics\nconnecting the various cone points.  An example of particular interest\nis (the double of) a domain exterior to one or more polygons in $\\RR^2$: diffractive\ngeodesics can move along edges of one polygon and also along lines\nconnecting vertices of two different polygons.\n\nTo what degree, one wonders, does this obstruct energy decay?  The\nfollowing theorem (which answers affirmatively a conjecture of\nChandler-Wilde--Graham--Langdon--Spence~\\cite{CWGLS:2012} for\npolygonal exterior domains) shows that the obstruction is very minor:\n\\begin{theorem}[Baskin--Wunsch \\cite{BaWu:13}]\\label{theorem:BaWu}\nAssume that no three cone points in $X$ are collinear and no two are\nconjugate.  Assume that geodesics missing the cone points escape to\ninfinity at a uniform rate.\n\nFor $\\chi \\in \\CcI(X),$ there exists $\\delta>0$ such that the cut-off resolvent\n$$\n\\chi (\\Lap-\\lambda^2)^{-1}\\chi\n$$\ncan be analytically continued from $\\Im \\lambda >0$ to the region\n$$\n\\Im \\lambda >-\\rho \\log \\smallabs{\\Re \\lambda},\\ \\smallabs{Re \\lambda} >\\rho^{-1}\n$$\nand for some $C,T>0$ enjoys the estimate\n$$\n\\norm{\\chi (\\Lap-\\lambda^2)^{-1}\\chi}_{L^{2}\\to L^{2}} \\leq C\n\\smallabs{\\lambda}^{-1} e^{T\\smallabs{\\Im \\lambda}}\n$$\nin this region.\n\\end{theorem}\nWe contrast this with the the standard result for smooth non-trapping\nperturbations of Euclidean space.  In that case the methods of Vainberg\n\\cite{Vainberg:Asymptotic} and Lax--Phillips \\cite{Lax-Phillips1} yield precisely the \\emph{same}\nresolvent estimate on $\\RR$ and a slightly stronger result on\nresonance-free regions: \\emph{any} region of the form $\\Im \\lambda>-\\rho \\smallabs{\\log \\Re\n\\lambda}$ is free of resonances outside a large disc.  Thus\nthe effect of diffractive trapping by cone points is extremely\nweak.  Previous results in this direction include energy decay results of \\cite{Cheeger-Taylor2}, Section\n6, in certain special cases of conic singularities; analogous results\nfor multiple inverse square potentials were previously proved by\nDuyckaerts \\cite{Duyckaerts1}.  Burq \\cite{Burq:Coin} gave a\nprecise description of the resonances in the closely related case of\ntwo convex analytic domains in the plane, one of which has a corner\nfacing the other.  The\ndiffractive trajectory here bounces back and forth between the corner\nand the other obstacle, and Burq showed the associated resonances lie\nalong a family of logarithmic curves.\n\nWe now briefly describe some results on evolution equations that follow from Theorem~\\ref{theorem:BaWu}.\nWe let $\\mathcal{D}_s$ denote the\ndomain of $\\Lap^{s\/2}$ (hence locally just $H^s$ away from cone points) and let\n$\\sin t\\sqrt\\Lap\/\\sqrt\\Lap$ be the wave propagator.  Let $\\chi$ equal $1$\non the set where $X$ is not isometric to $\\RR^n.$  In odd dimensions,\nthe resolvent is a meromorphic function of $\\lambda\\in \\CC$ (with no\ndifficulties at $\\lambda=0$) so in this case\nTheorem~\\ref{theorem:BaWu} shows that there are only finitely many\nresonances in any horizontal strip in $\\CC.$  This enables us to show\nthe following by a contour deformation argument:\n\\begin{corollary}\\label{corollary:resexp}\nLet $n$ be odd. Under the assumptions of Theorem~\\ref{theorem:BaWu}, for all $A>0,$\nsmall $\\ep>0,$ $t>0$ sufficiently large,  and $f \\in\n\\mathcal{D}_1,$\n$$\n\\chi \\frac{\\sin t\\sqrt{\\Lap}}{\\sqrt{\\Lap}} \\chi f=  \\sum_{\\substack{\\lambda_j \\in \\Res(\\Lap) \\\\ \\Im \\lambda\n  >-A}}\\sum_{m=0}^{M_j} e^{-it \\lambda_j} t^m w_{j,m} + E_A(t) f\n$$\nwhere the sum is of resonances of $\\Lap,$ i.e.\\ over the poles of the\nmeromorphic continuation of the resolvent, and the $w_{j,m}$ are the\nassociated resonant states corresponding to $\\lambda_j.$\nThe error satisfies\n$$\n\\norm{E_A(t)}_{\\mathcal{D}_{1}\\to L^{2}} \\leq C_\\ep e^{-(A-\\ep)t}.\n$$\n\nIn particular, since the resonances have imaginary part bounded above by a\nnegative constant, $\\chi \\frac{\\sin t\\sqrt{\\Lap}}{\\sqrt{\\Lap}} \\chi f$ is\nexponentially decaying in this case.\n\\end{corollary}\n\nAnother corollary is a local smoothing estimate for the Schr\\\"odinger\nequation.  As it comes from the resolvent estimate on $\\RR,$ this is\nagain lossless as compared to the situation on free $\\RR^n$:\n\\begin{corollary}\n  \\label{corollary:local-smoothing}\n  Suppose $u$ satisfies the Schr{\\\"o}dinger equation on $X$:\n  \\begin{align*}\n    i^{-1}\\pa_t u(t,z) + \\Lap  u(t,z) &= 0 \\\\\n    u(0,z) &= u_{0}(z)\\in L^{2}(X)\n  \\end{align*}\nUnder the assumptions of Theorem~\\ref{theorem:BaWu}, for all $\\chi \\in C^{\\infty}_{c}(X)$, $u$ satisfies the local smoothing estimate without loss:\n  \\begin{equation*}\n    \\int_{0}^{T}\\norm{\\chi u(t) }_{\\mathcal{D}_{1\/2}}^{2}\\, dt \\leq\n    C_{T} \\norm{u_{0}}_{L^2}^{2}.\n  \\end{equation*}\n\\end{corollary}\n\nThe elements of the proof of Theorem~\\ref{theorem:BaWu} are twofold.\nThe first step is to show that a \\emph{very weak Huygens principle}\nholds.  We recall that in nontrapping manifolds, a solution to the\nwave equation with compactly supported initial data is eventually\n\\emph{smooth}---this is the usual ``weak Huygens principle.''  Here we show instead that the solution eventually gets\n\\emph{as smooth as we like}:\n\\begin{proposition}\\label{proposition:Huygens}\nLet $\\chi \\in \\CcI(X).$   For any $s \\in \\RR,$ there exists $T_s\\gg 0$\nsuch that whenever $t>T_s,$\n$$\n\\chi U(t) \\chi: H^r \\to H^{r+s}\n$$\nfor all $r.$\n\\end{proposition}\n\nThe second part of the theorem is a modification of the celebrated\npa\\-ra\\-met\\-rix construction of Vainberg \\cite{Vainberg:Asymptotic} (see\nalso \\cite{Tang-Zworski1}).  This argument in its original form builds\na parametrix for the resolvent out of the fundamental solution to the\nwave equation, assuming that the latter satisfies the weak Huygens\nprinciple; the new variant, by contrast, makes the weaker assumption\nof the output of Proposition~\\ref{proposition:Huygens} and produces a very\nslightly weaker result (smaller resonance-free region).\n\nAmong the further applications of this line of reasoning is the following\ntheorem on Strichartz estimates for exterior polygonal domains (joint\nwork with Baskin and Marzuola) \\cite{BaskinMarzuolaWunsch:2014}):\nfor an exterior polygonal domain where the only trapped geodesics are\nstrictly diffractive (and where no three vertices are collinear) we\nfind that the same Strichartz estimates for the Schr\\\"odinger equation\nhold as on Euclidean space (locally in time for Neumann conditions,\nand globally for Dirichlet).\n\n\\section{The wave trace}\n\nIf $X$ is a compact Riemannian manifold without boundary let\n$$\n(\\phi_j, \\lambda_j^2)\n$$\ndenote the eigenfunctions and eigenvalues of $\\Lap.$\nOne might like to study\nthe ``inverse spectral problem'' of using the $\\lambda_j$ to\ncharacterize $X$ by forming a useful generating function out of the\n$\\lambda_j.$ An obvious but not directly useful one might be\n$$\n\\sum_j \\delta(\\lambda-\\lambda_j),\n$$\nbut a much more tractable one is the Fourier transform of this\nquantity,\n$$\n\\sum_j e^{-it\\lambda_j}.\n$$\nThe utility of this generating function stems from its identification\nas\n$$\n\\Tr U(t),\n$$\nwhere $$U(t)\\equiv e^{-it\\sqrt{\\Lap}}$$ is the ``half-wave'' evolution\noperator, mapping functions on $X$ to (certain) solutions to the wave\nequation.  If we can say something about the trace of $U(t)$ in terms\nof the geometry of $X,$ we can thus hope to learn something about\nspectral geometry.\n\nIn the setting of smooth\nboundaryless manifolds, we have the following classical results on the\nwave trace.  Let\n$$\n\\LSpec (X) =\\{0\\} \\bigcup \\big\\{\\pm \\text{lengths of periodic\n  geodesics on }X\\big\\}.\n$$\n\\begin{theorem}[Chazarain \\cite{Chazarain1}, Duistermaat--Guillemin\n  \\cite{Duistermaat-Guillemin1}; cf.\\ also Colin de Verdi\\`ere \\cite{Co:73a}, \\cite{Co:73b}]\\label{theorem:smoothpoisson}\n$$\\singsupp \\Tr U(t) \\subset \\LSpec (X).$$\n\\end{theorem}\nThis allows one to dream of ``hearing'' lengths of closed geodesics,\nbut does not rule out the possibility that the allowable singularities\ndo not, in fact, arise.  The presence of honest singularities is,\nhowever, guaranteed by:\n\\begin{theorem}[Duistermaat--Guillemin \\cite{Duistermaat-Guillemin1}]\nLet $L$ be the length of an nondegenerate periodic closed geodesic\n$\\gamma$ on\n$L$ that is isolated in the length spectrum.  Then near $t=L$ we have\n$$\n\\Tr U(t) \\sim \\frac{L_0}{2\\pi} i^{\\sigma} \\abs{I-P}^{-1\/2}(t-L)^{-1},\n$$\nwhere\n\\begin{itemize}\n\\item $L_0$ is the length of the primitive closed geodesic if \n $\\gamma$ is an iterate of a shorter one.\n\\item $\\sigma$ is the Morse index of the variational problem for a\n  periodic geodesic, evaluated at $\\gamma.$\n\\item $P$ is the linearized Poincar\\'e map, obtained as the\n  linearization at $\\gamma$ of the first return map to a hypersurface\n  of the phase space, transverse to $\\gamma.$\n\\end{itemize}\n\\end{theorem}\nNote that the nondegeneracy condition in the hypotheses is simply the condition that $I-P$ be nonsingular.\n\nThe generalization of Theorem~\\ref{theorem:smoothpoisson} to compact\nconic manifolds is straightforward: let $$\\DLSpec(X)=\n\\{0\\} \\bigcup \\big\\{\\pm \\text{lengths of periodic\n diffractive geodesics on }X\\big\\}.$$\n\\begin{theorem}[Wunsch \\cite{Wunsch2}]\nOn a conic manifold $X,$\n$$\n\\singsupp \\Tr U(t) \\subset \\DLSpec{X}.