Dissipative Spin Dynamics in Relativistic Matter
Samapan Bhadury, Wojciech Florkowski, Amaresh Jaiswal, Avdhesh Kumar, Radoslaw Ryblewski
DDissipative Spin Dynamics in Relativistic Matter
Samapan Bhadury, ∗ Wojciech Florkowski, † Amaresh Jaiswal, ‡ Avdhesh Kumar, § and Radoslaw Ryblewski ¶ School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India Institute of Theoretical Physics, Jagiellonian University ul. St.(cid:32)Lojasiewicza 11, 30-348 Krakow, Poland Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland (Dated: August 26, 2020)Using classical description of spin degrees of freedom, we extend recent formulation of the perfect-fluid hydrodynamics for spin-polarized fluids to the case including dissipation. Our work is basedon the analysis of classical kinetic equations for massive particles with spin / , with the collisionterms treated in the relaxation time approximation. The kinetic-theory framework determines thestructure of viscous and diffusive terms and allows to explicitly calculate a complete set of newkinetic coefficients that characterize dissipative spin dynamics. Keywords: perfect and viscous hydrodynamics with spin, energy-momentum and spin tensors, kinetic coef-ficients, relaxation time approximation
I. INTRODUCTION
In non-central ultra-relativistic heavy-ion collisions,the two colliding nuclei carry large amount of orbital an-gular momentum L . Soon after the initial impact, a sub-stantial part of L is deposited in the interaction zone andcan be further transformed to the spin part S (with thetotal angular momentum J = L + S being conserved).The latter can be reflected in the spin polarization ofthe particles emitted at freeze-out. To verify this phe-nomenon, the spin polarization of various particles (Λ, K ∗ , φ ) produced in relativistic heavy-ion collisions hasbeen recently measured by the STAR [1, 2], ALICE [3]and HADES [4] experiments.On the theoretical side, first predictions of a non-zeroglobal spin polarization of the Λ hyperons, based onperturbative-QCD calculations and the spin-orbit inter-action, were made in Refs. [5, 6] and [7], respectively(see also Ref. [8]). In these works, a substantial polariza-tion effect of the order of 10% was found. Subsequently,using relativistic hydrodynamics with local thermody-namic equilibrium of the spin degrees of freedom [9–18],a smaller polarization of about 1% was predicted, an ef-fect which was eventually confirmed by STAR [1, 2].Interestingly, the same hydrodynamic models [18, 19]are not able to describe the experimentally measured lon-gitudinal polarization of Λ’s [20, 21]. For example, theoscillation of the longitudinal polarization of the Λ hy-perons measured as a function of the azimuthal angle inthe transverse plane [20] has an opposite sign comparedto the results obtained with relativistic hydrodynamicswith thermalized spin degrees of freedom. This issue isat the moment the subject of very intensive investigations[16, 17, 22–39]. ∗ [email protected] † wojciech.fl[email protected] ‡ [email protected] § [email protected] ¶ [email protected] The relativistic hydrodynamic models (perfect or vis-cous) that have been used so far to describe the globalspin polarization of the Λ and ¯Λ hyperons [13, 15, 18]make use of the fact that spin polarization effects aregoverned by the thermal vorticity tensor (cid:36) µν = −
12 ( ∂ µ β ν − ∂ ν β µ ) . (1)Here the four-vector β µ is defined in the standard way asthe ratio of the fluid flow vector u µ and the local tem-perature T , i.e., β µ = u µ /T . One can notice that the useof (1) does not require any modifications of the existinghydrodynamic codes as spin effects are determined solelyby the form of u µ and T .However, on the general thermodynamic grounds [40],it is expected that the spin polarization effects may begoverned by the tensor ω µν (called below the spin polar-ization tensor) that can be independent of the thermalvorticity (1). This suggests that a completely new hydro-dynamic approach including spin dynamics can be con-structed, with the spin polarization tensor ω µν treatedas an independent hydrodynamical variable. In this con-text, the concept of local spin equilibrium also changesas one no longer requires that ω µν = (cid:36) µν to have zeroentropy production.First steps to formulate the perfect-fluid version of hy-drodynamics of spin polarized fluids that incorporatesthe spin polarization tensor ω µν have already been madein a series of publications [40–42], for a recent summarysee Ref. [43]. However, only in a very recent work [44],the dissipation effects in such systems have been explic-itly considered, see also Refs. [45–50].In this work we continue and significantly extend theresults obtained in Ref. [44]. In order to identify thestructure of dissipative terms, we use classical kinetictheory for particles with spin / . The collision termsare treated in the relaxation time approximation (RTA)according to the prescription defined in Ref. [44] and,for the sake of simplicity, we restrict our considerationsto the Boltzmann statistics. The kinetic-theory frame-work determines the structure of viscous and diffusive a r X i v : . [ nu c l - t h ] A ug terms and allows to explicitly calculate a set of new ki-netic coefficients that characterize dissipative spin dy-namics. These coefficients describe coupling between anon-equilibrium part of the spin tensor and thermody-namic forces such as the expansion tensor, shear flow ten-sor, the gradient of chemical potential divided by temper-ature, and, finally, the gradient of the spin polarizationtensor.The structure of the paper is as follows: In Sec. II werecall the formulation of the perfect-fluid hydrodynam-ics with spin. Our presentation is based on the classicalconcept of spin and classical distribution functions in anextended phase space. In Sec. III we introduce kineticequations with the collision terms treated in the relax-ation time approximation and derive the form of the dissi-pative corrections. This section contains also the explicitform of the new, spin-related kinetic coefficients. Weconclude and summarize in Sec. IV. The paper is closedwith several appendices where details of our straightfor-ward but quite lengthy calculations are given. We usenatural units and the metric tensor with the signature(+ − −− ). II. FORMULATION OF PERFECT FLUIDHYDRODYNAMICS FOR SPIN POLARIZEDFLUIDSA. Spin-dependent equilibrium distributionfunction
We start with the classical treatment of massive par-ticles with spin- / and introduce their internal angularmomentum s αβ [51]. It is connected with the particlefour-momentum p γ and spin four-vector s δ [52] by thefollowing relation s αβ = 1 m (cid:15) αβγδ p γ s δ , (2)where m is the mass of the particle. Equation (2) impliesthat s αβ = − s βα and p α s αβ = 0. Moreover, from Eq. (2)we find s α = 12 m (cid:15) αβγδ p β s γδ . (3)This implies that spin four-vector s α is orthogonal tofour-momentum p α , i.e., s · p = 0. In the particle restframe (PRF), where the four-momentum of a particle is p µ = ( m, , , s α has only spatialcomponents, i.e., s α = (0 , s ∗ ), with the length of the spinvector defined by − s = | s ∗ | = s = (cid:0) (cid:1) .Identification of the so-called collisional invariants ofthe Boltzmann equation allows us to construct the equi-librium distribution functions f ± s, eq ( x, p, s ) for particlesand antiparticles [43, 44], f ± s, eq ( x, p, s ) = f ± eq ( x, p ) exp (cid:20) ω µν ( x ) s µν (cid:21) . (4) Here f ± eq ( x, p ) = exp [ − p µ β µ ( x ) ± ξ ( x )] is the J¨uttner dis-tribution, with ξ and β µ traditionally defined as ratios ofchemical potential µ to temperature T and four-velocity u µ to temperature T , i.e., ξ = µ/T and β µ = u µ /T . The spin polarization tensor ω µν has been introduced inSec. I. It plays a crucial role in our formalism and canbe interpreted as the (tensor) potential conjugated to thespin angular momentum.Before we proceed further we note that in our approach s µν is dimensionless (measured in units of (cid:126) ) and so is ω µν . Consequently, we can make expansions in ω µν and,in fact, most of our results will be valid in the leadingorder of ω µν .Ordinary phase-space equilibrium distribution func-tions can be obtained by integrating out the spin degreesof freedom present in f ± s, eq ( x, p, s ), (cid:90) dS f ± s, eq ( x, p, s ) = f ± eq ( x, p ) , (5)where [43] dS = mπ s d s δ ( s · s + s ) δ ( p · s ) . (6)Different properties of spin integrals done with the inte-gration measure (6) are collected in Appendix A. B. Perfect fluid hydrodynamics for spin polarizedfluids