\n$$\n\\end{theorem}\nThe singularities at lengths of geodesics in $X^\\circ$ are easily seen\nto be described by the same formula given by Duistermaat--Guillemin,\nbut the geodesics interacting through conic points are not so simple.\nWe consider $\\gamma$ a closed, \\emph{strictly diffractive} geodesic\nundergoing $k$ diffractions and traversing geodesic segments\n$\\gamma_1,\\dots, \\gamma_k$ connecting cone points $Y_{i_1}, \\dots\nY_{i_k}.$ Recall that the hypothesis that the geodesic be strictly\ndiffractive means that it interacts with each cone point by entering\nand leaving on a pair of geodesics that cannot be uniformly locally\napproximated by geodesics in $X^\\circ.$ This is generically the case\nfor all closed geodesics.  Assume further that the length $L$ of\n$\\gamma$ is isolated in the length spectrum, and make the additional\nnondegeneracy hypothesis that no two cone points along the geodesic\nare conjugate to one another.\nNote that the following was previously known by work of Hillairet\n\\cite{Hillairet:2005} in the important special case of flat surfaces\nwith conic singularities (hence in particular for doubles of\npolygons).\n\\begin{theorem}[Ford--Wunsch \\cite{1411.6913}]\\label{theorem:FoWu}\nNear $t=L,$\n$$\n\\Tr U(t) \\sim \\int e^{i(t-L)\\xi} a(\\xi) \\, d\\xi\n$$\nwhere\n\\begin{equation}\\label{symbol}\n  a(\\xi) \\sim L_0 \\cdot (2\\pi)^{\\frac{kn}{2}} \\,  e^{\\frac{ik(n-3)\\pi}{4}} \\,\n  \\chi(\\xi) \\, \\xi^{-\\frac{k(n-1)}{2}} \\prod_{j=1}^k i^{-m_{\\gamma_j}} \\, \\mathcal{D}_j\n  \\, \\mathcal{W}_j \\ \\text{as $|\\xi| \\to \\infty$}. \n\\end{equation}\n\\end{theorem}\nHere, \n$\\chi\\in \\CI(\\RR)$ is $1$ for $\\xi>1$ and $0$ for\n  $\\xi<0.$  \nNote that the power of $\\xi$ is such that we obtain greater smoothness\nas the number of diffractions increases.  The leading order\nsingularity as a function of $t$ is proportional to $(t-L+i0)^{-1+k(n-1)\/2}$\n  (but is multiplied by $\\log(t-L+i0)$ if the power is an integer).\n\nAs before $L_0$ denotes the length of the\n``primitive'' geodesic if $\\gamma$ is an iterate of a shorter one.\nThe integers $m_{\\gamma_j}$ are simply the Morse indices of the\nvariational problems associated to traveling from one cone point to\nthe next, evaluated at $\\gamma_j.$\n\nWe will now explain the factors $\\mathcal{D}_j$ and $\\mathcal{W}_j.$\n\nThe terms $\\mathcal{D}_j$ are associated to the diffractions through each\nsuccessive cone point $Y_{i_j}.$  They are constructed as follows.\nEach cone point $Y_{i_j}$ is equipped with a metric $h_{i_j}\\equiv\nh\\rvert_{Y_{i_j}}.$  It thus has a Laplace-Beltrami operator\n$\\Lap_{i_j}$ and we may use the functional calculus to take functions\nof this operator.  In particular, let $$\\nu_{i_j} \\equiv \\sqrt{\n  \\Delta_{Y_{i_j}} + \\left( \\frac{2-n}{2} \\right)^2 }.$$\nWe then form the operator family\n$$\ne^{-it\\nu_{i_j}}: L^2(Y_{i_j})\\to L^2(Y_{i_j}).\n$$\nThis is essentially a ``half Klein Gordon propagator'' on the link of\nthe cone point (i.e., a boundary component).  Now let $\\kappa(\\bullet)$ denote the Schwartz\nkernel of an operator.  Supposing that the diffractive geodesic\n$\\gamma$ enters $Y_{i_j}$ at the point $y$ and leaves from point $y',$\nwe set\n$$\n\\mathcal{D}_j \\equiv \\kappa(e^{-i\\pi \\nu_{i_j}})[y,y'].\n$$\nThe propagator kernel is of course not continuous in general, however note\nthat the strictly diffractive nature of the geodesic ensures that $y$\nand $y'$ are not connected by a geodesic of length $\\pi$ in the link,\nwhich in turn precisely ensures, by propagation of singularities, that\nthe Schwartz kernel of the time-$\\pi$ Klein Gordon propagator is\nsmooth near $(y,y'),$ hence the evaluation of this distribution makes\nsense.\n\nNow we turn to $\\mathcal{W}_j$.  These quantities are associated to the\ngeodesic segments $\\gamma_j$ connecting successive cone points.  They\nare best described in terms of Jacobi fields, but can also be\nviewed as a proxy for a quantity involving the derivative of the\nexpenential map, hence a substitute for the term involving the\nPoincar\\'e map in the Duistermaat--Guillemin formula.   Note that the\nexponential map from one cone point to the next does not make sense,\nsince any small perturbation of the geodesic $\\gamma_j$ will miss the\nnext cone point entirely rather than simply hitting it at a different point.\nCorrespondingly, if we let $\\mathbf{J}$ be a set of Jacobi fields that are\northonormal to $\\gamma_j$ and at $\\gamma_j(0)$ give an orthonormal\nbasis of $TY_{i_j}$ then $\\mathbf{J}$ becomes \\emph{singular} as we approach\nthe end of $\\gamma_j$ at $Y_{i_{j+1}}.$  On the other hand, the metric\nis also singular at cone points, in the sense that it vanishes on\n$TY,$ so we can nonetheless make sense of the determinant\n$$\n\\det_g \\mathbf{J}\\rvert_{Y_{i_{j+1}}}.\n$$\nThen we have\n$$\n\\mathcal{W}_j \\equiv\\big\\lvert\\det_g \\mathbf{J}\\rvert_{Y_{i_{j+1}}}\\big\\rvert^{-1\/2}.\n$$\nThis quantity can be made to look more like the derivative of an\nexponential map as follows: we set\n\\begin{equation}\\label{thetaj}\n\\Theta_j=(\\length(\\gamma_j)^{-(n-1)})\\big\\lvert\\det_g \\mathbf{J}\\rvert_{Y_{i_{j+1}}}\\big\\rvert.\n\\end{equation}\nConsider the case in which $Y_j$ is a ``fictitious'' cone point\nobtained by blowing up a smooth point $p_0$ on a manifold.  Then Jacobi vector\nfields tangent to $Y_j$ are obtained as lifts under the blow-down map\nof Jacobi fields vanishing at $p_0,$ and \n$\\Theta_j$ becomes\na standard expression\nfor $\\det D\\exp_{Y_j} (\\bullet)$ in terms of Jacobi fields, at least\nwhen evaluated in $X^\\circ$ (cf.\\\n\\cite{Be:77}): in that case we simply have\n$$\n\\Theta_j=\\det_g \\big\\lvert D \\exp_{Y_j}(\\bullet)\\big\\rvert.\n$$\nSince $\\mathcal{W}_j=\\length(\\gamma_j)^{-(n-1)\/2} \\Theta_j^{-1\/2}$ we\nrecover the relationship with the exponential map in the case of a\ntrivial cone point.\n\nIn rough outline, the proof of Theorem~\\ref{theorem:FoWu} goes as\nfollows.  We know explicitly what the wave propagator look like on a\nmodel \\emph{product cone} $\\RR_+\\times Y_j$ endowed with the scale\ninvariant metric $dx^2 +x^2 h_0(y,dy)$---this is a computation of\nCheeger--Taylor \\cite{Cheeger-Taylor1}, \\cite{Cheeger-Taylor2}\ninvolving bravura use of the Hankel transform.  In particular, we\ncan evaluate the symbol of the diffracted wavefront explicitly in that\ncase.  More generally, in \\cite{Melrose-Wunsch1} the author and\nMelrose prove that near a cone point, the diffracted front of the wave\npropagator is guaranteed to be a conormal distribution.  The first new\nstep is therefore to show that in the non-product case, the principal\nsymbol of the diffracted front is still, modulo adjustments involving\ncomparing half-densities on the two spaces, given by the same\nexpression as in the product case where we use the model metric\n$dx^2+x^2 h\\rvert_{x=0} (y,dy).$ This involves comparing the two\npropagators and showing that the difference\nbetween model and exact propagators can be estimated by a\n\\emph{Morawetz inequality} near the cone tip.\n\nHaving understood the effect of a single diffraction, we then proceed\nas follows.  We take a microlocal partition of unity $A_j$ on $X,$\nwhere for technical reasons the $A_j$ are restricted to be simply\ncutoff functions near each boundary component $Y_i$ but are otherwise\nfully localized in phase space.  \nWe then decompose the wave trace as follows: fix\nsmall times $t_j$ with $\\sum t_j=T.$  Then by cyclicity of the trace\n$$\n\\Tr U(t) = \\sum_{i_0,\\dots, i_{N}}\\Tr \\sqrt{A_{i_0}} U(t-T)A_{i_1} U(t_1) A_{i_2}\\dots\nA_{t_{N}} U(t_N) \\sqrt{A_{i_0}}.\n$$\nBy propagation of singularities, this term is guaranteed to be trivial\nunless there is a diffractive geodesic successively passing through\nthe microsupports of the $A_{i_j}$'s, hence we may throw away most of\nthis sum.  The remaining terms are then computed by a stationary\nphase computation, gluing together the propagators for ``free''\npropagation through $X^\\circ$ with those for the diffractive\ninteraction with cone points (this was the same strategy previously\nused by Hillairet in \\cite{Hillairet:2005} as well as by the author in\n\\cite{Wunsch2}).\n\n\n\\section{Lower bounds for resonances}\n\nWhile $\\Tr U(t)$ only makes sense (even distributionally) on a \\emph{compact}\nmanifold, if we return to the setting of Section~\\ref{section:BW}\nwhere we have a \\emph{noncompact} manifold with Euclidean ends, we may still\nmake sense of an appropriately \\emph{renormalized} wave trace, and use\nthe diffractive trace formula (Theorem~\\ref{theorem:FoWu}) to obtain\nlower bounds on resonances.\n\nIn odd dimensions, we let $\\mathcal{A}$ denote the generator of\nthe wave group, and hence $e^{t \\mathcal{A}}$ the wave group itself;\nlikewise we let $\\mathcal{A}_0$ be the generator of the wave group on Euclidean space.\n We\nthen have the trace formula\n\\begin{equation}\\label{traceres}\n\\Tr (e^{t \\mathcal{A}} -e^{t \\mathcal{A}_0}) = \\sum_{\\lambda_j \\in \\Res} e^{-i \\lambda_j t},\\ t\n> 0\n\\end{equation}\nwhere the sum is over the resonances, counted with multiplicity (see\ne.g. \\cite{Sjostrand-Zworski5} for the details of how to makes sense of\nthis difference of operators in a wide variety of contexts).\nThis result in various settings was first proved by Bardos-Guillot-Ralston \\cite{Bardos-Guillot-Ralston1}, Melrose\n\\cite{MR83j:35128}, and Sj\\\"ostrand-Zworski \\cite{Sjostrand-Zworski5};\nan analogous result in even dimensions can be found in \\cite{Zw:99}.\n\nNow if we can actually guarantee the existence of singularities in the (renormalized) wave\ntrace, a Tauberian theorem of Sj\\\"ostrand-Zworski\n\\cite{Sjostrand-Zworski3} allows us to deduce from \\eqref{traceres} in\na lower bound on the number of resonances in logarithmic\nregions in $\\CC.$  Fortunately, Theorem~\\ref{theorem:FoWu} applies\nequally well in this context, and we obtain a lower bound on the\nnumber of resonances as follows.\nLet\n$$\nN_\\rho(r) =\\# \\{\\text{Resonances in } \\smallabs{\\lambda}<r,\\\n\\Im \\lambda \\geq -\\rho \\log \\smallabs{\\Re{\\lambda}}\\}.\n$$\nThen we have:\n\\begin{theorem}[Hillairet--Wunsch]\\label{theorem:HiWu}\nUnder the geometric assumptions of Theorem~\\ref{theorem:BaWu}, let $L$\nbe the length of a closed, strictly diffractive geodesic $\\gamma$\nundergoing $k$ diffractions.  Assume, in\nthe notation of Theorem~\\ref{theorem:FoWu}, that all the diffraction\ncoefficients $\\mathcal{D}_j$ are nonzero along $\\gamma;$ assume also that there are no closed diffractive geodesics\nbeside iterates of this one having length in $L\\NN.$ Then for all $\\ep>0,$\n$$\nN_\\rho(\\rho)\\geq C_{\\rho,\\epsilon} r^{1-\\epsilon}\n$$\nprovided\n$$\n\\rho > \\frac{(n-1)k}{2L}\n$$\n\\end{theorem}\nA detailed proof, which simply consists of using the trace formula\n(Theorem~\\ref{theorem:FoWu}) in \\eqref{traceres} together with the\nTauberian theorem of \\cite{Sjostrand-Zworski3}, can be found in\n\\cite{Ga:15}.  Note that the bound on $\\rho$ written here is that\nwhich we obtain by considering the whole sequence of singularities of\nthe wave trace obtained by considering arbitrary \\emph{iterates} of\nthe geodesic $\\gamma.$ We remark that the distinction between the\ntrace of the full wave group and $\\Tr U(t)$ is immaterial for this\npurpose since the former is twice the real part of the latter, and it\nis not difficult to verify from examination of \\eqref{symbol} that the\nsingularities arising from iterates of a given geodesic cannot all be\npurely imaginary.\n\nThe optimal $\\rho$ here is generally obtained by choosing $\\gamma$\nto be the geodesic that\ntraverses the longest geodesic segment connecting a pair of distinct cone\npoints, back and forth (assuming the diffraction coefficients are\nnonvanishing).  If\n$D_{\\text{max}}$ denotes the greatest distance between a pair of cone points, then we\nhave a closed geodesic of length $2 D_{\\text{max}}$ with $k=2,$ and we\nobtain the bound\n$$\n\\rho > \\frac{(n-1)}{2D_{\\text{max}}}.\n$$\n\nRemarkably, this theorem is essentially sharp, as was shown by\nGalkowski, who has produced an effective version of the Vainberg\nargument previously employed in \\cite{BaWu:13}:\n\\begin{theorem}[Galkowski \\cite{Ga:15}]\nLet $D_{\\text{max}}$ be the greatest distance between two cone points.  For any\n$\\epsilon>0$ the\nconstant $\\rho$ in Theorem~\\ref{theorem:BaWu} can be taken to be\n$(n-1)\/(2D_{\\text{max}})-\\epsilon,$ i.e.\\ $N_\\rho(r)$ is \\emph{bounded} for all $\\rho<(n-1)\/2D_{\\text{max}}.$\n\\end{theorem}\nSince $N_\\rho$ is bounded for $\\rho<(n-1)\/2D_{\\text{max}}$ and (subject to the\nnondegeneracy hypotheses of Theorem~\\ref{theorem:HiWu}) almost linearly\ngrowing for $\\rho>(n-1)\/2D_{\\text{max}},$ we find that in any set near the\ncritical curve $\\Im\\lambda=-((n-1)\/2D_{\\text{max}}) \\log \\smallabs{\\Re\n  \\lambda}$ of the form\n$$\n\\big(-\\frac{n-1}{2D_{\\text{max}}}-\\ep\\big) \\log \\smallabs{\\Re\n  \\lambda}<\\Im \\lambda<\\big(-\\frac{n-1}{2D_{\\text{max}}}+\\ep\\big)\n\\log \\smallabs{\\Re \\lambda},\\quad \\abs{\\lambda}>\\ep^{-1}\n$$\nthere are infinitely many resonances.  The intuition behind the\nimportance of the longest geodesic connecting two cone points is that\nrepeatedly traversing this segment back and forth is the way in which\na trapped singularity can diffract \\emph{least frequently}.  Since each\ndiffraction loses considerable energy owing to the smoothing effect of\ndiffraction, a resonant state propagating back and forth along this\ngeodesic is the one that loses energy to infinity at the\nslowest rate.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nIn non-central ultra-relativistic heavy-ion collisions, the two colliding nuclei carry large amount of orbital angular momentum ${\\boldsymbol L}$. Soon after the initial impact, a~substantial part of ${\\boldsymbol L}$ is deposited in the interaction zone and can be further transformed to the spin part ${\\boldsymbol S}$ (with the total angular momentum ${\\boldsymbol J}={\\boldsymbol L}+{\\boldsymbol S}$ being conserved). The latter can be reflected in the spin polarization of the particles emitted at freeze-out. To verify this phenomenon, the spin polarization of various particles ($\\Lambda$, $K^*$, $\\phi$) produced in relativistic heavy-ion collisions has been recently measured by the STAR~\\cite{STAR:2017ckg,Adam:2018ivw}, ALICE~\\cite{Acharya:2019vpe} and HADES~\\cite{Kornas:2019} experiments. \n\nOn the theoretical side, first predictions of a non-zero global spin polarization of the $\\Lambda$ hyperons, based on perturbative-QCD calculations and the spin-orbit interaction, were made in Refs.~\\cite{Liang:2004ph,Liang:2004xn} and \\cite{Voloshin:2004ha}, respectively (see also \\CIT{Betz:2007kg}). In these works, a substantial polarization effect of the order of 10\\% was found. Subsequently, using relativistic hydrodynamics with local thermodynamic equilibrium of the spin degrees of freedom~\\cite{Becattini:2007sr,Becattini:2013vja,Becattini:2013fla,Becattini:2007nd,Becattini:2016gvu,Becattini:2015ska,Karpenko:2016jyx,Xie:2017upb,Pang:2016igs,Becattini:2017gcx}, a~smaller polarization of about 1\\% was predicted, an effect which was eventually confirmed by STAR~\\cite{STAR:2017ckg,Adam:2018ivw}.\n\nInterestingly, the same hydrodynamic models~\\cite{Becattini:2017gcx,Becattini:2020ngo} are not able to describe the experimentally measured longitudinal polarization of $\\Lambda$'s~\\cite{Niida:2018hfw,Adam:2019srw}. For example, the oscillation of the longitudinal polarization of the $\\Lambda$ hyperons measured as a function of the azimuthal angle in the transverse plane~\\cite{Niida:2018hfw} has an opposite sign compared to the results obtained with relativistic hydrodynamics with thermalized spin degrees of freedom. This issue is at the moment the subject of very intensive investigations \\cite{Li:2017dan,Li:2017slc,Pang:2016igs,Xie:2017upb,Sun:2017xhx,Sun:2018bjl,Fang:2016vpj,Florkowski:2019qdp,Florkowski:2019voj,Gao:2020vbh,Li:2020vwh,Liu:2020bbd,Liu:2019krs,Ayala:2020ndx,Ivanov:2020qqe,Liu:2020ymh,Huang:2020xyr,Deng:2020ygd,Montenegro:2020paq,Ivanov:2019ern}. \n\nThe relativistic hydrodynamic models (perfect or viscous) that have been used so far to describe the global spin polarization of the $\\Lambda$ and $\\bar\\Lambda$ hyperons~\\cite{Becattini:2016gvu,Karpenko:2016jyx,Becattini:2017gcx} make use of the fact that spin polarization effects are governed by the thermal vorticity tensor\n\\begin{eqnarray}\n\\varpi_{\\mu \\nu}= -\\frac{1}{2} (\\partial_\\mu \\beta_\\nu-\\partial_\\nu \\beta_\\mu).\n\\label{eq:thermvor}\n\\end{eqnarray}\nHere the four-vector $\\beta_\\mu$ is defined in the standard way as the ratio of the fluid flow vector $u_\\mu$ and the local temperature $T$, i.e., $\\beta_\\mu = u_\\mu\/T$. One can notice that the use of \\EQn{eq:thermvor} does not require any modifications of the existing hydrodynamic codes as spin effects are determined solely by the form of $u^\\mu$ and $T$.\n\nHowever, on the general thermodynamic grounds~\\cite{Becattini:2018duy}, it is expected that the spin polarization effects may be  governed by the tensor $\\omega_{\\mu \\nu}$ (called below the spin polarization tensor) that can be independent of the thermal vorticity \\EQn{eq:thermvor}.  This suggests that a completely new hydrodynamic approach including spin dynamics can be constructed, with the spin polarization tensor $\\omega_{\\mu \\nu}$ treated as an independent hydrodynamical variable. In this context, the concept of local spin equilibrium also changes as one no longer requires that $\\omega_{\\mu \\nu} = \\varpi_{\\mu \\nu}$ to have zero entropy production. \n\nFirst steps to formulate the perfect-fluid version of hydrodynamics of spin polarized fluids that incorporates the spin polarization tensor $\\omega_{\\mu\\nu}$ have already been made in a series of publications~\\cite{Florkowski:2017ruc,Florkowski:2017dyn,Becattini:2018duy}, for a recent summary see~\\CIT{Florkowski:2018fap}. However, only in a very recent work~\\CITn{Bhadury:2020puc}, the dissipation effects in such systems have been explicitly considered, see also Refs.~\\cite{Weickgenannt:2020aaf,Speranza:2020ilk,Hattori:2019ahi,Yang:2020hri,Hattori:2019lfp, Shi:2020htn,Gallegos:2020otk}. \n\nIn this work we continue and significantly extend the results obtained in \\CIT{Bhadury:2020puc}. In order to identify the structure of dissipative terms, we use classical kinetic theory for particles with spin ${\\nicefrac{1}{2}}$.  The collision terms are treated in the relaxation time approximation (RTA) according to the prescription defined in \\CIT{Bhadury:2020puc} and, for the sake of simplicity, we restrict our considerations to the Boltzmann statistics. The kinetic-theory framework determines the structure of viscous and diffusive terms and allows to explicitly calculate a set of new kinetic coefficients that characterize dissipative spin dynamics. These coefficients describe coupling between a non-equilibrium part of the spin tensor and thermodynamic forces such as the expansion scalar, shear flow tensor, the gradient of chemical potential divided by temperature, and, finally, the gradient of the spin polarization tensor.  \n\n\\smallskip\nThe structure of the paper is as follows: In Sec. II we recall the formulation of the perfect-fluid hydrodynamics with spin. Our presentation is based on the classical concept of spin and classical distribution functions in an extended phase space. In Sec. III we introduce kinetic equations with the collision terms treated in the relaxation time approximation and derive the form of the dissipative corrections. This section contains also the explicit form of the new, spin-related kinetic coefficients. We conclude and summarize in Sec. IV. The paper is closed with several appendices where details of our straightforward but quite lengthy calculations are given. We use natural units and the metric tensor with the signature $(+---)$.\n\n\\section{Formulation of perfect fluid hydrodynamics for spin polarized fluids}\n\\label{sec:1}\n\\subsection{Spin-dependent equilibrium distribution function}\n\\label{subsec:1}\n\nWe start with the classical treatment of massive particles with spin-${\\nicefrac{1}{2}}$ and introduce their internal angular momentum $s^{\\alpha\\beta}$~\\cite{Mathisson:1937zz}. It is connected with the particle four-momentum $p_\\gamma$ and spin four-vector~$s_\\delta$~\\cite{Itzykson:1980rh}  by the following relation~\\footnote{We follow here the sign conventions used in our previous publications, e.g., in \\cite{Florkowski:2018fap}. We note that they are different from those used in \\cite{Weickgenannt:2019dks}. }\n\\begin{eqnarray}\ns^{\\alpha\\beta} = \\f{1}{m} \\epsilon^{\\alpha \\beta \\gamma \\delta} p_\\gamma s_\\delta,\n\\label{eq:salbe}\n\\end{eqnarray}\nwhere $m$ is the particle mass. Equation \\EQn{eq:salbe} implies that $s^{\\alpha\\beta} = -s^{\\beta\\alpha}$ and $p_\\alpha s^{\\alpha\\beta}  = 0$. Moreover,  assuming that the four-vectors $p$ and  $s$ are orthogonal to each other we find\n\\begin{eqnarray}\ns^{\\alpha} = \\f{1}{2m} \\epsilon^{\\alpha \\beta \\gamma \\delta} p_\\beta s_{\\gamma \\delta}.\n\\label{eq:sabinv}\n\\end{eqnarray}\nIn the particle rest frame (PRF), where the four-momentum of a particle is $p^\\mu = (m,0,0,0)$, the spin four-vector $s^\\alpha$ has only spatial components, i.e., $s^\\alpha = (0,{\\boldsymbol s}_*)$, with the length of the spin vector defined by $-s^2 = |{\\boldsymbol s}_*|^2={\\mathfrak{s}}^2 =  \\f{1}{2} \\left( 1+ \\f{1}{2}  \\right)$. \n\nIdentification of the so-called collisional invariants of the Boltzmann equation allows us to construct the equilibrium distribution functions $f^\\pm_{s, \\rm eq}(x,p,s)$ for particles and antiparticles~\\cite{Florkowski:2018fap,Bhadury:2020puc},\n\\begin{equation}\nf^\\pm_{s, \\rm eq}(x,p,s) =  f_{\\rm eq}^{\\pm}(x,p)\\exp\\left[\\frac{1}{2} \\omega_{\\mu\\nu} (x) s^{\\mu\\nu} \\right].\n\\label{eq:feqxps}\n\\end{equation}\nHere $f_{\\rm eq}^{\\pm}(x,p)=\\exp\\left[-p^\\mu\\beta_\\mu(x)\\pm\\xi(x)\\right]$ is the J\\\"uttner distribution, with $\\xi$ and $\\beta_{\\mu}$ traditionally defined as ratios of chemical potential $\\mu$ to temperature $T$ and four-velocity $u_\\mu$ to temperature $T$, i.e., $\\xi=\\mu\/T$ and $\\beta_\\mu=u_\\mu\/T$.~\\footnote{We note that since we always consider particles being on the mass shell ($p^0 = E_p = \\sqrt{{\\boldsymbol p}^2+m^2}$) the distribution $f(x,p,s)$ is in fact a function of ${\\boldsymbol p}$ only.} The spin polarization tensor $\\omega_{\\mu\\nu}$ has been introduced in Sec.~\\ref{intro}. It plays a crucial role in our formalism and can be interpreted as the (tensor) potential conjugated to the spin angular momentum.\n\nBefore we proceed further we note that in our approach $s^{\\mu\\nu}$ is dimensionless (measured in units of $\\hbar$) and so is $\\omega_{\\mu\\nu}$. Consequently, we can make expansions in $\\omega_{\\mu\\nu}$ and, in fact, most of our results will be valid in the leading order of $\\omega_{\\mu\\nu}$.\n\nOrdinary phase-space equilibrium distribution functions can be obtained by integrating out the spin degrees of freedom present in $f^\\pm_{s, \\rm eq}(x,p,s)$, \n\\begin{equation}\n\\int \\mathrm{dS} \\, f^\\pm_{s, \\rm eq}(x,p,s) = f^\\pm_{\\rm eq}(x,p), \n\\label{a}\n\\end{equation}\nwhere~\\cite{Florkowski:2018fap}\n\\begin{eqnarray}\n\\mathrm{dS} &=& \\frac{m}{\\pi {\\mathfrak{s}}} \\, \\mathrm{d}^4s~\\delta(s\\cdot s + {{\\mathfrak{s}}}^2)~\\delta(p\\cdot s)\\label{eq:dS}.\n\\end{eqnarray}\nDifferent properties of spin integrals done with the integration measure \\EQn{eq:dS} are collected in Appendix \\ref{sec:appspin}.\n\n\\subsection{Perfect fluid hydrodynamics for spin polarized fluids}\n\\label{subsec:2}\n\nFor a system of particles and anti-particles with spin degrees of freedom included only through degeneracy factors, the relevant conserved quantities are the energy-momentum tensor ($T^{\\mu\\nu}$) and charge current ($N^\\mu$). If spin is explicitly included, one has to consider an additional conserved quantity, namely, the angular-momentum tensor ($J^{\\lambda, \\mu\\nu}$)~\\cite{Florkowski:2018fap,Florkowski:2018ahw}. This is connected with the fact that the total angular momentum conservation law for particles with spin has a non-trivial form. \n\nThe total angular-momentum tensor ($J^{\\lambda, \\mu\\nu}$) can be written as a sum of the orbital ($L^{\\lambda, \\mu\\nu}$) and spin ($S^{\\lambda, \\mu\\nu}$) parts. The latter is known as the spin tensor. It is well known that there are various equivalent forms of the energy-momentum and spin tensors that can be used to define system's dynamics \\cite{Hehl:1976vr,Speranza:2020ilk,Tinti:2020gyh}. The forms used in this work agree with the definitions introduced by de Groot, van Leeuwen, and van Weert in \\cite{DeGroot:1980dk}. To emphasize this fact we sometimes use the acronym GLW.\n\nThe structures of $T^{\\mu\\nu}$, $N^\\mu$ and $S^{\\lambda, \\mu\\nu}$ can be connected to the behaviour of microscopic constituents of the system through the moments of the  phase-space distribution functions  $f_{\\rm eq}(x,p,s)$. Using the equilibrium distributions $f_{\\rm eq}(x,p,s)$ defined above, the hydrodynamic quantities such as charge current, energy-momentum tensor, and the spin tensor can be obtained in the similar way as in standard hydrodynamics. \n\n\\subsection{Charge current}\n\\label{subsubsec:1}\nThe equilibrium charge current is defined by the formula\n\\begin{eqnarray}\nN^\\mu_{\\rm eq} \n&=& \\!\\int \\! \\mathrm{dP}~\\mathrm{dS} \\, \\, p^\\mu \\, \\left[f^+_{s, \\rm eq}(x,p,s)\\!-\\!f^-_{s, \\rm eq}(x,p,s) \\right],\n\\label{eq:Neq-sp0}\n\\end{eqnarray}\nwhere the invariant momentum integration measure $\\mathrm{dP}$ is \n\\begin{eqnarray}\n\\mathrm{dP} &=& \\frac{d^3p}{(2 \\pi )^3 E_p},\n\\label{eq:dP}\n\\end{eqnarray}\nwhile the measure $\\mathrm{dS}$ is defined by \\EQ{eq:dS}. Using the equilibrium functions (\\ref{eq:feqxps}) we obtain\n\\begin{eqnarray}\nN^\\mu_{\\rm eq} = 2 \\sinh(\\xi) \\int \\mathrm{dP} \\, p^\\mu \ne^{- p \\cdot \\beta}\n\\int \\mathrm{dS} \\exp\\left( \\f{1}{2}  \\omega_{\\alpha \\beta} s^{\\alpha\\beta} \\right).\\nonumber\\\\\n\\label{eq:Neq-sp1}\n\\end{eqnarray}\nSince for large values of the spin polarization tensor the system becomes anisotropic in the momentum space and requires special treatment~\\cite{Florkowski:2010cf,Martinez:2010sc}, in most of our calculations we consider only the case of small values of $\\omega$. In this case the last exponential function in \\EQn{eq:Neq-sp1} can be expanded up to linear order and we find\n\\begin{eqnarray}\nN^\\mu_{\\rm eq} &=& 2 \\sinh(\\xi) \\int \\mathrm{dP} \\, p^\\mu \\, e^{- p \\cdot \\beta} \\int \\mathrm{dS} \\, \\left(1 +  \\f{1}{2}  \\omega_{\\alpha \\beta} s^{\\alpha\\beta}\\right). \\nonumber\\\\ \n\\label{eq:Neq-sp21}\n\\end{eqnarray}\nAfter carrying out integration first over spin and then over momentum we get\n\\bel{Nmu}\nN^\\alpha_{\\rm eq} = n u^\\alpha,\n\\end{eqnarray}\nwhere \n\\bel{nden}\nn = 4 \\, \\sinh(\\xi)\\, n_0(T)\n\\label{eq:n_eql}\n\\end{eqnarray}\nis the charge density~\\cite{Florkowski:2017ruc}.\nIn Eq.~(\\ref{nden}) the quantity $n_0(T)$ is the number density of spinless, neutral massive Boltzmann particles which is defined by the thermal average\n\\bel{avdef}\nn_0(T)= \\langle u\\cdot p\\rangle_0 ,\n\\end{eqnarray}\nwhere\n\\bel{avdef1.1}\n\\langle \\cdots \\rangle_0 \\equiv \\int \\mathrm{dP}  (\\cdots) \\,  e^{- \\beta \\cdot p}.\n\\end{eqnarray}\nThe explicit calculation gives\n\\begin{eqnarray}\nn_{0}(T) &=& \\int \\mathrm{dP} \\, (u\\cdot p)\\,  e^{- \\beta \\cdot p} =I_{{10}}^{(0)} \\nonumber \\\\\n&=& \\frac{1}{2 \\pi ^2} T^3 z^2  K_2 (z) \\,,\n\\end{eqnarray}\nwith $z\\equiv m\/T$. Thermodynamic integrals $I_{nq}^{(r)}$ are defined in Appendix \\ref{sec:thermint}.\n\n\\subsection{Energy-momentum tensor}\n\\label{subsubsec:2}\n\nThe energy-momentum tensor is defined as the second moment in momentum space,\n\\begin{eqnarray}\nT^{\\mu \\nu}_{\\rm eq}\n&=& \\int \\mathrm{dP}~\\mathrm{dS} \\, \\, p^\\mu p^\\nu \\, \\left[f^+_{s, \\rm eq}(x,p,s) + f^-_{s, \\rm eq}(x,p,s) \\right]. \\nonumber\\\\\n\\label{eq:Teq-sp02}\n\\end{eqnarray}\nUsing Eq.