For a system of particles and anti-particles with spindegrees of freedom included only through degeneracy fac-tors, the relevant conserved quantities are the energy-momentum tensor ( T µν ) and charge current ( N µ ). If spinis explicitly included, one has to consider an additionalconserved quantity, namely, the angular-momentum ten-sor ( J λ,µν ) [43, 53]. This is connected with the fact thatthe total angular momentum conservation law for parti-cles with spin has a non-trivial form.The total angular-momentum tensor ( J λ,µν ) can bewritten as a sum of the orbital ( L λ,µν ) and spin ( S λ,µν )parts. The latter is known as the spin tensor. It iswell known that there are various equivalent forms ofthe energy-momentum and spin tensors that can be usedto define system’s dynamics [46, 54, 55]. The forms usedin this work agree with the definitions introduced by deGroot, van Leeuwen, and van Weert in [56]. To empha-size this fact we sometimes use the acronym GLW.The structures of T µν , N µ and S λ,µν can be connectedto the behaviour of microscopic constituents of the sys-tem through the moments of the phase-space distributionfunctions f eq ( x, p, s ). Using the equilibrium distributions We note that since we always consider particles being on themass shell ( p = E p = (cid:112) p + m ) the distribution f ( x, p, s ) isin fact a function of p only. f eq ( x, p, s ) defined above, the hydrodynamic quantitiessuch as charge current, energy-momentum tensor, andthe spin tensor can be obtained in the similar way as instandard hydrodynamics. C. Charge current
The equilibrium charge current is defined by the for-mula N µ eq = (cid:90) dP dS p µ (cid:2) f + s, eq ( x, p, s ) − f − s, eq ( x, p, s ) (cid:3) , (7)where the invariant momentum integration measure dPis dP = d p (2 π ) E p , (8)while the measure dS is defined by Eq. (6). Using theequilibrium functions (4) we obtain N µ eq = 2 sinh( ξ ) (cid:90) dP p µ e − p · β (cid:90) dS exp (cid:18) ω αβ s αβ (cid:19) . (9)Since for large values of the spin polarization tensor thesystem becomes anisotropic in the momentum space andrequires special treatment [57, 58], in most of our cal-culations we consider only the case of small values of ω .In this case the last exponential function in (9) can beexpanded up to linear order and we find N µ eq = 2 sinh( ξ ) (cid:90) dP p µ e − p · β (cid:90) dS (cid:18) ω αβ s αβ (cid:19) . (10)After carrying out integration first over spin and thenover momentum we get N α eq = nu α , (11)where n = 4 sinh( ξ ) n ( T ) (12)is the charge density [41]. In Eq. (12) the quantity n ( T )is the number density of spinless, neutral massive Boltz-mann particles which is defined by the thermal average n ( T ) = (cid:104) u · p (cid:105) , (13)where (cid:104)· · · (cid:105) ≡ (cid:90) dP( · · · ) e − β · p . (14)The explicit calculation gives n ( T ) = (cid:90) dP ( u · p ) e − β · p = I (0)10 = 12 π T z K ( z ) , (15)with z ≡ m/T . Thermodynamic integrals I ( r ) nq are definedin Appendix B. D. Energy-momentum tensor
The energy-momentum tensor is defined as the secondmoment in momentum space, T µν eq = (cid:90) dP dS p µ p ν (cid:2) f + s, eq ( x, p, s ) + f − s, eq ( x, p, s ) (cid:3) . (16)Using Eq. (4) we can rewrite this formula as T µν eq = 2 cosh( ξ ) (cid:90) dP p µ p ν e − p · β (cid:90) dS exp (cid:18) ω αβ s αβ (cid:19) . (17)Considering the case of small ω and carrying out integra-tion over spin and momentum space we get T αβ eq ( x ) = εu α u β − P ∆ αβ , (18)where ε = 4 cosh( ξ ) ε ( T ) (19)and P = 4 cosh( ξ ) P ( T ) , (20)respectively [41]. The auxiliary quantities ε ( T ) and P ( T ) are defined as follows ε ( T ) = (cid:104) ( u · p ) (cid:105) (21)and P ( T ) = − (1 / (cid:104) p · p − ( u · p ) (cid:105) . (22)Similarly to n ( T ), they describe the energy density andpressure of spinless, neutral massive Boltzmann particles.In Eq. (18), the tensor ∆ αβ = g αβ − u α u β is an opera-tor projecting on the space orthogonal to the fluid four-velocity u µ . For the reader’s convenience, the propertiesof this and other projectors are listed in Appendix C.With the help of thermodynamic integrals I ( r ) nq definedin Appendix B one obtains ε ( T ) = (cid:90) dP ( u · p ) e − β · p = I (0)20 = 12 π T z [3 K ( z ) + zK ( z )] (23)and P ( T ) = −
13 ∆ µν (cid:90) dP p µ p ν e − β · p = − (cid:90) dP (cid:2) p · p − ( u · p ) (cid:3) e − β · p = − I (0)21 = 12 π T z K ( z ) = n ( T ) T. (24) E. Spin tensor
Now we come to the fundamental object in our for-malism, namely, the spin tensor. We adopt the followingdefinition [43] S λ,µν eq = (cid:90) dP dS p λ s µν (cid:2) f + s, eq ( x, p, s ) + f − s, eq ( x, p, s ) (cid:3) = 2 cosh( ξ ) (cid:90) dP p λ exp ( − p · β ) (25) × (cid:90) dS s µν exp (cid:18) ω αβ s αβ (cid:19) . Expanding the exponential function in the last line, inthe leading order in ω we obtain (cid:90) dS s µν exp (cid:18) ω αβ s αβ (cid:19) = (cid:90) dS s µν (cid:18) ω αβ s αβ (cid:19) = 23 m s (cid:16) m ω µν + 2 p α p [ µ ω ν ] α (cid:17) . (26)Using Eq. (26) in Eq. (25) we find S λ,µν eq = 4 s m cosh( ξ ) (cid:90) dP p λ e − p · β (cid:16) m ω µν +2 p α p [ µ ω ν ] α (cid:17) . (27)It is interesting to observe that the last result agrees withthe formula S λ,µν GLW obtained in the semiclassical expansionof the Wigner functions [53]. This fact supports our useof the definition (25).After carrying out the momentum integration we get S λ,µν eq = S λ,µν GLW = C (cid:16) n ( T ) u λ ω µν + S λ,µν ∆GLW (cid:17) . (28)Here C = (4 / s cosh( ξ ) and the auxiliary tensor S λ,µν ∆GLW is given by the expression S α,βγ ∆GLW = A u α u δ u [ β ω γ ] δ (29)+ B (cid:16) u [ β ∆ αδ ω γ ] δ + u α ∆ δ [ β ω γ ] δ + u δ ∆ α [ β ω γ ] δ (cid:17) , where B = − z ε ( T ) + P ( T ) T = − z s ( T ) (30)and A = 6 z s ( T ) + 2 n ( T ) = − B + 2 n ( T ) , (31)with s being the entropy density of spinless, neutral,massive Boltzmann particles satisfying thermodynamicrelation s = ( ε + P ) /T .We note that since our energy-momentum tensor issymmetric, the spin tensor is separately conserved. Theconservation of the spin tensor gives six additional equa-tions which are required to determine the space-time evo-lution of ω . We note that this situation may change ifnon-local effects are included, for a very recent discussionof this point see Refs. [45, 46]. F. Entropy Current
To construct the entropy current we adopt the Boltz-mann definition H µ = − (cid:90) dP dS p µ (cid:2) f + s, eq (cid:0) ln f + s, eq − (cid:1) + f − s, eq (cid:0) ln f − s, eq − (cid:1)(cid:3) . (32)Using Eqs. (4), (7), (16), and (25), we find H µ = β α T µα eq − ω αβ S µ,αβ eq − ξN µ eq + N µ eq (33)where N µ eq = cosh( ξ )sinh( ξ ) N µ eq . (34)Using Eq. (33) as well as the conservation laws for charge,energy-momentum and spin we obtain the following ex-pression, ∂ µ H µ = ( ∂ µ β α ) T µα eq −