~(\\ref{eq:feqxps}) we can rewrite this formula as\n\\begin{eqnarray}\nT^{\\mu \\nu}_{\\rm eq}\n&=& 2 \\cosh(\\xi) \\int \\mathrm{dP} \\, p^\\mu p^\\nu \\, e^{- p \\cdot \\beta}\n\\int \\mathrm{dS} \\exp\\left(  \\f{1}{2}  \\omega_{\\alpha \\beta} s^{\\alpha\\beta} \\right). \\nonumber\\\\\n\\label{eq:Teq-sp03}\n\\end{eqnarray}\nConsidering the case of small $\\omega$ and carrying out integration over spin and momentum space we get\n\\bel{Tmn}\nT^{\\alpha\\beta}_{\\rm eq}(x) &=& \\varepsilon u^\\alpha u^\\beta - P \\Delta^{\\alpha\\beta},\n\\end{eqnarray}\nwhere\n\\bel{enden}\n\\varepsilon = 4 \\,\\cosh(\\xi) \\, \\varepsilon_{0}(T)\n\\end{eqnarray}\nand\n\\bel{prs}\nP = 4 \\, \\cosh(\\xi) \\, P_{0}(T),\n\\label{eq:P_eql}\n\\end{eqnarray}\nrespectively~\\cite{Florkowski:2017ruc}. The auxiliary quantities $\\varepsilon_{0}(T)$ and $P_{0}(T)$ are defined as follows\n\\bel{enden0}\n\\varepsilon_{0}(T) &=& \\langle(u\\cdot p)^2\\rangle_0\n\\end{eqnarray}\nand\n\\bel{prs0}\nP_{0}(T) = -(1\/3) \\langle \\,  p\\cdot p - (u\\cdot p)^2\\,   \\rangle_0. \n\\end{eqnarray}\nSimilarly to $n_0(T)$, they describe the energy density and pressure of spinless, neutral massive Boltzmann particles. In Eq.~(\\ref{Tmn}), the tensor $\\Delta^{\\alpha\\beta} = g^{\\alpha\\beta} - u^{\\alpha} u^{\\beta}$ is an operator projecting on the space orthogonal to the fluid four-velocity $u^{\\mu}$. For the reader's convenience, the properties of this and other projectors are listed in Appendix~\\ref{sec:projectors}.\n\nWith the help of thermodynamic integrals $I_{nq}^{(r)}$  defined in Appendix \\ref{sec:thermint} one obtains\n\\begin{eqnarray}\n\\varepsilon_0(T) &=& \\int \\mathrm{dP}\\,(u\\cdot p)^2 e^{- \\beta \\cdot p}=I^{(0)}_{20} \\nonumber \\\\\n&=& \\frac{1}{2 \\pi ^2} T^4 z^2 \\left[3 K_2 (z)+z K_1 (z)\\right]\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nP_0(T) &=& -\\frac{1}{3}\\Delta_{\\mu\\nu} \\int \\mathrm{dP}\\, p^{\\mu}p^{\\nu} e^{- \\beta \\cdot p} \\nonumber \\\\ \n&=& -\\frac{1}{3}\\int \\mathrm{dP}\\,\\left[ p\\cdot p-(u\\cdot p)^2\\right] e^{- \\beta \\cdot p}=-I^{(0)}_{21} \\nonumber \\\\\n&=&  \\frac{1}{2 \\pi ^2}{T^4 z^2  K_2 (z)}= n_0 (T) T.\n\\label{eq:p0n0T}\n\\end{eqnarray}\n\n\\subsection{Spin tensor}\n\\label{subsubsec:3}\n\nNow we come to the fundamental object in our formalism, namely, the spin tensor. We adopt\nthe following definition \\cite{Florkowski:2018fap}\n\\begin{eqnarray}\nS^{\\lambda, \\mu\\nu}_{\\rm eq} &=& \\int  \\mathrm{dP}~\\mathrm{dS} \\, \\, p^\\lambda \\, s^{\\mu \\nu} \n\\left[f^+_{s, \\rm eq}(x,p,s) + f^-_{s, \\rm eq}(x,p,s) \\right] \\nonumber \\\\\n&=& 2 \\cosh(\\xi) \\int  \\mathrm{dP} \\, p^\\lambda \\exp\\left( - p \\cdot \\beta \\right) \\label{eq:Seq-sp01} \\\\\n&& \\hspace{1cm} \\times\n\\int  \\mathrm{dS} \\, s^{\\mu \\nu} \\, \\exp\\left( \\f{1}{2}  \\omega_{\\alpha \\beta} s^{\\alpha\\beta} \\right). \\nonumber\n\\end{eqnarray}\nExpanding the exponential function in the last line, in the leading order in $\\omega$ we obtain\n\\begin{eqnarray}\n&&\\hspace{-0.cm}  \\int  \\mathrm{dS} \\, s^{\\mu \\nu}  \\, \\exp\\left(  \\f{1}{2}  \\omega_{\\alpha \\beta} s^{\\alpha\\beta}\\right) =\n\\int  \\mathrm{dS} \\, s^{\\mu \\nu}  \\, \\left(1 +   \\f{1}{2}  \\omega_{\\alpha \\beta} s^{\\alpha\\beta}\\right)\n\\nonumber \\\\\n&&\\hspace{1.5cm}=\\f{2}{3 m^2} {{\\mathfrak{s}}}^2 \\left( m^2 \\omega^{\\mu\\nu} + 2 p^\\alpha p^{[\\mu} \\omega^{\\nu ]}_{\\,\\,\\alpha} \\right).\n\\label{eq:spinint3}\n\\end{eqnarray}\nUsing \\EQ{eq:spinint3} in \\EQ{eq:Seq-sp01} we find\n\\begin{eqnarray}\nS^{\\lambda, \\mu\\nu}_{\\rm eq} \n&=& \\f{4 {{\\mathfrak{s}}}^2}{3 m^2} \\cosh(\\xi) \\!\\!\\int \\!\\! \\mathrm{dP} \\, p^\\lambda \\, e^{ - p \\cdot \\beta }\n\\left( m^2 \\omega^{\\mu\\nu} \\!+\\! 2 p^\\alpha p^{[\\mu} \\omega^{\\nu ]}_{\\,\\,\\alpha} \\right). \\nonumber\\\\\n\\label{eq:Seq-sp1}\n\\end{eqnarray}\nIt is interesting to observe that the last result agrees with the formula $S^{\\lambda , \\mu \\nu }_{\\rm GLW}$ obtained in the semiclassical expansion of the Wigner functions~\\CITn{Florkowski:2018ahw}. This fact supports our use of the definition \\EQn{eq:Seq-sp01}.\n\n\nAfter carrying out the momentum integration we get\n\\begin{eqnarray}\nS^{\\lambda, \\mu\\nu}_{\\rm eq}=S^{\\lambda , \\mu \\nu }_{\\rm GLW}\n&=&  {\\cal C} \\left( n_{0}(T) u^\\lambda \\omega^{\\mu\\nu}  +  S^{\\lambda , \\mu \\nu }_{\\Delta{\\rm GLW}} \\right).\n\\label{eq:Smunulambda_de_Groot2}\n\\end{eqnarray} \nHere ${\\cal C}= (4\/3)\\mathfrak{s}^2 \\ch({\\xi)}$ and the auxiliary tensor $S^{\\lambda , \\mu \\nu }_{\\Delta{\\rm GLW}}$ is given by the expression\n\\begin{eqnarray}\nS^{\\alpha, \\beta\\gamma}_{\\Delta{\\rm GLW}} \n&=&  {\\cal A}_{0} \\, u^\\alpha u^\\delta u^{[\\beta} \\omega^{\\gamma]}_{~\\delta} \\label{SDeltaGLW} \\\\\n&+&{\\cal B}_{0} \\, \\Big( \nu^{[\\beta} \\Delta^{\\alpha\\delta} \\omega^{\\gamma]}_{~\\delta}\n+ u^\\alpha \\Delta^{\\delta[\\beta} \\omega^{\\gamma]}_{~\\delta}\n+ u^\\delta \\Delta^{\\alpha[\\beta} \\omega^{\\gamma]}_{~\\delta}\\Big),\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \n{\\cal B}_{0} &=&-\\frac{2}{z^2}  \\frac{\\varepsilon_{0}(T)+P_{0}(T)}{T}=-\\frac{2}{z^2} s_{0}(T)\\label{coefB}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\cal A}_{0} &=&\\frac{6}{z^2} s_{0}(T) +2 n_{0} (T) = -3{\\cal B}_{0} +2 n_{0}(T),\\nonumber\\\\\n\\label{coefA}\n\\end{eqnarray}\nwith $s_0$ being the entropy density of spinless, neutral, massive Boltzmann particles satisfying thermodynamic relation $s_0 =\\LR \\varepsilon_0+ P_0\\right) \/ T$. \n\nWe note that since our energy-momentum tensor is symmetric, the spin tensor is separately conserved. The conservation of the spin tensor gives six additional equations which are required to determine the space-time evolution of $\\omega$. We note that this situation may change if non-local effects are included, for a very recent discussion of this point see Refs.~\\cite{Weickgenannt:2020aaf,Speranza:2020ilk}.\n\n\\subsection{Entropy Current}\n\\label{subsubsec:4}\n\nTo construct the entropy current we adopt the Boltzmann definition\n\\begin{eqnarray}\nH^\\mu &=& -\\!\\!\\int \\! \\mathrm{dP}~\\mathrm{dS} \\, p^\\mu\n\\left[ \nf^+_{s, \\rm eq} \\left( \\ln f^+_{s, \\rm eq} -1 \\right) \n\\right. \\nonumber \\\\\n&& \\left. \\hspace{1.5cm}\n+ \nf^-_{s, \\rm eq} \\left( \\ln f^-_{s, \\rm eq} -1 \\right) \\right].\n\\label{eq:H1}\n\\end{eqnarray} \nUsing Eqs.~(\\ref{eq:feqxps}), (\\ref{eq:Neq-sp0}), (\\ref{eq:Teq-sp02}), and  (\\ref{eq:Seq-sp01}), we find\n\\begin{eqnarray}\nH^\\mu = \\beta_\\alpha T^{\\mu \\alpha}_{\\rm eq} - \\f{1}{2} \\omega_{\\alpha\\beta} S^{\\mu, \\alpha \\beta}_{\\rm eq} - \\xi N^\\mu_{\\rm eq} + P \\beta^\\mu.\n\\label{eq:H2}\n\\end{eqnarray} \nUsing Eqs.~\\eqref{eq:n_eql}, \\eqref{eq:P_eql} and \\eqref{eq:p0n0T}, one can obtain the relation, $P \\beta^\\mu = (\\cosh(\\xi)\/\\sinh(\\xi))  N^\\mu_{\\rm eq}$.\nUsing Eq.~(\\ref{eq:H2}) as well as the conservation laws for charge, energy-momentum and spin we obtain the following expression,\n\\begin{eqnarray}\n\\partial_\\mu H^\\mu &=& \\left( \\partial_\\mu \\beta_\\alpha \\right) T^{\\mu \\alpha}_{\\rm eq} \n-\\f{1}{2} \\left( \\partial_\\mu \\omega_{\\alpha\\beta} \\right) S^{\\mu, \\alpha \\beta}_{\\rm eq}\n\\nonumber \\\\\n&& \n- \\left(\\partial_\\mu \\xi \\right)  N^\\mu_{\\rm eq} + \\partial_\\mu (P \\beta^\\mu) .\\nonumber\\\\\n\\label{eq:H3}\n\\end{eqnarray} \nNow starting from the definition (\\ref{eq:Neq-sp0}), applying the conservation laws for charge current, and using Eqs. (\\ref{eq:Neq-sp0}), (\\ref{eq:Teq-sp02}), (\\ref{eq:Seq-sp01}), and the definition of $P\\beta^{\\mu}$ given above, one can easily show that the right-hand side of the last equation is zero (see Appendix \\ref{sec:entropy} for details of the proof), i.e., the entropy current is conserved,\n\\begin{eqnarray} \n\\partial_\\mu H^\\mu = 0.\n\\label{eq:entcon}\n\\end{eqnarray}\nIt should be emphasized that the last result is exact in the sense that it does not depend on the expansion in $\\omega$. Moreover, we see that the contributions to the entropy production coming from the spin polarization tensor are quadratic. This means that there is no effect on the entropy production from the polarization in the linear order. This suggests that we can neglect the effects of polarization on the global evolution of matter, provided we restrict our considerations to the linear terms. For both the conserved charge and the energy-momentum tensor the corrections start with the second order, hence, as long as we restrict ourselves to the linear terms in $\\omega$, we can first solve the system of standard hydrodynamic equations (which are not affected by polarization in the linear order) and subsequently determine the spin evolution (linear in $\\omega)$ on top of such a hydrodynamic background.\n\\section{Formulation of dissipative hydrodynamics for spin polarized fluids}\n\\label{sec:3}\n\nThe formalism presented in the previous section is already well established and may be treated as the definition of the perfect-fluid hydrodynamics with spin. In the next section, we include dissipation effects. This will be done with the help of the relaxation time approximation used for the collision terms in the classical kinetic equations, as originally introduced in Ref.~\\cite{Bhadury:2020puc}.\n\n\\subsection{Classical RTA kinetic equation}\n\\label{sec:2.1}\nIn the absence of mean fields, the distribution function satisfies the equation\n\\begin{eqnarray}\np^\\mu \\partial_\\mu f^\\pm_s(x,p,s) =C[f^\\pm_s(x,p,s)],\n\\label{RTA_spin}\n\\end{eqnarray}\nwhere $C[f^\\pm_s(x,p,s)]$ is the collision term. In the relaxation time approximation, the collision term has the form~\\CITn{Bhadury:2020puc}\n\\begin{equation}\nC[f^\\pm_s(x,p,s)] =  p \\cdot u\n\\, \\frac{f^\\pm_{s, \\rm eq}(x,p,s)-f_s^\\pm(x,p,s)}{\\tau_{\\rm eq}}.