12 ( ∂ µ ω αβ ) S µ,αβ eq − ( ∂ µ ξ ) N µ eq + ∂ µ N µ eq . (35)Now using the conservation laws for charge, energy andmomentum one can easily show that the entropy currentis conserved, namely ∂ µ H µ = 0 . (36)It should be emphasized that the last result is exact inthe sense that it does not depend on the expansion in ω .Moreover, we see that the contributions to the entropyproduction coming from the spin polarization tensor arequadratic. This means that there is no effect on the en-tropy production from the polarization in the linear or-der. This suggests that we can neglect the effects of po-larization on the global evolution of matter, provided werestrict our considerations to the linear terms. For boththe conserved charge and the energy-momentum tensorthe corrections start with the second order, hence, as longas we restrict ourselves to the linear terms in ω , we canfirst solve the system of standard hydrodynamic equa-tions (which are not affected by polarization in the lin-ear order) and subsequently determine the spin evolution(linear in ω ) on top of such a hydrodynamic background. III. FORMULATION OF DISSIPATIVEHYDRODYNAMICS FOR SPIN POLARIZEDFLUIDS
The formalism presented in the previous section is al-ready well established and may be treated as the defi-nition of the perfect-fluid hydrodynamics with spin. Inthe next section, we include dissipation effects. This willbe done with the help of the relaxation time approxima-tion used for the collision terms in the classical kineticequations, as originally introduced in Ref. [44].
A. Classical RTA kinetic equation
In the absence of mean fields, the distribution functionsatisfies the equations p µ ∂ µ f ± s ( x, p, s ) = C [ f ± s ( x, p, s )] , (37)where C [ f ± s ( x, p, s )] is the collision term. In the re-laxation time approximation, the collision term has theform [44] C [ f ± s ( x, p, s )] = p · u f ± s, eq ( x, p, s ) − f ± s ( x, p, s ) τ eq . (38)We consider now a simple Chapman-Enskog expansionof the single particle distribution function about its equi- librium value in powers of space-time gradients f ± s ( x, p, s ) = f ± s, eq ( x, p, s ) + δf ± s ( x, p, s ) , (39)Using Eqs. (38) and (39) in Eq. (37) we get p µ ∂ µ f ± s, eq ( x, p, s ) = − p · u δf ± s ( x, p, s ) τ eq . (40)After substituting equilibrium distribution function (4)in Eq. (40) we obtain (in linear order in ω ) δf ± s = − τ eq ( u · p ) e ± ξ − p · β (cid:20)(cid:16) ± p µ ∂ µ ξ − p λ p µ ∂ µ β λ (cid:17)(cid:18) s αβ ω αβ (cid:19) + 12 p µ s αβ ( ∂ µ ω αβ ) (cid:21) . (41)The corrections δf ± s result in dissipative effects in theconserved quantities such as charge current, energy-momentum tensor, and spin tensor. We discuss themnow starting from the simplest case of the charge cur-rent. The details of rather lengthy calculations are givenin Appendix D. B. Conserved hydrodynamic quantities anddissipative corrections
Taking the appropriate moments of the transport equa-tion (37), the following equations for the charge current( N µ ), energy-momentum tensor ( T µν ) and spin tensor( S λ,µν ) can be obtained ∂ µ N µ ( x ) = − u µ (cid:18) N µ ( x ) − N µ eq ( x ) τ eq (cid:19) , (42) ∂ µ T µν ( x ) = − u µ (cid:18) T µν ( x ) − T µν eq ( x ) τ eq (cid:19) , (43) ∂ λ S λ,µν ( x ) = − u λ (cid:32) S λ,µν ( x ) − S λ,µν eq ( x ) τ eq (cid:33) , (44)respectively.Conservation of the charge current ( ∂ µ N µ = 0),energy-momentum tensor ( ∂ µ T µν = 0), and spin tensor( ∂ λ S λ,µν = 0) implies that the quantities on the right-hand sides of Eqs. (42)–(44) should be zero, i.e., we musthave u µ δN µ = 0 , (45) u µ δT µν = 0 , (46) u λ δS λ,µν = 0 , (47) where δN µ , δT µν , and δS λ,µν are defined in terms of thenon-equilibrium parts of the distribution functions: δN µ = (cid:90) dP dS p µ ( δf + s − δf − s ) , (48) δT µν = (cid:90) dP dS p µ p ν ( δf + s + δf − s ) , (49) δS λ,µν = (cid:90) dP dS p λ s µν ( δf + s + δf − s ) . (50)Note that Eqs. (45) and (46), satisfied by the corrections δN µ and δT µν , are known in the literature as the Lan-dau matching conditions. They are used (and needed) todetermine the values of the chemical potential, tempera-ture, and three independent components of the flow four-vector appearing in the equilibrium distributions definedby Eq. (4) — altogether Eqs. (45) and (46) are five inde-pendent equations for five unknown functions. A novelfeature of our approach is that we introduce an additionalmatching condition given by Eq. (47). These are in factsix equations that allow us to determine six independentcomponents of the spin polarization tensor ω µν . Belowwe refer to the complete set of Eqs. (45)–(47) as to theLandau matching conditions.The conserved quantities obtained from the momentsof the transport equations (37) can be further tensor de-composed in terms of the hydrodynamic degrees of free-dom. The charge current is decomposed into two parts N µ = (cid:90) dP dS p µ (cid:2) f + ( x, p, s ) − f − ( x, p, s ) (cid:3) = N µ eq + δN µ = nu µ + ν µ . (51)In this decomposition, the quantity ν µ is known as thecharge diffusion current. The presence of the dissipa-tive corrections implies that the form of the energy-momentum tensor is T µν = (cid:90) dP dS p µ p ν (cid:2) f + ( x, p, s ) + f − ( x, p, s ) (cid:3) = T µν eq + δT µν = εu µ u ν − P ∆ µν + π µν − Π∆ µν . (52)In this decomposition, ε, P, π µν , and Π are energy den-sity, equilibrium pressure, shear stress tensor, and bulkpressure, respectively. We use here the Landau frame,where T µν u ν = εu µ . Finally, we define the correction tothe spin tensor by the decomposition S λ,µν = (cid:90) dP dS p λ s µν (cid:2) f + ( x, p, s ) + f − ( x, p, s ) (cid:3) = S λ,µν eq + δS λ,µν . (53)The non-equilibrium quantities n , ε , P can be obtainedby the Landau matching conditions, namely n = n eq = u µ N µ eq (54)= u µ (cid:90) dP dS p µ (cid:2) f +eq ( x, p, s ) − f − eq ( x, p, s ) (cid:3) ,ε = ε eq = u µ u ν T µν eq (55)= u µ u ν (cid:90) dP dS p µ p ν (cid:2) f +eq ( x, p, s ) + f − eq ( x, p, s ) (cid:3) and P = P eq = −
13 ∆ µν T µν eq (56)= − ∆ µν (cid:90) dP dS p µ p ν (cid:2) f +eq ( x, p, s ) + f − eq ( x, p, s ) (cid:3) . After carrying out integration over spin and momen-tum, Eqs. (54), (55), and (56) yield the same results asEqs. (12), (19), and (20). Here we also note that thechoice of Landau frame and matching conditions enforcesthe following constraints on the dissipative currents u µ ν µ = 0 ,u µ π µν = 0 . (57) C. Convective derivatives of hydrodynamicvariables
An intermediate step in the calculation of standardkinetic coefficients is the derivation of expressions for theconvective derivatives of the hydrodynamic variables ξ , β , and u µ . The convective derivatives are space-timederivatives taken along the streamlines of the fluid. Wedenote them by a dot or the letter D , for example,˙ ξ = Dξ = u µ ∂ µ ξ. (58)With spin degrees of freedom included, one has to cal-culate the convective derivative of the spin polarization tensor ω µν as well. In this section we describe the nec-essary steps needed to determine all those derivatives.The details of the calculations, which are quite lengthydue to complicated tensor structures, are given in theAppendices E-F.Using the conservation laws for energy and momentum( ∂ µ T µν = 0) as well as charge ( ∂ µ N µν = 0), we get thefollowing equations that dictate the evolution of T , u µ ,and µ , ˙ ε + ( ε + P + Π) θ − π µν σ µν = 0 , (59)( ε + P ) ˙ u α − ∇ α P + ∆ αµ ∂ ν π µν = 0 , (60)˙ n + nθ + ∂ µ n µ = 0 . (61)Here we use the following notation: θ = ∂ µ u µ is theexpansion scalar, ∇ µ = ∆ µν ∂ ν denotes the transversegradient, and σ µν = ( ∇ µ u ν + ∇ ν u µ ) − ∆ µν ( ∇ λ u λ (cid:1) is the shear flow tensor. In order to determine the space-time evolution of the spin polarization tensor, the abovesystem