\n\\label{eq:col}\n\\end{equation}\nWe consider now a simple Chapman-Enskog expansion of the single particle distribution function about its equilibrium value in powers of space-time gradients \n\\begin{equation}\nf^\\pm_s(x,p,s) =f^\\pm_{s, \\rm eq}(x,p,s)+\\delta f^\\pm_s(x,p,s).\n\\label{cha-ensk}\n\\end{equation}\nIn the above equation $\\delta f^\\pm_s(x,p,s)$ is a deviation from the equilibrium single-particle distribution function and, in principle, can be of any order in space-time gradients. \n\n Using Eqs.~(\\ref{eq:col}) and~(\\ref{cha-ensk}) in Eq.~(\\ref{RTA_spin}), and keeping only the first-order terms in space-time gradients, we get\n\n\\begin{equation}\\label{RTA_spin2}\np^\\mu \\partial_\\mu f^\\pm_{s, \\rm eq}(x,p,s) =-p \\cdot u\n\\, \\frac{\\delta f^\\pm_s(x,p,s)}{\\tau_{\\rm eq}}.\n\\end{equation}\nAfter substituting equilibrium distribution function (\\ref{eq:feqxps})  in Eq.~(\\ref{RTA_spin2}) we obtain (in linear order in $\\omega$)\n\\begin{widetext}\n\\begin{eqnarray}\n\\delta f^\\pm_s &=&- \\frac{\\tau_{\\rm eq}}{(u \\cdot p)}e^{\\pm\\xi - p\\cdot \\beta} \\bigg[ \\Big(\\pm p^\\mu\\partial_\\mu\\xi - p^\\lambda p^\\mu\\partial_\\mu\\beta_\\lambda\\Big) \\bigg(1 + \\frac{1}{2}s^{\\alpha\\beta}\\omega_{\\alpha\\beta}\\bigg)+\\frac{1}{2} p^\\mu s^{\\alpha\\beta}(\\partial_\\mu\\omega_{\\alpha\\beta})\\bigg].\\label{del feq lim}\n\\end{eqnarray}\n\\end{widetext}\nThe corrections $\\delta f^\\pm_s$ result in dissipative effects in the conserved quantities such as charge current, energy-momentum tensor, and spin tensor. We discuss them now starting from the simplest case of the charge current. The details of rather lengthy calculations are given in Appendix~\\ref{sec:appD}.\n\n\\subsection{Conserved hydrodynamic quantities and dissipative corrections}\n\nTaking the appropriate moments of  the transport equation (\\ref{RTA_spin}), the following equations for the charge current ($N^{\\mu}$), energy-momentum tensor ($T^{\\mu\\nu}$) and spin tensor ($S^{\\lambda , \\mu \\nu}$) can be obtained\n\\begin{eqnarray}\n\\partial_\\mu N^{\\mu}(x)&=&-u_{\\mu}\\left(\\frac{N^{\\mu}(x)-N^{\\mu}_{\\rm eq}(x)}{\\tau_{\\rm eq}}\\right), \\label{eq:cc2}\\\\\n\\partial_\\mu T^{\\mu\\nu}(x)&=&-u_{\\mu}\\left(\\frac{T^{\\mu\\nu}(x)-T^{\\mu\\nu}_{\\rm eq}(x)}{\\tau_{\\rm eq}}\\right), \\label{eq:emt2}\\\\\n\\partial_\\lambda S^{\\lambda , \\mu \\nu}(x)&=&-u_{\\lambda}\\left(\\frac{S^{\\lambda , \\mu \\nu}(x)-S^{\\lambda , \\mu \\nu}_{\\rm eq}(x)}{\\tau_{\\rm eq}}\\right),\\label{eq:st2}\n\\end{eqnarray}\nrespectively.\n\nConservation of the charge current ($\\partial_\\mu N^{\\mu}=0$), energy-momentum tensor ($\\partial_\\mu T^{\\mu\\nu}=0$), and spin tensor ($\\partial_\\lambda S^{\\lambda , \\mu \\nu}=0$) implies that the quantities on the right-hand sides of Eqs.~(\\ref{eq:cc2})--(\\ref{eq:st2}) should be zero, i.e., we must have\n\\begin{eqnarray}\nu_{\\mu}\\delta N^{\\mu}&=&0,\\label{eq:lm1}\\\\\n\tu_{\\mu}\\delta T^{\\mu\\nu}&=&0,\\label{eq:lm2}\\\\\n\tu_{\\lambda}\\delta S^{\\lambda , \\mu \\nu}&=& 0, \\label{eq:lm3}\n\\end{eqnarray}\nwhere  $\\delta N^\\mu$, $\\delta T^{\\mu\\nu}$, and $\\delta S^{\\lambda,\\mu\\nu}$ are defined in terms of the non-equilibrium parts of the distribution functions:\n\\begin{align}\n&\\delta N^\\mu = \\int \\mathrm{dP}~\\mathrm{dS}~p^\\mu (\\delta f^+_s-\\delta f^-_s),\\label{eq:dN}\\\\\n&\\delta T^{\\mu\\nu} = \\int \\mathrm{dP}~\\mathrm{dS}~p^\\mu p^\\nu (\\delta f^+_s+\\delta f^-_s)\\label{eq:dT},\\\\\n&\\delta S^{\\lambda,\\mu\\nu}=\\int \\mathrm{dP~dS}~p^{\\lambda} s^{\\mu\\nu} (\\delta f^+_s+\\delta f^-_s).\\label{eq_dspin}\n\\end{align}\nNote that Eqs.~(\\ref{eq:lm1}) and  (\\ref{eq:lm2}),  satisfied by the corrections $\\delta N^\\mu$ and $\\delta T^{\\mu\\nu}$, are known in the literature as the Landau matching conditions. They are used (and needed) to determine the values of the chemical potential, temperature, and three independent components of the flow four-vector appearing in the equilibrium distributions defined by Eq.~(\\ref{eq:feqxps}) --- altogether Eqs.~(\\ref{eq:lm1}) and  (\\ref{eq:lm2}) are five independent equations for five unknown functions. A novel feature of our approach is that we introduce an additional matching condition given by Eq.~(\\ref{eq:lm3}). These are in fact six equations that allow us to determine six independent components of the spin polarization tensor $\\omega_{\\mu\\nu}$. Below we refer to the complete set of  Eqs.~(\\ref{eq:lm1})--(\\ref{eq:lm3}) as to the Landau matching conditions. \n\n\nThe conserved quantities obtained from the moments of the transport equations (\\ref{RTA_spin}) can be further tensor decomposed in terms of the hydrodynamic degrees of freedom. The charge current is decomposed into two parts\n\\begin{eqnarray}\nN^\\mu &=&\\int  \\mathrm{dP}~ \\mathrm{dS}~p^{\\mu} \\left[f^+_s(x, p, s) - f^-_s(x, p, s)\\right]\\nonumber\\\\&=&N^\\mu_{\\rm eq}+\\delta N^{\\mu}=nu^{\\mu}+\\nu^\\mu\\label{Nmu1}.\n\\end{eqnarray}\nIn this decomposition, the quantity $\\nu^\\mu$ is known as the charge diffusion current. The presence of the dissipative corrections implies that the form of the energy-momentum tensor is\n\\begin{eqnarray}\nT^{\\mu \\nu}&=&\\int  \\mathrm{dP}~ \\mathrm{dS}~p^{\\mu} \np^{\\nu} \\left[f^+_s(x, p, s) + f^-_s(x, p, s)\\right]\\nonumber\\\\\n&=& T^{\\mu \\nu}_{\\rm eq}+\\delta T^{\\mu \\nu} \\nonumber \\\\\n&=& \\varepsilon u^\\mu u^\\nu - P \\Delta^{\\mu\\nu}+\\pi^{\\mu\\nu}-\\Pi \\Delta^{\\mu\\nu}. \\label{Tmunu1}\n\\end{eqnarray}\nIn this decomposition, $\\varepsilon, P, \\pi^{\\mu\\nu}$, and $\\Pi$ are energy density, equilibrium pressure, shear stress tensor, and bulk pressure,  respectively. We use here the Landau frame, where $T^{\\mu\\nu}u_{\\nu}=\\varepsilon u^\\mu$. Finally, we define the correction to the spin tensor by the decomposition\n\\begin{eqnarray}\nS^{\\lambda,\\mu\\nu} &=&\\int \\mathrm{dP~dS}~p^{\\lambda} s^{\\mu\\nu}\\left[f^+_s(x, p, s) + f^-_s(x, p, s)\\right]\\nonumber\\\\&=&S^{\\lambda,\\mu\\nu}_{\\rm eq}+\\delta S^{\\lambda,\\mu\\nu}.\\label{Slmunu}\n\\end{eqnarray}\n\nThe non-equilibrium quantities $n$, $\\varepsilon$, $P$ can be obtained by the Landau matching conditions, namely\n\\begin{eqnarray}\nn &=& n_{\\rm {eq}} = u_{\\mu } N_{\\rm {eq}}^{\\mu } \\label{eq:no_den} \\\\\n&=&u_{\\mu } \\int  \\mathrm{dP}~\\mathrm{dS} \\, p^{\\mu }\\left[f^{+}_{s, \\rm {eq}}(x,p,s)-f^{-}_{s, \\rm {eq}}(x,p,s)\\right],  \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\varepsilon &=& \\varepsilon_{\\rm {eq}} = u_{\\mu }u_{\\nu} T^{\\mu\\nu}_{\\rm {eq}}  \\label{eq:end} \\\\\n&=&u_{\\mu} u_{\\nu} \\int  \\mathrm{dP}~\\mathrm{dS} \\, \np^{\\mu }p^{\\nu}\\left[f^{+}_{s, \\rm {eq}}(x,p,s) + f^{-}_{s, \\rm {eq}}(x,p,s)\\right] \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nP &=& P_{\\rm {eq}} = -\\frac{1}{3}\\Delta_{\\mu\\nu} \nT^{\\mu\\nu}_{\\rm {eq}} \\label{eq:P} \\\\\n&=&-\\frac{\\Delta_{\\mu\\nu}}{3}\n\\int  \\mathrm{dP}~\\mathrm{dS} \\, p^{\\mu}p^{\\nu}\\left[f^{+}_{s, \\rm {eq}}(x,p,s) + f^{-}_{s, \\rm {eq}}(x,p,s)\\right].\\nonumber\n\\end{eqnarray}\nAfter carrying out integration over spin and momentum, Eqs.~(\\ref{eq:no_den}), (\\ref{eq:end}), and (\\ref{eq:P}) yield the same results as Eqs.~(\\ref{nden}), (\\ref{enden}), and (\\ref{prs}). Here we also note that the choice of Landau frame and matching conditions enforces the following constraints on the dissipative currents\n\\begin{eqnarray}\nu_\\mu \\nu^\\mu&=&0, \\nonumber\\\\\nu_\\mu \\pi^{\\mu\\nu}&=&0. \\nonumber\\\\\nu_\\lambda \\delta S^{\\lambda,\\mu\\nu}&=&0. \\label{matching_conditions}\n\\end{eqnarray}\n\\subsection{Convective derivatives of hydrodynamic variables}\n\\label{Sbc}\n\nAn intermediate step in the calculation of standard kinetic coefficients is the derivation of expressions for the convective derivatives of the hydrodynamic variables $\\xi$, $\\beta$, and $u^\\mu$. The convective derivatives are space-time derivatives taken along the streamlines of the fluid. We denote them by a dot or the letter $D$, for example, \n\\begin{eqnarray}\n{\\dot \\xi} = D \\xi \n= u^\\mu \\partial_\\mu \\xi.\n\\end{eqnarray}\nWith spin degrees of freedom included, one has to calculate the convective derivative of the spin polarization tensor $\\omega_{\\mu\\nu}$ as well. In this section we describe the necessary steps needed to determine all those derivatives. The details of the calculations, which are quite lengthy due to complicated tensor structures, are given in the Appendices \\ref{Ac}-\\ref{sec:LanCon}.\n\nUsing the conservation laws for energy and momentum ($\\partial_\\mu T^{\\mu \\nu}= 0$) as well as charge  ($\\partial_\\mu N^{\\mu \\nu}= 0$), we get the following equations that dictate the evolution of $T$, $u^\\mu$, and $\\mu$,~\\footnote{Equations~ (\\ref{eq:con_e})--(\\ref{eq:con_n}) do not include the spin polarization tensor, if we consider only linear terms in $\\omega$.}\n\\begin{align}\n&\\dot{\\varepsilon} + (\\varepsilon + P + \\Pi)~\\theta -\\pi^{\\mu\\nu}\\sigma_{\\mu\\nu} = 0, \\label{eq:con_e}\\\\\n&(\\varepsilon + P) \\dot{u}^\\alpha - \\nabla^\\alpha P + \\Delta^\\alpha_\\mu \\partial_\\nu \\pi^{\\mu\\nu} = 0, \\label{eq:con_P}\\\\\n&\\dot{n} + n\\theta + \\partial_\\mu n^\\mu = 0. \\label{eq:con_n}\n\\end{align}\nHere we use the following notation:  $\\theta=\\partial_{\\mu}u^{\\mu}$ is the expansion scalar, $\\nabla^\\mu = \\Delta^{\\mu\\nu} \\partial_\\nu$ denotes the transverse gradient, and $\\sigma ^{\\mu\\nu}=\\frac{1}{2}\\left(\\nabla ^{\\mu }u^{\\nu }+\\nabla ^{\\nu }u^{\\mu}\\right)-\\frac{1}{3}\\Delta ^{\\mu\\nu}\\left.(\\nabla ^{\\lambda }u_{\\lambda }\\right)$ is the shear flow tensor. In order to determine the space-time evolution of the spin polarization tensor, the above system of equations should be supplemented by the conservation of the spin tensor, \n\\begin{align}\n\\partial_\\lambda S^{\\lambda,\\mu\\nu}  = 0. \\label{eq:con_S}\n\\end{align}\n\nKeeping only the terms up to the first order in velocity gradients,  the conservation equations (\\ref{eq:con_e}), (\\ref{eq:con_P}), (\\ref{eq:con_n}), and (\\ref{eq:con_S}) are reduced to\n\\begin{align}\n&\\dot{\\varepsilon} + (\\varepsilon + P)~\\theta= 0,\\label{eq:con_e1}\\\\\n&(\\varepsilon + P) \\dot{u}^\\alpha - \\nabla^\\alpha P = 0,\\label{eq:con_P1}\\\\\n&\\dot{n} + n\\theta = 0,\\label{eq:con_n1}\\\\\n&\\partial_\\lambda S^{\\lambda,\\mu\\nu}_{\\rm {eq}}  = 0, \\label{eq:con_S1}\n\\end{align}\nrespectively.\nFurthermore, from Eqs.~(\\ref{nden}) and (\\ref{enden}) we obtain\n\\begin{eqnarray}\n\\dot{n}&=& 4 \\cosh (\\xi )\\dot{\\xi } I_{10}^{(0)} +4 \\sinh (\\xi )\\dot{I}_{10}^{(0)}, \\label{eq:dotn}\\\\\n\\dot{\\varepsilon }&=&4 \\sinh (\\xi)\\dot{\\xi} I_{20}^{(0)}+4\\cosh (\\xi)\\dot{I}_{20}^{(0)}.\\label{eq:enddot}\n\\end{eqnarray}\nUsing Eq.~(\\ref{eq:ir3}) that connects derivatives of the thermodynamic integrals, the above equations can be written as\n\\begin{eqnarray}\n\\dot{n}&=& 4 \\cosh (\\xi )\\dot{\\xi } I_{10}^{(0)} -4 \\sinh (\\xi) \\dot{\\beta } I_{20}^{(0)} , \\label{eq:dotn1.1}\\\\\n\\dot{\\varepsilon }&=&4 \\sinh (\\xi)\\dot{\\xi} I_{20}^{(0)}-4\\cosh (\\xi) \\dot{\\beta }I_{30}^{(0)}. \\label{eq:enddot1.1}\n\\end{eqnarray}\nSubstituting $n$, $\\varepsilon$, $P$, $\\dot{n}$ and $\\dot{\\varepsilon }$ from Eqs. (\\ref{nden}), (\\ref{enden}), (\\ref{prs}), (\\ref{eq:dotn1.1}) and (\\ref{eq:enddot1.1})  in Eqs. (\\ref{eq:con_e1}) and (\\ref{eq:con_n1}) we get\n\\begin{eqnarray}\n  \\sinh (\\xi) \\dot{\\xi} I_{20}^{(0)}\\!-\\!  \\cosh (\\xi) \\dot{\\beta }  I_{30}^{(0)} &=&\\!- \\cosh (\\xi)\\left(I_{20}^{(0)}\\!-\\!I_{21}^{(0)}\\right)\\!\\theta, \\nonumber\\\\\\label{eq:enddot1}\\\\\n  \\cosh (\\xi ) \\dot{\\xi } I_{10}^{(0)}\\!-\\!  \\sinh (\\xi) \\dot{\\beta}I_{20}^{(0)}&=&-  \\sinh (\\xi)I_{10}^{(0)} \\theta. \\label{eq:dotn1}\n\\end{eqnarray}\nUsing the relations: $I_{20}^{(0)}=\\varepsilon_{0}$, $I_{21}^{(0)}=-P_{0}=-n_0 T$, $ I_{10}^{(0)}=n_{0}$, and $I_{30}^{(0)}=\\frac{1}{\\beta }{\\left(3 \\left(P_0+\\varepsilon _0\\right)+z^2 P_0\\right)}$, and solving  Eqs.~(\\ref{eq:enddot1}) and (\\ref{eq:dotn1}) for $\\dot{\\xi}$ and $\\dot{\\beta}$ we can get\n\\begin{eqnarray}\n\t\\dot{\\xi} &=&\\xi_\\theta \\, \\theta, \n\t\\label{eq:xidot1} \\\\\n\t\t\\dot{\\beta}&=& \\beta_\\theta \\, \\theta, \\label{eq:betadot1}\n\t\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\xi_\\theta &=&\\frac{\\sinh (\\xi ) \\cosh (\\xi ) \\left[\\varepsilon _0^2-n_0 T \\left(\\left(3+z^2\\right)P_0+2 \\varepsilon _0\\right)\\right]}{n_0 T \\cosh ^2(\\xi ) \\left( \\left(3+z^2\\right)P_0+3 \\varepsilon _0\\right)-\\varepsilon _0^2 \\sinh ^2(\\xi )}, \\nonumber\\\\ \\label{eq:xith} \\\\\n\\beta_\\theta&=& \\frac{n_0 \\left(\\cosh ^2(\\xi )P_0 +\\varepsilon _0\\right)}{n_0 T \\cosh ^2(\\xi ) \\left(\\left(3+z^2\\right)P_0+3 \\varepsilon _0\\right)-\\varepsilon _0^2 \\sinh ^2(\\xi )}.\\nonumber\\\\ \\label{eq:Dmunu}\n\\end{eqnarray}\nSubstituting into Eq.~(\\ref{eq:con_P1}) the energy density $\\varepsilon$ and  pressure $P$ defined by Eqs.~(\\ref{eq:end}) and (\\ref{eq:P})  we get\n\\begin{eqnarray}\n  \\cosh(\\xi)\\left(I_{20}^{(0)}-I_{21}^{(0)}\\right) \\dot{u}^{\\alpha }&=&- \\sinh (\\xi )\\left(\\nabla ^{\\alpha }\\xi \\right)I_{21}^{(0)}\\nonumber\\\\&&\\hspace{-0.5cm}- \\cosh (\\xi)\\left(\\nabla ^{\\alpha }I_{21}^{(0)}\\right). \\label{eq:dotu1.1}\n\\end{eqnarray}\nNow we can write\n\\begin{eqnarray}\n\\nabla ^{\\alpha }I_{21}^{(0)}&=&\\nabla ^{\\alpha }\\left(\\frac{1}{3}\\Delta _{\\mu \\nu }\\int {\\rm dP} \\,p^{\\mu } p^{\\nu } e^{-p^{\\lambda }\\beta _{\\lambda }}\\right)\\nonumber\\\\\n&=&\\frac{1}{3}\\Delta _{\\mu \\nu }\\left(-\\nabla ^{\\alpha }\\beta_{\\lambda}\\right)\\int {\\rm dP}\\, p^{\\mu } p^{\\nu } p^{\\lambda} e^{-p^{\\lambda }\\beta _{\\lambda }}\\nonumber\\\\\n&=&-\\frac{1}{3}\\Delta _{\\mu \\nu }\\left(\\frac{1}{T}\\nabla ^{\\alpha }u_{\\lambda }-\\frac{u_{\\lambda }}{T^2}\\nabla ^{\\alpha }T\\right) \\bigg[I_{30}^{(0)} u^{\\lambda } u^{\\mu} u^{\\nu}\\nonumber\\\\&+&I_{31}^{(0)} \\left(\\Delta ^{\\lambda \\mu } u^{\\nu }+\\Delta ^{\\nu \\lambda } u^{\\mu }+\\Delta ^{\\mu \\nu } u^{\\lambda }\\right)\\bigg]\\nonumber\\\\\n&=&-\\left(-\\frac{u_{\\lambda }}{T^2}\\nabla ^{\\alpha }T\\right) u^{\\lambda }I_{31}^{(0)}\\nonumber\\\\\n&=&\\left(-\\nabla^{\\alpha}\\beta\\right)I_{31}^{(0)}\\label{eq:I21r}\n\\end{eqnarray}\nand using Eq.~(\\ref{eq:I21r}) in Eq.~(\\ref{eq:dotu1.1}) we obtain\n\\begin{eqnarray}\n  \\cosh(\\xi)\\left(I_{20}^{(0)}-I_{21}^{(0)}\\right) \\dot{u}^{\\alpha }&=&-\\, \\sinh (\\xi )\\left(\\nabla ^{\\alpha }\\xi \\right)I_{21}^{(0)}\\nonumber\\\\&&\\hspace{-0.5cm}+\\, \\cosh (\\xi)\\left(\\nabla^{\\alpha}\\beta\\right)I_{31}^{(0)}.\n \\label{eq:dotu1.2}\n\\end{eqnarray}\nNow from the recurrence relation (\\ref{eq:ir2}) we obtain\n\\begin{eqnarray}\nI_{31}^{(0)}&=&-\\frac{1}{\\beta }\\left(I_{20}^{(0)}-I_{21}^{(0)}\\right)=-\\frac{1}{\\beta }{(\\varepsilon _0 + P_0)}, \\nonumber\\\\ I_{21}^{(0)}&=&-P_0=-\\frac{n_0}{\\beta}.  \n\\end{eqnarray}\nUsing the above expressions for $I_{31}^{(0)}$ and $I_{21}^{(0)}$ in Eq.~(\\ref{eq:dotu1.2}), the following equation for $\\dot{u}^{\\mu}$ can be derived\n\\begin{eqnarray}\n\\beta  \\dot{u}^{\\alpha }&=&\\frac{n_0 \\tanh (\\xi )}{ \\varepsilon _0+P_0 }\\left(\\nabla ^{\\alpha }\\xi \\right)-\\left(\\nabla ^{\\alpha }\\beta \\right). \\label{eq:udot1}\n\\end{eqnarray}\n\n\nNow we turn to the equilibrium spin tensor. With the help of  Eq.~(\\ref{eq:Smunulambda_de_Groot2}) it can be written as\n\\begin{eqnarray}\n&& S^{\\lambda, \\mu\\nu}_{\\rm eq} =\\frac{4 \\mathfrak{s}^2}{3}\\cosh (\\xi )I_{10}^{(0)}u^{\\lambda }\\omega ^{\\mu \\nu }  \\\\\n&& \n+\\frac{4 \\mathfrak{s}^2}{3 m^2}\\cosh (\\xi )\\bigg[2I_{30}^{(0)}u^{\\lambda }u^{\\alpha }u^{[\\mu }\\omega ^{\\nu ]}{}_{\\alpha }\\nonumber\\\\\n&& \n+ 2I_{31}^{(0)}\\left(\\Delta ^{\\lambda \\alpha }u^{[\\mu }\\omega ^{\\nu ]}{}_{\\alpha }+u^{\\lambda }\\Delta ^{\\alpha [\\mu }\\omega ^{\\nu ]}{}_{\\alpha }+u^{\\alpha }\\Delta ^{\\lambda [\\mu }\\omega ^{\\nu ]}{}_{\\alpha }\\right)\\bigg]. \\nonumber\n\\end{eqnarray} \nThe above equation can further be simplified as\n\\begin{eqnarray}\nS^{\\lambda, \\mu\\nu}_{\\rm eq}&=&\\frac{4 \\mathfrak{s}^2}{3}\\cosh (\\xi )I_{10}^{(0)}u^{\\lambda }\\omega ^{\\mu \\nu }\n\\label{eq:Smunulambda_de_Groot2.1}\\\\\n&+&\\frac{8 \\mathfrak{s}^2}{3 m^2}\\cosh (\\xi )\\bigg[\\left(I_{30}^{(0)}-3 I_{31}^{(0)}\\right)u^{\\lambda }u^{\\alpha }u^{[\\mu }\\omega ^{\\nu ]}{}_{\\alpha }\\nonumber\\\\&+&I_{31}^{(0)}\\left(u^{[\\mu }\\omega ^{\\nu ]\\lambda }-\\omega ^{\\mu \\nu } u^{\\lambda }+u^{\\alpha }g^{\\lambda [\\mu }\\omega ^{\\nu ]}{}_{\\alpha }\\right)\\bigg].\n\\nonumber\n\\end{eqnarray} \nSubstituting Eq.~(\\ref{eq:Smunulambda_de_Groot2.1}) into Eq.~(\\ref{eq:con_S1}), and using Eqs.~(\\ref{eq:xidot1}), (\\ref{eq:betadot1}), and (\\ref{eq:udot1}), the following dynamical equation for the spin polarization tensor $\\omega^{\\mu\\nu}$ can be obtained \n\\begin{eqnarray}\n\\dot{\\omega }^{\\mu \\nu}&=& D_{\\Pi }^{\\mu \\nu }\\theta +\\left(\\nabla ^{\\alpha }\\xi \\right)D_n^{[\\mu \\nu ]}{}_{\\alpha }+D_{\\pi }^{[\\nu }{}_{\\lambda }\\sigma ^{\\lambda\\mu ]}\\nonumber\\\\&&+D_{\\text{$\\Sigma $1}}^{\\alpha }\\nabla ^{[\\mu }\\omega ^{\\nu ]}{}_{\\alpha }+D_{\\text{$\\Sigma $2}}^{[\\mu \\nu ]\\alpha }\\nabla ^{\\lambda }\\omega _{\\alpha \\lambda }. \\label{eq:dotomega}\n\\end{eqnarray}\nFor details see Appendix \\ref{Ac}, where the explicit expressions for various $D$-coefficients are given.\n\nNote that while deriving the dynamical equation (\\ref{eq:dotomega}), we initially encounter the term $u_{\\nu}\\dot{\\omega }^{\\mu \\nu }$ in the expression for $\\dot{\\omega }^{\\mu \\nu }$. To eliminate this term we derive another dynamical equation for $u_{\\nu}\\dot{\\omega }^{\\mu \\nu }$ by taking projection of Eq.~(\\ref{eq:con_S1}) along $u_\\nu$. The dynamical equation for $u_{\\nu}\\dot{\\omega }^{\\mu \\nu }$ is given by the expression\n\\begin{eqnarray}\nu_{\\nu }\\dot{\\omega }^{\\mu \\nu }&=&C_{\\Pi }^{\\mu }\\theta +C_{n{}{\\lambda }}^{\\mu }{}(\\nabla ^{\\lambda }\\xi )+C_{\\pi{}{\\alpha }}\\sigma ^{\\alpha \\mu }+C_{\\text{$\\Sigma$} {}{\\nu }}^{\\mu } \\nabla _{\\lambda }\\omega ^{\\nu \\lambda }. \\nonumber\\\\\n\\end{eqnarray}\nThe explicit expression for various $C$-coefficients appearing above are also given in Appendix \\ref{Ac}. See also Appendix \\ref{sec:LanCon}, where the Landau matching conditions are presented in more detail.\n\n\n\\subsection{Transport coefficients}\n\\label{sec:trcoeff}\n\nThe dissipative forces arise due to non-zero gradients in the system. In the present case, we will confine ourselves only to first order in gradients and hence the dissipative parts of $T^{\\mu\\nu}$, $N^\\mu$, and $S^{\\lambda,\\mu\\nu}$, i.e., $\\delta T^{\\mu\\nu}$, $\\delta N^\\mu$, and $\\delta S^{\\lambda,\\mu\\nu}$, respectively, must be first order in gradients too. The shear stress $(\\pi^{\\mu\\nu})$, bulk viscous pressure ($\\Pi$) and particle diffusion current $(n^\\mu)$ can be found from $\\delta T^{\\mu\\nu}$ and $\\delta N^\\mu$ as:\n\\begin{align}\n\\pi^{\\mu\\nu} = \\Delta^{\\mu\\nu}_{\\alpha\\beta}\\,\\delta T^{\\alpha\\beta}, \\quad \\Pi = -\\frac{1}{3}\\Delta_{\\alpha\\beta}\\,\\delta T^{\\alpha\\beta}, \\quad \\nu^\\mu = \\Delta^\\mu_\\alpha~\\delta N^\\alpha.\n\\end{align}\nHence, using Eqs.~(\\ref{eq:dT}) and (\\ref{eq:dN}), the above  dissipative quantities can be written as:\n\\begin{align}\n&\\pi^{\\mu\\nu} = \\Delta^{\\mu\\nu}_{\\alpha\\beta} \\int \\mathrm{dP}~\\mathrm{dS}~p^\\alpha p^\\beta (\\delta f^+_s+\\delta f^-_s),\\label{eq:shear}\\\\\n&\\Pi = - \\frac{1}{3}\\Delta_{\\alpha\\beta} \\int \\mathrm{dP}~\\mathrm{dS}~p^\\alpha p^\\beta (\\delta f^+_s+\\delta f^-_s),\\label{eq:bulk}\\\\\n&\\nu^\\mu = \\Delta^\\mu_\\alpha \\int \\mathrm{dP}~\\mathrm{dS}~p^\\alpha (\\delta f^+_s-\\delta f^-_s).\\label{eq:nu}\n\\end{align}\nEvaluating the expressions defined by Eqs.~(\\ref{eq:shear}), (\\ref{eq:bulk}), and (\\ref{eq:nu}),  the dissipative quantities are found to be (see Appendix \\ref{Ag})\n\\begin{align}\n\\pi^{\\mu\\nu} &= 2 \\tau_{\\rm eq}\\,\\beta_\\pi \\sigma^{\\mu\\nu}, \\nonumber \\\\ \\Pi &= - \\tau_{\\rm eq}\\,\\beta_\\Pi \\theta, \\nonumber \\\\\n\\nu^\\mu &= \\tau_{\\rm eq}~\\beta_n \\nabla^{\\mu} \\xi.\n\\end{align}\nHere, coefficients, $\\beta_\\pi$, $\\beta_\\Pi$ and $\\beta_n$ are the first-order transport coefficients which for massive particles with finite chemical potential are found to be\n\\begin{widetext}\n\\begin{align}\n&\\beta_\\pi = 4~I^{(1)}_{42} \\cosh({\\xi}),\\\\\n&\\beta _{\\Pi }=4\\Bigg\\{\\frac{n_0 \\cosh (\\xi )}{\\beta} \\left[ \\frac{ {\\sinh}^2(\\xi ) \\left(\\varepsilon _0 \\left(P_0+\\varepsilon _0\\right)-n_0 T \\left(P_0 \\left(z^2+3\\right)+3 \\varepsilon _0\\right)\\right)}{ \\varepsilon _0^2 \\sinh ^2(\\xi )-n_0 T \\cosh ^2(\\xi ) \\left(P_0 \\left(z^2+3\\right)+3 \\varepsilon _0\\right) }\\right]\\nonumber\\\\\n&~~~-\\frac{n_0 \\cosh (\\xi )}{\\beta } \\left[\\frac{ \\left(P_0+\\varepsilon _0\\right) \\left(P_0 \\cosh ^2(\\xi )+\\varepsilon _0\\right)}{n_0 T \\cosh ^2(\\xi ) \\left(P_0 \\left(z^2+3\\right)+3 \\varepsilon _0\\right)-\\varepsilon _0^2 \\sinh ^2(\\xi )}\\right]+\\frac{5\\beta }{3}I^{(1)}_{42}\\Bigg\\},\\\\\n&\\beta_n = 4 \\bigg[\\bigg(\\frac{n_0 \\tanh(\\xi)}{\\varepsilon_0 + P_0}\\bigg)~I^{(0)}_{21}~\\sinh({\\xi}) - I^{(1)}_{21}~\\cosh({\\xi})\\bigg].\n\\end{align}\nSimilarly, using Eq.~(\\ref{del feq lim}) in (\\ref{eq_dspin}) and then carrying out integration over spin and momentum variables we get, \t\n\\begin{eqnarray}\n\\delta S^{\\lambda,\\mu\\nu} &=& \\tau_{\\rm eq} \\Big[ B^{\\lambda,\\mu\\nu}_{\\Pi}\\, \\theta + B^{\\kappa\\lambda,\\mu\\nu}_{n}\\, (\\nabla_\\kappa \\xi) + B_{\\pi }^{(\\kappa\\delta) \\lambda, \\mu \\nu }\\sigma _{\\kappa \\delta } + B_{\\Sigma }^{\\eta \\beta \\gamma\\lambda ,\\mu \\nu }\\nabla _{\\eta }\\omega _{\\beta \\gamma }\\Big]. \n\\label{deltaS}\n\\end{eqnarray}\n Different coefficients appearing on the right-hand side of Eq.~(\\ref{deltaS}) are the kinetic coefficients for spin. They have tensor structures expressed in terms of $u^\\mu,~g^{\\mu\\nu}$, and $\\omega^{\\mu\\nu}$.  Explicit forms of these coefficients are as follows:\n\\begin{eqnarray}\nB^{\\lambda,\\mu\\nu}_{\\Pi} &=&\nB_{\\Pi }^{(1)}u^{[\\mu }\\omega ^{\\nu ]\\lambda } + B_{\\Pi }^{(2)}u^{\\lambda }u^{\\alpha }u^{[\\mu }\\omega ^{\\nu ]}{}_{\\alpha } +\nB_{\\Pi }^{(3)}\\Delta ^{\\lambda [\\mu }u_{\\alpha }\\omega ^{\\nu ]\\alpha },\n\\label{eq:betaPi}\\\\\nB_{\\pi}^{\\lambda \\kappa \\delta ,\\mu \\nu }&=&B_{\\pi }^{(1)}\\Delta^{[\\mu \\kappa }\\Delta^{\\lambda \\delta }u_{\\alpha }\\omega ^{\\nu ]\\alpha }+B_{\\pi }^{(2)}\\Delta^{\\lambda \\delta }u^{[\\mu }\\omega ^{\\nu ]\\kappa }+B_{\\pi }^{(3)}u^{[\\mu }\\Delta^{\\nu ]\\delta }\\Delta^{\\lambda}_{\\alpha}\\omega ^{\\alpha \\kappa } + B_{\\pi }^{(4)}\\Delta ^{\\lambda [\\mu }\\omega ^{\\rho \\kappa } u_{\\rho }\\Delta^{\\nu ]\\delta },\n\\label{eq:betapi}\\\\\nB_n^{\\lambda \\kappa ,}{}^{\\mu \\nu }&=&B_n^{(1)}\\Delta ^{\\lambda \\kappa } \\omega ^{\\mu \\nu }+B_n^{(2)}\\Delta ^{\\lambda \\kappa }u^{\\alpha }u^{[\\mu }\\omega ^{\\nu ]}{}_{\\alpha }+B_n^{(3)}\\Delta ^{\\lambda \\alpha }\\Delta ^{[\\mu \\kappa }\\omega ^{\\nu ]}{}_{\\alpha }+B_n^{(4)}u^{[\\mu }\\Delta ^{\\nu ]\\kappa }u^{\\rho }\\omega ^{\\lambda }{}_{\\rho }+B_n^{(5)}\\Delta ^{\\lambda [\\mu }\\omega ^{\\nu ]\\kappa }\\nonumber\\\\&&+B_n^{(6)}\\Delta ^{\\lambda [\\mu }u^{\\nu ]}u_{\\alpha }\\omega ^{\\alpha \\kappa},\n\\label{eq:betan}\\\\\nB_{\\Sigma }^{\\eta \\beta \\gamma \\lambda ,\\mu \\nu }&=&B_{\\Sigma}^{(1)}\\Delta^{\\lambda \\eta }g^{[\\mu \\beta }g^{\\nu ]\\gamma }+B_{\\Sigma }^{(2)} u^{\\gamma }\\Delta ^{\\lambda \\eta }u^{[\\mu }\\Delta ^{\\nu ]\\beta } + B_{\\Sigma}^{(3)}\\left(\\Delta ^{\\lambda \\eta }\\Delta ^{\\gamma [\\mu }g^{\\nu ]\\beta }+\\Delta ^{\\lambda \\gamma }\\Delta ^{[\\mu \\eta }g^{\\nu ]\\beta }+\\Delta ^{\\gamma \\eta }\\Delta ^{\\lambda [\\mu }g^{\\nu ]\\beta }\\right)\\nonumber\\\\&&+B_{\\Sigma}^{(4)}\\Delta ^{\\gamma \\eta }\\Delta ^{\\lambda [\\mu }\\Delta^{\\nu ]\\beta}+B_{\\Sigma}^{(5)}u^{\\gamma }\\Delta ^{\\lambda \\beta }u^{[\\mu }\\Delta ^{\\nu ]\\eta }\\label{eq:betaSig},\n\\end{eqnarray}\nwhere the scalar coefficients $B_{X}^{(i)}$ are explicitly defined in Appendix~\\ref{Ac}. \n\n\\end{widetext}\n\nEquation~(\\ref{deltaS}) is our main result. It shows that the dissipative spin effects are connected with the presence of expansion scalar, gradient of the ratio of chemical potential and temperature, the shear-flow tensor, and the gradient of the spin polarization tensor. All these quantities may be interpreted as ``thermodynamic forces'' that trigger dissipative currents. The first three among them are well known --- they lead to appearance of bulk pressure, diffusive current, and shear stress tensor. Interestingly, in the considered case, they also induce the dissipative part of the spin tensor. The fourth term in Eq.~(\\ref{deltaS}) describes the induction of the disspative spin tensor by the gradient of the spin polarization tensor, hence, may be treated as a direct non-equilibrium interaction between spin degrees of freedom. \n\nFinally, we note that all the kinetic coefficients obtained from Eq.~(\\ref{eq:col}) are proportional to the same relaxation time $\\tau_{\\rm eq}$. This implies that the equilibration times for momenta and spin degrees of freedom are the same. In phenomenological applications it is conceivable to vary the values of the relaxation times that appear in different kinetic coefficients, arguing that they describe independent physical phenomena. However, such modifications require further studies.\n\n\n\n\n\n\\section{Summary and Conclusions}\n\nIn this paper we have significantly extended the results obtained in Ref.~\\cite{Bhadury:2020puc}. We used classical kinetic theory for particles with spin ${\\nicefrac{1}{2}}$ with Boltzmann statistics to obtain the structure of dissipative terms and the associated transport coefficients. We considered the relaxation time approximation for collision term in order to account for the interactions. This kinetic-theory framework was used to determine the structure of spin-dependent viscous and diffusive terms and explicitly evaluate a set of new kinetic coefficients that characterize dissipative spin dynamics.\n\nOur main result is given by Eq.~(\\ref{deltaS}), together with the explicit expressions for the kinetic coefficients $B$ given in the appendices. Equation~(\\ref{deltaS}) shows that a non-equilibrium part of the spin tensor is produced by the thermodynamic forces such as expansion scalar, gradient of the ratio of chemical potential and temperature, the shear-flow tensor, and the gradient of the spin polarization tensor. Thus, the spin dissipative phenomena are connected with those leading to formation of bulk pressure, diffusion current, and the shear stress tensor. Probably, the most interesting term in Eq.~(\\ref{deltaS}) is the last one, which describes induction of a non-equilibrium spin tensor by a gradient of the spin polarization tensor. In the future investigations, it would be interesting to analyze the role played by various coeficients appearing in Eq.~(\\ref{deltaS}) and to find out which kind of corrections they imply for the spin tensor. The complicated tensor structure of the spin kinetic coefficients may lead to various interesting phenomena. \n\n\\begin{acknowledgments}\nW.F. and R.R. acknowledge the hospitality of National Institute of Science Education and Research where most of this work was done. S.B., A.J. and  A.K. would like to acknowledge the kind hospitality of Jagiellonian University and Institute of Nuclear Physics, Krakow, where part of this work was completed. A.J. was supported in part by the DST-INSPIRE faculty award under Grant No. DST\/INSPIRE\/04\/2017\/000038. A. K. was \nsupported in part by the Department of Science and Technology, Government\nof INDIA under the SERB National Post-Doctoral Fellowship Reference No.\nPDF\/2020\/000648.\nW.F. and R.R. were supported in part by the Polish National Science Centre Grants  No. 2016\/23\/B\/ST2\/00717 and No. 2018\/30\/E\/ST2\/00432.\n\\end{acknowledgments}\n\n\\onecolumngrid\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}