of equations should be supplemented by the con-servation of the spin tensor, ∂ λ S λ,µν = 0 . (62)Keeping only the terms up to the first order in veloc-ity gradients, the conservation equations (59), (60), (61),and (62) are reduced to˙ ε + ( ε + P ) θ = 0 , (63)( ε + P ) ˙ u α − ∇ α P = 0 , (64)˙ n + nθ = 0 , (65) ∂ λ S λ,µν eq = 0 , (66)respectively. Furthermore, from Eqs. (12) and (19) weobtain ˙ n = 4 cosh( ξ ) ˙ ξI (0)10 + 4 sinh( ξ ) ˙ I (0)10 , (67)˙ ε = 4 sinh( ξ ) ˙ ξI (0)20 + 4 cosh( ξ ) ˙ I (0)20 . (68)Using Eq. (B16) that connects derivatives of the Besselfunctions, the above equations can be written as˙ n = 4 cosh( ξ ) ˙ ξI (0)10 − ξ ) ˙ βI (0)20 , (69)˙ ε = 4 sinh( ξ ) ˙ ξI (0)20 − ξ ) ˙ βI (0)30 . (70)Substituting n , ε , P , ˙ n and ˙ ε from Eqs. (12), (19), (20),(69) and (70) in Eqs. (63) and (65) we getsinh( ξ ) ˙ ξI (0)20 − cosh( ξ ) ˙ βI (0)30 = − cosh( ξ ) (cid:16) I (0)20 − I (0)21 (cid:17) θ, (71)cosh( ξ ) ˙ ξI (0)10 − sinh( ξ ) ˙ βI (0)20 = − sinh( ξ ) I (0)10 θ. (72) Equations (59)–(61) do not include the spin polarization tensor,if we consider only linear terms in ω . Using the relations: I (0)20 = ε , I (0)21 = − P = − n T , I (0)10 = n , and I (0)30 = β (cid:0) P + ε ) + z P (cid:1) , and solvingEqs. (71) and (72) for ˙ ξ and ˙ β we can get˙ ξ = ξ θ θ, (73)˙ β = β θ θ, (74)where ξ θ = sinh( ξ ) cosh( ξ ) (cid:2) ε − n T (cid:0)(cid:0) z (cid:1) P + 2 ε (cid:1)(cid:3) n T cosh ( ξ ) ((3 + z ) P + 3 ε ) − ε sinh ( ξ ) , (75) β θ = n (cid:0) cosh ( ξ ) P + ε (cid:1) n T cosh ( ξ ) ((3 + z ) P + 3 ε ) − ε sinh ( ξ ) . (76)Substituting into Eq. (64) the energy density ε and pres-sure P defined by Eqs. (55) and (56) we getcosh( ξ ) (cid:16) I (0)20 − I (0)21 (cid:17) ˙ u α = − sinh( ξ ) ( ∇ α ξ ) I (0)21 − cosh( ξ ) (cid:16) ∇ α I (0)21 (cid:17) . (77)Now we can write ∇ α I (0)21 = ∇ α (cid:18)
13 ∆ µν (cid:90) dP p µ p ν e − p λ β λ (cid:19) = 13 ∆ µν ( −∇ α β λ ) (cid:90) dP p µ p ν p λ e − p λ β λ = −
13 ∆ µν (cid:18) T ∇ α u λ − u λ T ∇ α T (cid:19) (cid:20) I (0)30 u λ u µ u ν + I (0)31 (cid:0) ∆ λµ u ν + ∆ νλ u µ + ∆ µν u λ (cid:1) (cid:21) = − (cid:16) − u λ T ∇ α T (cid:17) u λ I (0)31 = ( −∇ α β ) I (0)31 (78)and using Eq. (78) in Eq. (77) we obtaincosh( ξ ) (cid:16) I (0)20 − I (0)21 (cid:17) ˙ u α = − sinh( ξ ) ( ∇ α ξ ) I (0)21 + cosh( ξ ) ( ∇ α β ) I (0)31 . (79)Now from the recurrence relation (B15) we obtain I (0)31 = − β (cid:16) I (0)20 − I (0)21 (cid:17) = − β ( ε + P ) ,I (0)21 = − P = − n β . (80)Using the above expressions for I (0)31 and I (0)21 in Eq. (79),the following equation for ˙ u µ can be derived β ˙ u α = n tanh( ξ ) ε + P ( ∇ α ξ ) − ( ∇ α β ) . (81) Now we turn to the equilibrium spin tensor. With thehelp of Eq. (28) it can be written as S λ,µν eq = 4 s ξ ) I (0)10 u λ ω µν (82)+ 4 s m cosh( ξ ) (cid:20) I (0)30 u λ u α u [ µ ω ν ] α +2 I (0)31 (cid:16) ∆ λα u [ µ ω ν ] α + u λ ∆ α [ µ ω ν ] α + u α ∆ λ [ µ ω ν ] α (cid:17) (cid:21) . The above equation can further be simplified as S λ,µν eq = 4 s ξ ) I (0)10 u λ ω µν (83)+ 8 s m cosh( ξ ) (cid:20) (cid:16) I (0)30 − I (0)31 (cid:17) u λ u α u [ µ ω ν ] α + I (0)31 (cid:16) u [ µ ω ν ] λ − ω µν u λ + u α g λ [ µ ω ν ] α (cid:17) (cid:21) . Substituting Eq. (83) into Eq. (66), and using Eqs. (73),(74), and (81), the following dynamical equation for thespin polarization tensor ω µν can be obtained˙ ω µν = D µν Π θ + ( ∇ α ξ ) D [ µν ] n α + D [ νπ λ σ λµ ] + D α Σ1 ∇ [ µ ω ν ] α + D [ µν ] α Σ2 ∇ λ ω αλ . (84)For details see Appendix E, where the explicit expressionsfor various D -coefficients are given.Note that while deriving the dynamical equation (84),we initially encounter the term u ν ˙ ω µν in the expressionfor ˙ ω µν . To eliminate this term we derive another dynam-ical equation for u ν ˙ ω µν by taking projection of Eq. (66)along u ν . The dynamical equation for u ν ˙ ω µν is given bythe expression u ν ˙ ω µν = C µ Π θ + C µnλ ( ∇ λ ξ ) + C πα σ αµ + C µ Σ ν ∇ λ ω νλ . (85)The explicit expression for various C -coefficients appear-ing above are also given in Appendix E. See also Ap-pendix F, where the Landau matching conditions are pre-sented in more detail. D. Transport coefficients
The dissipative forces arise due to non-zero gradients inthe system. In the present case, we will confine ourselvesonly to first order in gradients and hence the dissipativeparts of T µν , N µ , and S λ,µν , i.e., δT µν , δN µ , and δS λ,µν ,respectively, must be first order in gradients too. Theshear stress ( π µν ), bulk viscous pressure (Π) and particlediffusion current ( n µ ) can be found from δT µν and δN µ as: π µν = ∆ µναβ δT αβ , Π = −
13 ∆ αβ δT αβ , ν µ = ∆ µα δN α . (86)Hence, using Eqs. (49) and (48), the above dissipativequantities can be written as: π µν = ∆ µναβ (cid:90) dP dS p α p β ( δf + s + δf − s ) , (87)Π = −
13 ∆ αβ (cid:90) dP dS p α p β ( δf + s + δf − s ) , (88) ν µ = ∆ µα (cid:90) dP dS p α ( δf + s − δf − s ) . (89)Evaluating the expressions defined by Eqs. (87), (88),and (89), the dissipative quantities are found to be (see Appendix G) π µν = 2 τ eq β π σ µν , Π = − τ eq β Π θ,ν µ = τ eq β n ∇ µ ξ. (90)Here, coefficients, β π , β Π and β n are the first-order trans-port coefficients which for massive particles with finitechemical potential are found to be β π = 4 I (1)42 cosh( ξ ) , (91) β Π = 4 (cid:40) n cosh( ξ ) β (cid:34) sinh ( ξ ) (cid:0) ε ( P + ε ) − n T (cid:0) P (cid:0) z + 3 (cid:1) + 3 ε (cid:1)(cid:1) ε sinh ( ξ ) − n T cosh ( ξ ) ( P ( z + 3) + 3 ε ) (cid:35) − n cosh( ξ ) β (cid:34) ( P + ε ) (cid:0) P cosh ( ξ ) + ε (cid:1) n T cosh ( ξ ) ( P ( z + 3) + 3 ε ) − ε sinh ( ξ ) (cid:35) + 5 β I (1)42 (cid:41) , (92) β n = 4 (cid:20)(cid:18) n tanh( ξ ) ε + P (cid:19) I (0)21 sinh( ξ ) − I (1)21 cosh( ξ ) (cid:21) . (93)Similarly, using Eq. (41) in (50) and then carrying out integration over spin and momentum variables we get, δS λ,µν = τ eq (cid:104) B λ,µν Π θ + B κλ,µνn ( ∇ κ ξ ) + B ( κδ ) λ,µνπ σ κδ + B ηβγλ,µν Σ ∇ η ω βγ (cid:105) . (94)Different coefficients appearing on the right-hand side of Eq. (94) are the kinetic coefficients for spin. They have atensor structures expressed in terms of u µ , g µν , and ω µν . Explicit forms of these coefficients are as follows: B λ,µν Π = B (1)Π u [ µ ω ν ] λ + B (2)Π u λ u α u [ µ ω ν ] α + B (3)Π ∆ λ [ µ u α ω ν ] α , (95) B λκδ,µνπ = B (1) π ∆ [ µκ ∆ λδ u α ω ν ] α + B (2) π ∆ λδ u [ µ ω ν ] κ + B (3) π u [ µ ∆ ν ] δ ∆ λα ω ακ + B (4) π ∆ λ [ µ ω ρκ u ρ ∆ ν ] δ , (96) B λκ,n µν = B (1) n ∆ λκ ω µν + B (2) n ∆ λκ u α u [ µ ω ν ] α + B (3) n ∆ λα ∆ [ µκ ω ν ] α + B (4) n u [ µ ∆ ν ] κ u ρ ω λρ + B (5) n ∆ λ [ µ ω ν ] κ + B (6) n ∆ λ [ µ u ν ] u α ω ακ , (97) B ηβγλ,µν Σ = B (1)Σ ∆ λη g [ µβ g ν ] γ + B (2)Σ u γ ∆ λη u [ µ ∆ ν ] β + B (3)Σ (cid:16) ∆ λη ∆ γ [ µ g ν ] β + ∆ λγ ∆ [ µη g ν ] β + ∆ γη ∆ λ [ µ g ν ] β (cid:17) + B (4)Σ ∆ γη ∆ λ [ µ ∆ ν ] β + B (5)Σ u γ ∆ λβ u [ µ ∆ ν ] η , (98)where the scalar coefficients B ( i ) X are explicitly defined in Appendix E.Equation (94) is our main result. It shows that the dis-sipative spin effects are connected with the presence ofexpansion scalar, gradient of the ratio of chemical poten-tial and temperature, the shear-flow tensor, and the gra-dient of the spin polarization tensor. All these quantitiesmay be interpreted as “thermodynamic forces” that trig-ger dissipative currents. The first three among them arewell known — they lead to appearance of bulk pressure,diffusive current, and shear stress tensor. Interestingly,in the considered case, they also induce the dissipativepart of the spin tensor. The fourth term in Eq. (94) de-scribes the induction of the disspative spin tensor by thegradient of the spin polarization tensor, hence, may be treated as a direct non-equilibrium interaction betweenspin degrees of freedom.Finally, we note that all the kinetic coefficients ob-tained from Eq. (38) are proportional to the same relax-ation time τ eq . This implies that the equilibration timesfor momenta and spin degrees of freedom are the same.In phenomenological applications it is conceivable to varythe values of the relaxation times that appear in differentkinetic coefficients, arguing that they describe indepen-dent physical phenomena. However, such modificationsrequire further studies. IV. SUMMARY AND CONCLUSIONS
In this paper we have significantly extended the resultsobtained in Ref. [44]. We used classical kinetic theory forparticles with spin / with Boltzmann statistics to ob-tain the structure of dissipative terms and the associatedtransport coefficients. We considered the relaxation timeapproximation for collision term in order to account forthe interactions. This kinetic-theory framework was usedto determine the structure of spin-dependent viscous anddiffusive terms and explicitly evaluate a set of new kineticcoefficients that characterize dissipative spin dynamics.Our main result is given by Eq. (94), together with theexplicit expressions for the kinetic coefficients B givenin the appendices. Equation (94) shows that a non-equilibrium part of the spin tensor is produced by thethermodynamic forces such as expansion scalar, gradi-ent of the ratio of chemical potential and temperature,the shear-flow tensor, and the gradient of the spin po-larization tensor. Thus, the spin dissipative phenomenaare connected with those leading to formation of bulkpressure, diffusion current, and the shear stress tensor. Probably, the most interesting term in Eq. (94) is thelast one, which describes induction of a non-equilibriumspin tensor by a gradient of the spin polarization tensor.In the future investigations, it would be interesting toanalyze the role played by various coeficients appearingin Eq. (94) and to find out which kind of corrections theyimply for the spin tensor. The complicated tensor struc-ture of the spin kinetic coefficients may lead to variousinteresting phenomena. ACKNOWLEDGMENTS
W.F. and R.R. acknowledge the hospitality of Na-tional Institute of Science Education and Research wheremost of this work was done. S.B., A.J. and A.K. wouldlike to acknowledge the kind hospitality of JagiellonianUniversity and Institute of Nuclear Physics, Krakow,where part of this work was completed. A.J. was sup-ported in part by the DST-INSPIRE faculty award un-der Grant No. DST/INSPIRE/04/2017/000038. W.F.and R.R. were supported in part by the Polish NationalScience Center Grants No. 2016/23/B/ST2/00717 andNo. 2018/30/E/ST2/00432.
Appendix A: Spin-space integrals
In this appendix several integrals over the spin space are explicitly done. The results obtained here are usedthroughout the paper in the calculations of the charge current, energy-momentum tensor, and the spin tensor.
1. Normalization of spin integration measure
We start with the calculation of the normalization of the spin integration measure [43]. Since it is a Lorentz invariantquantity depending on the (external) momentum p , the calculations can be done in the particle rest frame (PRF)where p µ = ( m, , ,
0) and s µ = (0 , s ∗ ), (cid:90) dS = mπ s (cid:90) d s δ ( s · s + s ) δ ( p · s ) = mπ s (cid:90) d s (cid:90) d | s ∗ || s ∗ | (cid:90) dΩ δ ( | s ∗ | − s ) δ ( ms ) . (A1)With the normalization (cid:82) dΩ = (cid:82) sin θ d θ (cid:82) d φ = 4 π we obtain (cid:90) dS = 4 ππ s (cid:90) d | s ∗ || s ∗ | δ ( | s ∗ | − s ) = 2 . (A2)The factor of 2 reflects here the two possibilities of the spin- / projection.
2. Spin average of s µν While expanding the spin-dependent equilibrium distribution function in powers of ω , we encounter the integralsof the form (cid:90) dS s µν = 1 m (cid:90) dS (cid:15) µναβ p α s β = 1 m (cid:15) µναβ p α (cid:90) dS s β . (A3)Since the last integral can be a function of momentum p β only, we can write (cid:90) dS s β = c p β . (A4)0After contraction with p , this equation gives (cid:90) dS ( p · s ) = c m , (A5)which implies that the constant c equals zero, as p · s = 0. Hence, throughout the paper we can use the property (cid:90) dS s µν = 0 . (A6)
3. Spin average of s µν s αβ In the second order of expansions in ω we deal with the integrals of the form (cid:90) dS s µν s αβ = 1 m (cid:90) dS (cid:15) µνρσ p ρ s σ (cid:15) αβγδ p γ s δ = 1 m (cid:15) µνρσ (cid:15) αβγδ p ρ p γ (cid:90) dS s σ s δ . (A7)Since the last integral can be a function of momenta and the metric tensor, we write (cid:90) dS s σ s δ = a g σδ + b p σ p δ , (A8)where a and b are scalar coefficients. Multiplying Eq. (A8) by p σ p δ in the first case and contracting the indices inEq. (A8) in the second case, we obtain two equations (cid:90) dS ( p · s ) = a m + b m (A9)and (cid:90) dS s = 4 a + b m . (A10)The left-hand sides of Eqs. (A9) and (A10) yield (cid:90) dS ( p · s ) = 0 , (cid:90) dS s = mπ s (cid:90) d s ( s · s ) δ ( s · s + s ) δ ( p · s )= − mπ s (cid:90) d s (cid:90) d | s ∗ || s ∗ | (cid:90) dΩ δ ( | s ∗ | − s ) δ ( ms )= − mπ s (cid:90) d s δ ( s ) m (cid:90) d | s ∗ || s ∗ | δ ( | s ∗ | − s ) 4 π = − mπ s πm s − s . (A11)Thus, from Eqs. (A9) and (A10) we get a m + b m = 0 , (A12)4 a + b m = − s . (A13)Solving these two equations we get a = − s / b = 2 s / (3 m ). Hence we have (cid:90) dS s σ s δ = − s (cid:18) g σδ − p σ p δ m (cid:19) (A14)and (cid:90) dS s µν s αβ = − s m (cid:15) µνρσ (cid:15) αβγδ p ρ p γ (cid:18) g σδ − p σ p δ m (cid:19) . (A15)1 Appendix B: Thermodynamic integrals
Thermodynamic integrals considered in this work are given by the following expression I ( r ) nq = 1(2 q + 1)!! (cid:90) dP ( u · p ) n − q − r (∆ αβ p α p β ) q e − β · p . (B1)From the above formula, as the special cases, we obtain: I (0)10 = T z π K ( z ) , (B2) I (0)20 = T z π [3 K ( z ) + zK ( z )] , (B3) I (0)21 = − T z π K ( z ) , (B4) I (0)30 = T z π [ K ( z ) + K ( z ) − K ( z )] , (B5) I (0)31 = − T z π [ K ( z ) − K ( z ) + 2 K ( z )] , (B6) I (0)40 = T z π [ K ( z ) + 2 K ( z ) − K ( z ) − K ( z )] , (B7) I (0)41 = − T z π [ K ( z ) − K ( z ) − K ( z ) + 2 K ( z )] , (B8) I (0)42 = T z π [ K ( z ) − K ( z ) + 15 K ( z ) − K ( z )] . (B9)Here K n ( z ) denotes the modified Bessel functions of the second kind with the argument z = m/T . They are definedby the integral K n ( z ) = (cid:90) ∞ d x cosh( nx ) e − z cosh x . (B10)The other thermodynamic integrals which are needed in our calculation are given by the expressions I (1)21 = − T z π (cid:20) K ( z ) − K ( z ) + K i, ( z ) (cid:21) , (B11) I (1)42 = T z π (cid:104) K ( z ) − K ( z ) + K ( z ) − K i, ( z ) (cid:105) , (B12)where K i, ( z ) = (cid:90) ∞ d x sech x e − z cosh x = π (cid:104) − z K ( z ) L − ( z ) − z K ( z ) L ( z ) (cid:105) (B13)is the first-order Bickley-Naylor function with L i being the modified Struve function.Note that here we have not listed the functions I (1)20 , I (1)30 , I (1)31 , I (1)40 , I (1)50 , I (1)51 and I (1)52 as they all can be written interms of the integrals listed above, using the following recurrence relations I ( r ) n,q = I ( r − n − ,q ; n ≥ q, (B14) I (0) n,q = 1 β (cid:104) ( n − q ) I (0) n − ,q − I (0) n − ,q − (cid:105) , (B15)˙ I (0) n,q = − ˙ βI (0) n +1 ,q . (B16)2 Appendix C: Properties of the projection operators
Herein we list useful relations involving the projection operators and the differential operator ∇ µ ≡ ∆ µν ∂ ν :∆ µν ∆ µν = 3 , u µ ∆ µν = u µ ∆ νµ = 0 , ∆ µν ∆ λν = ∆ µλ , u µ ∇ µ = 0 , (C1)∆ αβµν = 12 (cid:18) ∆ αµ ∆ βν + ∆ αν ∆ βµ −
23 ∆ αβ ∆ µν (cid:19) , (C2) u µ ∆ αβµν = u µ ∆ αβνµ = u α ∆ αβµν = u α ∆ βαµν = 0 , (C3)∆ µν ∆ αβµν = ∆ µν ∆ µναβ = 0 , (C4)∆ µλ ∆ αβµν = ∆ αβλν , (C5)∆ αβλρ ∇ ρ u λ = 12 (cid:18) ∆ αλ ∆ βρ + ∆ αρ ∆ βλ −
23 ∆ αβ ∆ λρ (cid:19) ∇ ρ u λ = 12 (cid:18) ∇ β u α + ∇ α u β −
23 ∆ αβ ∇ λ u λ (cid:19) ≡ σ αβ . (C6) Appendix D: Calculation of the dissipative corrections δT µν , δN ν , δS λ,µν
1. Dissipative corrections δN ν The dissipative part of the baryon current can be written as δN µ = (cid:90) dP dS p µ (cid:0) δf + s − δf − s (cid:1) . (D1)From Eq. (41) we obtain δf + s − δf − s = − τ eq u · p e + ξ − β · p (cid:20) (cid:18) s αβ ω αβ (cid:19) (cid:0) p µ ∂ µ ξ − p λ p µ ∂ µ β λ (cid:1) + 12 s αβ p µ ∂ µ ω αβ (cid:21) − τ eq u · p e − ξ − β · p (cid:20) (cid:18) s αβ ω αβ (cid:19) (cid:0) p µ ∂ µ ξ + p λ p µ ∂ µ β λ (cid:1) − s αβ p µ ∂ µ ω αβ (cid:21) = − τ eq u · p e − β · p (cid:20) (cid:18) s αβ ω αβ (cid:19)(cid:0) cosh ξ p µ ∂ µ ξ − sinh ξ p λ p µ ∂ µ β λ (cid:1) + 12 sinh ξ s αβ p µ ∂ µ ω αβ (cid:21) . (D2)Substituting Eq. (D2) into Eq. (D1) we get δN µ = − τ eq (cid:90) dP dS p µ u · p e − β · p (cid:34)(cid:18) s αβ ω αβ (cid:19)(cid:0) cosh ξ p ρ ∂ ρ ξ − sinh ξ p λ p ρ ∂ ρ β λ (cid:1) + 12 sinh ξ s αβ p ρ ∂ ρ ω αβ (cid:35) . (D3)Now using Eqs. (A2) and (A6) we can easily carry out the integration over spin variables, therefore, δN µ = − τ eq (cid:90) dP p µ u · p e − β · p (cid:20) cosh ξ p ρ ∂ ρ ξ − sinh ξ p λ p ρ ∂ ρ β λ (cid:21) = − τ eq cosh ξ ∂ ρ ξ (cid:90) dP p µ p ρ u · p e − β · p + 4 τ eq sinh ξ ∂ ρ β λ (cid:90) dP p µ p λ p ρ u · p e − β · p . (D4)Momentum integration can be carried out using the following useful integral formulas, I µ µ ... µ n ( r ) = (cid:90) dP( u · p ) r (cid:0) p µ p µ ... p µ n e − β · p (cid:1) = I ( r ) n u µ u µ ... u µ n + I ( r ) n (∆ µ µ u µ u µ ... u µ n + permutations) + ... . (D5)3Thus, in the cases which are of interest for us, the formula (D5) gives: I µρ ( r ) = I ( r )20 u µ u ρ + I ( r )21 ∆ µρ , (D6) I µνρ ( r ) = I ( r )30 u µ u ν u ρ + I ( r )31 (∆ µν u ρ + ∆ µρ u ν + ∆ νρ u µ ) , (D7) I µνλρ ( r ) = I ( r )40 u µ u ν u λ u ρ + I ( r )41 (cid:0) ∆ µν u λ u ρ + ∆ µλ u ν u ρ + ∆ νλ u µ u ρ + ∆ µρ u ν u λ + ∆ νρ u µ u λ + ∆ λρ u µ u ν (cid:1) + I ( r )42 (cid:0) ∆ µν ∆ λρ + ∆ µλ ∆ νρ + ∆ µρ ∆ νλ (cid:1) , (D8) I µνλρσ ( r ) = I ( r )50 u µ u ν u λ u ρ u σ + I ( r )51 (cid:16) ∆ µν u λ u ρ u σ + ∆ µλ u ν u ρ u σ + ∆ νλ u µ u ρ u σ + ∆ µρ u ν u λ u σ + ∆ νρ u µ u λ u σ + ∆ µσ u ν u λ u ρ + ∆ νσ u µ u λ u ρ + ∆ λρ u µ u ν u σ + ∆ λσ u µ u ν u ρ + ∆ ρσ u µ u ν u λ (cid:17) + I ( r )52 (cid:104) u µ (cid:0) ∆ νλ ∆ ρσ + ∆ νρ ∆ λσ + ∆ νσ ∆ λρ (cid:1) + u ν (cid:0) ∆ µλ ∆ ρσ + ∆ µρ ∆ λσ + ∆ µσ ∆ λρ (cid:1) + u λ (∆ µν ∆ ρσ + ∆ µρ ∆ νσ + ∆ µσ ∆ νρ ) + u ρ (cid:0) ∆ µν ∆ λσ + ∆ µλ ∆ νσ + ∆ µσ ∆ νλ (cid:1) + u σ (cid:0) ∆ µν ∆ λρ + ∆ µλ ∆ νρ + ∆ µρ ∆ νλ (cid:1) (cid:105) . (D9)Using the integral formula (D5) in Eq. (D4) we get δN µ = − τ eq cosh ξ I µρ (1) ∂ ρ ξ + 4 τ eq sinh ξ I µλρ (1) ∂ ρ β λ , = − τ eq cosh ξ (cid:16) I (1)20 u µ u ρ + I (1)21 ∆ µρ (cid:17) ∂ ρ ξ + 4 τ eq sinh ξ (cid:104) I (1)30 u µ u λ u ρ + I (1)31 (cid:0) ∆ µλ u ρ + ∆ µρ u λ + ∆ λρ u µ (cid:1)(cid:105) ∂ ρ β λ . (D10)One can express the space-like (transverse) derivative operator as ∇ ρ = ∆ αρ ∂ α = ( g αρ − u ρ u α ) ∂ α = ∂ ρ − u ρ D. (D11)Using Eq. (D11) we can write ∂ ρ ξ = ( ∇ ρ + u ρ D ) ξ = ∇ ρ ξ + u ρ ˙ ξ, (D12) ∂ ρ ω µν = ( ∇ ρ + u ρ D ) ω µν = ∇ ρ ω µν + u ρ ˙ ω µν , (D13) ∂ ρ β λ = ∂ ρ ( βu λ ) = β∂ ρ u λ + u λ ∂ ρ β. (D14)Again using Eq. (D11) in Eq. (D14) we get ∂ ρ β λ = β ( ∇ ρ + u ρ D ) u λ + u λ ( ∇ ρ + u ρ D ) β = β ∇ ρ u λ + βu ρ ˙ u λ + u λ ∇ ρ β + u λ u ρ ˙ β. (D15)Using Eqs. (D12) and (D15) in Eq. (D10) one gets δN µ = − τ eq cosh ξ (cid:16) I (1)20 u µ u ρ + I (1)21 ∆ µρ (cid:17) (cid:16) ∇ ρ ξ + u ρ ˙ ξ (cid:17) + 4 τ eq sinh ξ (cid:104) I (1)30 u µ u λ u ρ + I (1)31 (cid:0) ∆ µλ u ρ + ∆ µρ u λ + ∆ λρ u µ (cid:1) (cid:105) (cid:16) β ∇ ρ u λ + βu ρ ˙ u λ + u λ ∇ ρ β + u λ u ρ ˙ β (cid:17) . (D16)
2. Dissipative corrections δT µν The dissipative part of the energy-momentum tensor can be written as δT µν = (cid:90) dP dS p µ p ν (cid:0) δf + s + δf − s (cid:1) . (D17)From Eq. (41) we find the sum of the out-of-equilibrium corrections to the distribution functions for particles andantiparticles δf + s + δf − s = − τ eq u · p e + ξ − β · p (cid:20) (cid:18) s αβ ω αβ (cid:19) (cid:0) p ρ ∂ ρ ξ − p λ p ρ ∂ ρ β λ (cid:1) + 12 s αβ p ρ ∂ ρ ω αβ (cid:21) + τ eq u · p e − ξ − β · p (cid:20) (cid:18) s αβ ω αβ (cid:19) (cid:0) p ρ ∂ ρ ξ + p λ p ρ ∂ ρ β λ (cid:1) − s αβ p ρ ∂ ρ ω αβ (cid:21) = − τ eq u · p e − β · p (cid:20) (cid:18) s αβ ω αβ (cid:19) (cid:0) sinh ξ p ρ ∂ ρ ξ − cosh ξ p λ p ρ ∂ ρ β λ (cid:1) + 12 cosh ξ s αβ p ρ ∂ ρ ω αβ (cid:21) . (D18)4Substituting Eq. (D18) in Eq. (D17), we obtain δT µν = − τ eq (cid:90) dP dS p µ p ν u · p e − β · p (cid:34)(cid:18) s αβ ω αβ (cid:19)(cid:0) sinh ξ p ρ ∂ ρ ξ − cosh ξ p λ p ρ ∂ ρ β λ (cid:1) + 12 cosh ξs αβ p ρ ∂ ρ ω αβ (cid:35) . (D19)Integration over spin variables and using Eqs. (A2) and (A6) leads to δT µν = − τ eq (cid:90) dP p µ p ν u · p e − β · p (cid:0) sinh ξp ρ ∂ ρ ξ − cosh ξp λ p ρ ∂ ρ β λ (cid:1) = − τ eq sinh ξ ∂ ρ ξ (cid:90) dP p µ p ν p ρ u · p e − β · p + 4 τ eq cosh ξ ∂ ρ β λ (cid:90) dP p µ p ν p λ p ρ u · p e − β · p . (D20)Using the integral formula (D5) in the above equation we obtain δT µν = − τ eq sinh ξ I µνρ (1) ∂ ρ ξ + 4 τ eq cosh ξ I µνλρ (1) ∂ ρ β λ . (D21)Subsequently, using Eqs. (D7) and (D8), we find δT µν = − τ eq sinh ξ ∂ ρ ξ (cid:104) I (1)30 u µ u ν u ρ + I (1)31 (∆ µν u ρ + ∆ µρ u ν + ∆ νρ u µ ) (cid:105) + 4 τ eq cosh ξ ∂ ρ β λ (cid:20) I (1)40 u µ u ν u λ u ρ + I (1)41 (cid:0) ∆ µν u λ u ρ + ∆ µλ u ν u ρ + ∆ νλ u µ u ρ + ∆ µρ u λ u ν + ∆ νρ u λ u µ + ∆ λρ u µ u ν (cid:1) + I (1)42 (cid:0) ∆ µν ∆ λρ + ∆ µλ ∆ νρ + ∆ µρ ∆ νλ (cid:1) (cid:21) . (D22)Furthermore, using Eqs. (D12) and (D15) in the above equation we get δT µν = − τ eq sinh ξ (cid:104) I (1)30 u µ u ν u ρ + I (1)31 (∆ µν u ρ + ∆ µρ u ν + ∆ νρ u µ ) (cid:105) (cid:16) ∇ ρ ξ + u ρ ˙ ξ (cid:17) + 4 τ eq cosh ξ (cid:20) I (1)40 u µ u ν u λ u ρ + I (1)41 (cid:0) ∆ µν u λ u ρ + ∆ µλ u ν u ρ + ∆ νλ u µ u ρ + ∆ µρ u λ u ν + ∆ νρ u λ u µ + ∆ λρ u µ u ν (cid:1) + I (1)42 (cid:0) ∆ µν ∆ λρ + ∆ µλ ∆ νρ + ∆ µρ ∆ νλ (cid:1) (cid:21) (cid:16) β ∇ ρ u λ + βu ρ ˙ u λ + u λ ∇ ρ β + u λ u ρ ˙ β (cid:17) . (D23)
3. Dissipative corrections δS λ,µν Dissipative part of the spin-tensor is given by the formula δS λ,µν = (cid:90) dP dS p λ s µν ( δf + s + δf − s ) . (D24)Using Eq. (D18) in Eq. (D24) one gets δS λ,µν = − τ eq (cid:90) dP dS p λ s µν u · p e − β · p (cid:34)(cid:18)
1+ 12 s αβ ω αβ (cid:19)(cid:0) sinh ξ p ρ ∂ ρ ξ − cosh ξ p κ p ρ ∂ ρ β κ (cid:1) + 12 cosh ξ s αβ p ρ ∂ ρ ω αβ (cid:35) . (D25)With the help of Eqs. (A6) and (A15) the integration over the spin degrees of freedom in the above equation can beeasily performed giving δS λ,µν = − s τ eq m (cid:90) dP p λ u · p e − β · p (cid:18) sinh ξ p ρ ∂ ρ ξ − cosh ξ p κ p ρ ∂ ρ β κ + cosh ξ p ρ ∂ ρ (cid:19)(cid:0) m ω µν + 2 p α p [ µ ω ν ] α (cid:1) . (D26)The above equation can further be written as a sum of six terms δS λ,µν = − s τ eq sinh ξ ∂ ρ ξ (cid:90) dP p λ p ρ ω µν u · p e − β · p (cid:124) (cid:123)(cid:122) (cid:125) I + 4 s τ eq cosh ξ ∂ ρ β κ (cid:90) dP p λ p κ p ρ ω µν u · p e − β · p (cid:124) (cid:123)(cid:122) (cid:125) II − s τ eq cosh ξ ∂ ρ ω µν (cid:90) dP p λ p ρ u · p e − β · p (cid:124) (cid:123)(cid:122) (cid:125) III − s m τ eq sinh ξ ∂ ρ ξ (cid:90) dP p λ p ρ p α p [ µ ω ν ] α u · p e − β · p (cid:124) (cid:123)(cid:122) (cid:125) IV + 8 s m τ eq cosh ξ ∂ ρ β κ (cid:90) dP p λ p κ p ρ p α p [ µ ω ν ] α u · p e − β · p (cid:124) (cid:123)(cid:122) (cid:125) V − s m τ eq cosh ξ (cid:90) dP p λ p ρ p α p [ µ ∂ ρ ω ν ] α u · p e − β · p (cid:124) (cid:123)(cid:122) (cid:125) V I . (D27)5Now we evaluate one by one each of the terms appearing in this expression. Term I: I = 4 s τ eq sinh ξ ∂ ρ ξ ω µν (cid:90) dP p λ p ρ u · p e − β · p . (D28)Using Eqs. (D12), (D5) and (D6), we can get I = 4 s τ eq sinh ξ ( ∇ ρ ξ + u ρ ˙ ξ ) ω µν I λρ (1) = 4 s τ eq sinh ξ ( ∇ ρ ξ + ˙ ξu ρ ) ω µν (cid:16) I (1)20 u λ u ρ + I (1)21 ∆ λρ (cid:17) = 4 s τ eq sinh ξω µν (cid:16) I (1)21 ∇ λ ξ + ˙ ξu λ I (1)20 (cid:17) . (D29) Term II: II = 4 s τ eq cosh ξ ∂ ρ β κ (cid:90) dP p λ p κ p ρ ω µν u · p e − β · p . (D30)Using Eqs. (D15), (D5), and (D7) this term can be written as II = 4 s τ eq cosh ξ ω µν (cid:16) β ∇ ρ u κ + βu ρ ˙ u κ + u κ ∇ ρ β + u κ u ρ ˙ β (cid:17)(cid:104) I (1)30 u λ u κ u ρ + I (1)31 (cid:0) ∆ λκ u ρ + ∆ λρ u κ + ∆ κρ u λ (cid:1)(cid:105) . (D31)This expression simplifies to II = 4 s τ eq cosh ξ ω µν (cid:104) I (1)30 ˙ βu λ + I (1)31 (cid:0) βu λ θ + β ˙ u λ + ∇ λ β (cid:1)(cid:105) . (D32) Term III:
III = 4 s τ eq cosh ξ ∂ ρ ω µν (cid:90) dP p λ p ρ u · p e − β · p . (D33)Using Eqs. (D13), (D5), and (D6) we can write III = 4 s τ eq cosh ξ ( ∇ ρ ω µν + u ρ ˙ ω µν ) I λρ (1) = 4 s τ eq cosh ξ ( ∇ ρ ω µν + u ρ ˙ ω µν ) (cid:16) I (1)20 u λ u ρ + I (1)21 ∆ λρ (cid:17) = 4 s τ eq cosh ξ (cid:16) I (1)21 ∇ λ ω µν + I (1)20 u λ ˙ ω µν (cid:17) . (D34) Term IV: IV = 8 s m τ eq sinh ξ ∂ ρ ξ (cid:90) dP p λ p ρ p α p [ µ ω ν ] α u · p e − β · p . (D35)Using Eqs. (D12), (D5), and (D8) we find IV = 8 s m τ eq sinh ξ ( ∇ ρ ξ + u ρ ˙ ξ ) I λρα [ µ (1) ω ν ] α = 8 s m τ eq sinh ξ ( ∇ ρ ξ + u ρ ˙ ξ ) (cid:34) I (1)40 u λ u ρ u α u [ µ + I (1)41 (cid:16) ∆ λρ u α u [ µ + ∆ λα u ρ u [ µ + ∆ λ [ µ u ρ u α + ∆ ρα u λ u [ µ + ∆ ρ [ µ u λ u α + ∆ α [ µ u λ u ρ (cid:17) + I (1)42 (cid:16) ∆ λρ ∆ α [ µ + ∆ λα ∆ ρ [ µ + ∆ λ [ µ ∆ ρα (cid:17)(cid:35) ω ν ] α = 8 s m τ eq sinh ξ (cid:34) I (1)41 (cid:16) u α u [ µ ω ν ] α ∇ λ ξ + u λ u [ µ ω ν ] α ∇ α ξ + u λ u α ω [ να ∇ µ ] ξ (cid:17) + I (1)42 (cid:16) ∆ α [ µ ω ν ] α ∇ λ ξ + ∆ λα ω [ να ∇ µ ] ξ + ∆ λ [ µ ω ν ] α ∇ α ξ (cid:17) + ˙ ξ (cid:110) I (1)40 u λ u α u [ µ ω ν ] α + I (1)41 (cid:16) ∆ λα u [ µ ω ν ] α + ∆ λ [ µ u α ω ν ] α + ∆ α [ µ u λ ω ν ] α (cid:17)(cid:111) (cid:35) . (D36)6 Term V: V = 8 s m τ eq cosh ξ ∂ ρ β κ (cid:90) dP p λ p κ p ρ p α p [ µ ω ν ] α u · p e − β · p . (D37)Using Eqs. (D15), (D5) and (D9) this term can be written as V = 8 s m τ eq cosh ξ (cid:16) β ∇ ρ u κ + βu ρ ˙ u κ + u κ ∇ ρ β + u κ u ρ ˙ β (cid:17) I λκρα [ µ (1) ω ν ] α = 8 s m τ eq cosh ξ (cid:16) β ∇ ρ u κ + βu ρ ˙ u κ + u κ ∇ ρ β + u κ u ρ ˙ β (cid:17)(cid:34) I (1)50 u λ u κ u ρ u α u [ µ + I (1)51 (cid:16) ∆ λκ u ρ u α u [ µ + ∆ λρ u κ u α u [ µ + ∆ λα u κ u ρ u [ µ + ∆ λ [ µ u κ u ρ u α + ∆ κρ u λ u α u [ µ + ∆ κα u λ u ρ u [ µ + ∆ κ [ µ u λ u ρ u α + ∆ ρα u λ u κ u [ µ + ∆ ρ [ µ u λ u κ u α + ∆ α [ µ u λ u κ u ρ (cid:17) + I (1)52 (cid:26) u λ (cid:16) ∆ κρ ∆ α [ µ + ∆ κα ∆ ρ [ µ + ∆ κ [ µ ∆ ρα (cid:17) + u κ (cid:16) ∆ λρ ∆ α [ µ + ∆ λα ∆ ρ [ µ + ∆ λ [ µ ∆ ρα (cid:17) + u ρ (cid:16) ∆ λκ ∆ α [ µ + ∆ λα ∆ κ [ µ + ∆ λ [ µ ∆ κα (cid:17) + u α (cid:16) ∆ λκ ∆ ρ [ µ + ∆ λρ ∆ κ [ µ + ∆ λ [ µ ∆ κρ (cid:17) + u [ µ (cid:0) ∆ λκ ∆ ρα + ∆ λρ ∆ κα + ∆ λα ∆ κρ (cid:1)(cid:27)(cid:35) ω ν ] α = 8 s m τ eq cosh ξ (cid:34) I (1)50 ˙ βu λ u α u [ µ ω ν ] α + I (1)51 (cid:18) βθu λ u α u [ µ ω ν ] α + β ˙ u λ u α u [ µ ω ν ] α + β ˙ u α u λ u [ µ ω ν ] α + β ˙ u [ µ u λ u α ω ν ] α + u α u [ µ ω ν ] α ∇ λ β + u λ u [ µ ω ν ] α ∇ α β + u λ u α ω [ να ∇ µ ] β + ˙ β ∆ λα u [ µ ω ν ] α + ˙ β ∆ λ [ µ u α ω ν ] α + ˙ β ∆ α [ µ u λ ω ν ] α (cid:19) + I (1)52 (cid:18) βθu λ ∆ α [ µ ω ν ] α + βu λ ω [ να ∇ µ ] u α + βu λ ω [ να ∇ α u µ ] + βu α ω [ να ∇ µ ] u λ + βu α ω [ να ∇ λ u µ ] + u α ∆ λ [ µ ω ν ] α β θ + βu [ µ ω ν ] α ∇ α u λ + βu [ µ ω ν ] α ∇ λ u α + βθu [ µ ∆ λα ω ν ] α + β ˙ u λ ∆ α [ µ ω ν ] α + β ∆ λα ˙ u [ µ ω ν ] α + β ˙ u α ∆ λ [ µ ω ν ] α + ∆ α [ µ ω ν ] α ∇ λ β + ∆ λα ω [ να ∇ µ ] β + ∆ λ [ µ ω ν ] α ∇ α β (cid:19)(cid:35) . (D38) Term VI:
V I = 8 s m τ eq cosh ξ (cid:90) dP p λ p ρ p α p [ µ ∂ ρ ω ν ] α u · p e − β · p . (D39)Using Eqs. (D13), (D5) and (D8) we obtain V I = 8 s m τ eq cosh ξ I λρα [ µ (1) ∂ ρ ω ν ] α = 8 s m τ eq cosh ξ (cid:20) I (1)40 u λ u ρ u α u [ µ + I (1)41 (cid:16) ∆ λρ u α u [ µ + ∆ λα u ρ u [ µ + ∆ λ [ µ u ρ u α + ∆ ρα u λ u [ µ + ∆ ρ [ µ u λ u α + ∆ α [ µ u λ u ρ (cid:17) + I (1)42 (cid:16) ∆ λρ ∆ α [ µ + ∆ λα ∆ ρ [ µ + ∆ λ [ µ ∆ ρα (cid:17) (cid:21) (cid:16) ∇ ρ ω ν ] α + u ρ ˙ ω ν ] α (cid:17) = 8 s m τ eq cosh ξ (cid:20) I (1)40 u λ u α u [ µ ˙ ω ν ] α + I (1)41 (cid:16) u α u [ µ ∇ λ ω ν ] α + u λ u [ µ ∇ α ω ν ] α + u λ u α ∇ [ µ ω ν ] α + ∆ λα u [ µ ˙ ω ν ] α + ∆ λ [ µ u α ˙ ω ν ] α + ∆ α [ µ u λ ˙ ω ν ] α (cid:17) + I (1)42 (cid:16) ∆ α [ µ ∇ λ ω ν ] α + ∆ λα ∇ [ µ ω ν ] α + ∆ λ [ µ ∇ α ω ν ] α (cid:17) (cid:21) . (D40)Now substituting Eqs. (D29), (D32), (D34), (D36), (D38), and (D40) into Eq. (D27) one can obtain the following7expression for the dissipative correction to the spin tensor δS λ,µν = 4 s τ eq (cid:34) − sinh ξ (cid:26) I (1)21 ω µν ∇ λ ξ + I (1)20 ˙ ξ u λ ω µν + 2 m (cid:18) I (1)41 (cid:16) u α u [ µ ω ν ] α ∇ λ ξ + u λ u [ µ ω ν ] α ∇ α ξ + u λ u α ω [ να ∇ µ ] ξ (cid:17) + I (1)42 (cid:16) ∆ α [ µ ω ν ] α ∇ λ ξ + ∆ λα ω [ να ∇ µ ] ξ + ∆ λ [ µ ω ν ] α ∇ α ξ (cid:17) + ˙ ξ I (1)40 u λ u α u [ µ ω ν ] α + ˙ ξ I (1)41 (cid:16) ∆ λα u [ µ ω ν ] α + ∆ λ [ µ u α ω ν ] α + ∆ α [ µ u λ ω ν ] α (cid:17) (cid:19)(cid:27) + cosh ξ (cid:26) I (1)30 ˙ β u λ ω µν + I (1)31 (cid:0) β θ u λ + β ˙ u λ + ∇ λ β (cid:1) ω µν + 2 m ˙ β I (1)50 u λ u α u [ µ ω ν ] α + 2 m I (1)51 (cid:18) β θ u λ u α u [ µ ω ν ] α + (cid:0) β ˙ u λ + ∇ λ β (cid:1) u α u [ µ ω ν ] α + ( β ˙ u α + ∇ α β ) u λ u [ µ ω ν ] α + (cid:16) β ˙ u [ µ + ∇ [ µ β (cid:17) ω ν ] α u λ u α + ˙ β (cid:16) ∆ λα u [ µ ω ν ] α + ∆ λ [ µ u α ω ν ] α + ∆ α [ µ u λ ω ν ] α (cid:17) (cid:19) + 2 m I (1)52 (cid:18) β θ u λ ∆ α [ µ ω ν ] α + βu λ ω [ να ∇ µ ] u α + βu λ ω [ να ∇ α u µ ] + β u α ω [ να ∇ µ ] u λ + β u α ω [ να ∇ λ u µ ] + u α ∆ λ [ µ ω ν ] α β θ + β u [ µ ω ν ] α ∇ α u λ + β u [ µ ω ν ] α ∇ λ u α + βθu [ µ ∆ λα ω ν ] α + ∆ α [ µ (cid:0) β ˙ u λ + ∇ λ β (cid:1) ω ν ] α + ∆ λα (cid:16) β ˙ u [ µ + ∇ [ µ β (cid:17) ω ν ] α + ∆ λ [ µ ( β ˙ u α + ∇ α β ) ω ν ] α (cid:19) − (cid:16) I (1)21 ∇ λ ω µν + I (1)20 u λ ˙ ω µν (cid:17) − m (cid:18) I (1)40 u λ u α u [ µ ˙ ω ν ] α + I (1)41 (cid:16) u α u [ µ ∇ λ ω ν ] α + u λ u [ µ ∇ α ω ν ] α + u λ u α ∇ [ µ ω ν ] α + ∆ λα u [ µ ˙ ω ν ] α + ∆ λ [ µ u α ˙ ω ν ] α + ∆ α [ µ u λ ˙ ω ν ] α (cid:17) + I (1)42 (cid:16) ∆ α [ µ ∇ λ ω ν ] α + ∆ λα ∇ [ µ ω ν ] α + ∆ λ [ µ ∇ α ω ν ] α (cid:17) (cid:19)(cid:27)(cid:35) . (D41) Appendix E: Eliminating ˙ ξ , ˙ β , ˙ u µ and ˙ ω µν from δS λ,µν Note that the derivation of equations that specify the convective derivatives ˙ ξ , ˙ β , and ˙ u µ has already been donein Sec. III C and our results are reported in Eqs. (73), (74), and (81). Here we present important steps needed forderivation of the dynamical equation for ˙ ω µν . Substituting Eq. (83) in Eq. (66) we can get˙ ω µν = − (cid:16) I (0)10 − m I (0)31 (cid:17) (cid:16) I (0)10 θ ω µν + I (0)10 ˙ ξω µν tanh ξ + ˙ I (0)10 ω µν (cid:17) − (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:34) tanh ξ (cid:16)(cid:16) I (0)30 − I (0)31 (cid:17) ˙ ξu α u [ µ ω ν ] α + I (0)31 ∂ λ ξ (cid:16) u [ µ ω ν ] λ − ω µν u λ + u α g λ [ µ ω ν ] α (cid:17)(cid:17) + (cid:16) ˙ I (0)30 − I (0)31 (cid:17) u α u [ µ ω ν ] α + (cid:16) u [ µ ω ν ] λ ∂ λ I (0)31 − ˙ I (0)31 ω µν + u α g λ [ µ ω ν ] α ∂ λ I (0)31 (cid:17) + (cid:16) I (0)30 − I (0)31 (cid:17) θ u α u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) ˙ u α u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) u α ˙ u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) u α u [ µ ˙ ω ν ] α + I (0)31 (cid:16) ω [ νλ ∂ λ u µ ] + u [ µ ∂ λ ω ν ] λ − θ ω µν + g λ [ µ ω ν ] α ∂ λ u α + u α ∂ [ µ ω ν ] α (cid:17)(cid:35) . (E1)8Using the relations ˙ I (0)10 = − ˙ βI (0)20 , ˙ I (0)30 = − ˙ βI , ˙ I (0)31 = − ˙ βI (0)41 , ∂ λ I (0)30 = − ( ∂ λ β ) I (0)40 , ∂ λ I (0)31 = − ( ∂ λ β ) I (0)41 andsubstituting ∂ λ = ∇ λ + u λ D in the above equation we obtain˙ ω µν = − (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:16) I (0)30 − I (0)31 (cid:17) u α u [ µ ˙ ω ν ] α − (cid:16) I (0)10 − m I (0)31 (cid:17) (cid:16) I (0)10 θ ω µν + I (0)10 ˙ ξω µν tanh ξ − ˙ βI (0)20 ω µν (cid:17) − (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:34) tanh ξ (cid:16) I (0)30 − I (0)31 (cid:17) ˙ ξ u α u [ µ ω ν ] α + tanh ξ I (0)31 (cid:16) ∇ λ ξ + ˙ ξu λ (cid:17) (cid:16) u [ µ ω ν ] λ − ω µν u λ + u α g λ [ µ ω ν ] α (cid:17) − ˙ β (cid:16) I (0)40 − I (0)41 (cid:17) u α u [ µ ω ν ] α − (cid:16)(cid:16) ∇ λ β + ˙ βu λ (cid:17) I (0)41 u [ µ ω ν ] λ − ˙ βI (0)41 ω µν + (cid:16) ∇ λ β + ˙ βu λ (cid:17) I (0)41 u α g λ [ µ ω ν ] α (cid:17) + (cid:16) I (0)30 − I (0)31 (cid:17) θ u α u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) ˙ u α u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) u α ˙ u [ µ ω ν ] α + I (0)31 (cid:16) ω [ νλ ∇ λ u µ ] + u [ µ ∇ λ ω ν ] λ − θ ω µν + ω [ να ∇ µ ] u α + u α ∇ [ µ ω ν ] α (cid:17) (cid:35) . (E2)We first eliminate u α (cid:0) u [ µ ˙ ω ν ] α (cid:1) from the above expression. Contracting the resulting equation with u ν and using I (0)30 − I (0)31 = − βI (0)41 , I (0)30 − I (0)31 = I (0)31 − βI (0)41 at appropriate places we obtain u ν ˙ ω µν = − m m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:16) I (0)10 θ ω µν u ν + I (0)10 ˙ ξ ω µν tanh ξ u ν − I (0)20 ˙ β ω µν u ν (cid:17) − m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:20) − tanh ξ (cid:16) I (0)30 + I (0)31 (cid:17) ˙ ξ ω µν u ν − tanh ξI (0)31 ∆ µν ω νλ ∇ λ ξ + ˙ β (cid:16) I (0)40 + I (0)41 (cid:17) ω µν u ν + ( β ˙ u α + ∇ α β ) I (0)41 ∆ µν ω να + (cid:16)(cid:16) I (0)41 β − I (0)31 (cid:17) θ ω µν u ν + I (0)31 (cid:0) ω νλ u ν ∇ λ u µ − ∆ µν ∇ λ ω νλ + u ν ω να ∇ µ u α + u α u ν ∇ µ ω να (cid:1)(cid:17) (cid:21) . (E3)Now eliminating ˙ ξ , ˙ β , and ˙ u µ (with the help of Eqs. (73), (74) and (81)) the above equation can be written as u ν ˙ ω µν = C µ Π θ + C µnλ ∇ λ ξ + C πα σ αµ + C µ Σ ν ∇ λ ω νλ . (E4)Various C − coefficients appearing in the above equation are as follows: C µ Π = C Π u ν ω µν , (E5) C µnλ = C n ∆ µν ω νλ , (E6) C πα = C π u ν ω να , (E7) C µ Σ ν = C Σ ∆ µν , (E8)where C Π = − m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:34) m ξ θ tanh ξI (0)10 − m β θ I (0)20 + m I (0)10 − tanh ξ (cid:16) I (0)30 + I (0)31 (cid:17) ξ θ + β θ (cid:16) I (0)40 + I (0)41 (cid:17) + β I (0)41 − I (0)31 (cid:35) , (E9) C n = tanh ξm I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33) , (E10) C π = − I (0)31 m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) , (E11) C Σ = I (0)31 m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) . (E12)9Using Eq. (E4) and the recurrence relation I (0)30 − I (0)31 = − βI (0)41 in (E2) and then eliminating ˙ ξ , ˙ β , and ˙ u µ (usingEqs. (73), (74), and (81)) we obtain˙ ω µν = D µν Π θ + D [ µν ] n α ( ∇ α ξ ) + D [ νπ λ σ λµ ] + D α Σ1 ∇ [ µ ω ν ] α + D [ µν ] α Σ2 ∇ λ ω αλ . (E13)The various D -coefficients appearing in the above equation are given by the following expressions D µν Π = D Π1 ω µν + D Π2 u α u [ µ ω ν ] α , (E14) D [ µν ] n α = − D n (cid:16) u [ µ ω ν ] α + g [ µα u κ ω ν ] κ (cid:17) − D n u [ µ ∆ ν ] ρ ω ρα , (E15) D [ µπ λ = − ω [ µλ I (0)31 ( m I (0)10 − I (0)31 ) − u [ µ u α ω αλ I (0)30 − I (0)31 ) I (0)31 ( m I (0)10 − I (0)31 ) (cid:2) m I (0)10 − ( I (0)30 + I (0)31 ) (cid:3) , (E16) D α Σ1 = − u α I (0)31 ( m I (0)10 − I (0)31 ) , (E17) D [ µν ] α Σ2 = − u [ µ g ν ] α I (0)31 (cid:16) m I (0)10 − I (0)31 (cid:17) − u [ µ ∆ ν ] α I (0)30 − I (0)31 ) I (0)31 (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:104) m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17)(cid:105) , (E18)where D Π1 = − (cid:16) I (0)10 − m I (0)31 (cid:17) (cid:32) ξ θ tanh ξ I (0)10 − β θ I (0)20 + I (0)10 − m ξ θ tanh ξ I (0)31 + 2 β θ I (0)41 m − I (0)31 m (cid:33) , (E19) D Π2 = 2 m I (0)10 − I (0)31 (cid:34) β θ (cid:16) I (0)40 − I (0)41 (cid:17) − ξ θ (cid:16) I (0)30 − I (0)31 (cid:17) tanh ξ − (cid:18) I (0)30 − I (0)31 (cid:19) + (cid:16) I (0)30 − I (0)31 (cid:17) m I (0)10 − I (0)30 − I (0)31 × (cid:18) m ξ θ tanh ξ I (0)10 − m β θ I (0)20 + m I (0)10 − ξ θ (cid:16) I (0)30 + I (0)31 (cid:17) tanh ξ + β θ (cid:16) I (0)40 + I (0)41 (cid:17) + βI (0)41 − I (0)31 (cid:19)(cid:35) , (E20) D n = 2 tanh ξ (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33) , (E21) D n = tanh ξm I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33) (cid:16) I (0)30 − I (0)31 (cid:17)(cid:16) m I (0)10 − I (0)31 (cid:17) . (E22)Using Eqs. (73), (74), (81), (E4), (E13), and (D41), we finally obtain δS λ,µν = τ eq (cid:104) B λ,µν Π θ + B λκ,µνn ( ∇ κ ξ ) + B λκδ,µνπ σ κδ + B ηβγλ,µν Σ ∇ η ω βγ (cid:105) , (E23)where different coefficients appearing on the right-hand side of Eq. (E23) are the kinetic coefficients for spin-relatedphenomena. These coefficients are listed in Eqs. (95), (96), (97), and (98) where: B (1)Π = 4 s (cid:32) − m ξ θ sinh ξ I (1)41 + 2 m I (1)51 β θ cosh ξ + 103 m I (1)52 β cosh ξ − m I (1)41 cosh ξ D Π1 (cid:33) , (E24) B (2)Π = 4 s (cid:34) − m ξ θ sinh ξI (1)40 + 4 m ξ θ sinh ξ I (1)41 + 2 m I (1)50 β θ cosh ξ + 2 m I (1)51 β cosh ξ − m I (1)51 β θ cosh ξ − m I (1)52 β cosh ξ − (cid:18) I (1)20 − m I (1)41 (cid:19) cosh ξ D Π2 − m (cid:16) I (1)40 − I (1)41 (cid:17) cosh ξ C Π (cid:35) , (E25) B (3)Π = 4 s (cid:32) − m ξ θ sinh ξI (1)41 + 2 m I (1)51 β θ cosh ξ + 103 m I (1)52 β cosh ξ − m I (1)41 cosh ξ C Π (cid:33) , (E26)0 B (1) π = 16 s m β cosh ξ I (0)42 , (E27) B (2) π = 16 s m cosh ξ (cid:32) β I (0)42 − I (1)41 I (0)31 m I (0)10 − I (0)31 (cid:33) , (E28) B (3) π = 16 s m cosh ξ (cid:32) I (1)41 I (0)31 m I (0)10 − I (0)31 (cid:33) , (E29) B (4) π = 16 s m cosh ξ I (1)41 I (0)31 m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) , (E30) B (1) n = 4 s m cosh ξ (cid:20) − tanh ξ (cid:16) m I (1)21 − I (1)42 (cid:17) + (cid:18) n tanh( ξ ) ε + P (cid:19) (cid:16) m I (1)31 − I (1)52 (cid:17)(cid:21) , (E31) B (2) n = 8 s m cosh ξ (cid:34) − tanh ξ (cid:16) I (1)41 − I (1)42 (cid:17) + (cid:18) n tanh ξε + P (cid:19)(cid:16) I (1)51 − I (1)52 (cid:17) − I (1)41 tanh ξ (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33)(cid:35) , (E32) B (3) n = 8 s m cosh ξ (cid:20) − tanh ξ I (1)42 + (cid:18) n tanh ξε + P (cid:19) I (1)52 (cid:21) , (E33) B (4) n = 8 s m cosh ξ I (1)41 tanh ξ (cid:16) m I (0)10 − I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33) , (E34) B (5) n = 8 s m cosh ξ (cid:34) − tanh ξ I (1)42 + (cid:18) n tanh ξε + P (cid:19) I (1)52 − I (1)41 tanh ξm I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33) (cid:35) , (E35) B (6) n = 8 s m cosh ξ (cid:34) I (1)41 tanh ξm I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) (cid:32) I (0)31 − n I (0)41 ε + P (cid:33) (cid:35) , (E36) B (1)Σ = − s ξ I (1)21 , (E37) B (2)Σ = − s m cosh ξ (cid:32) I (1)41 + I (1)41 I (0)31 m I (0)10 − I (0)31 (cid:33) , (E38) B (3)Σ = − s m cosh ξ I (1)42 , (E39) B (4)Σ = − s m cosh ξ I (1)41 I (0)31 m I (0)10 − (cid:16) I (0)30 + I (0)31 (cid:17) , (E40) B (5)Σ = 8 s m cosh ξ (cid:32) I (1)41 I (0)31 m I (0)10 − I (0)31 (cid:33) . (E41) Appendix F: Landau matching Conditions
In this section we show that δN µ , δT µν , and δS λ,µν given by Eqs. (D16), (D23), and (E23) satisfy the relations(45), (46), and (47).
1. Proving u µ δN µ = 0 Projecting Eq. (D16) along u µ we obtain u µ δN µ = − I (1)20 ˙ ξτ eq cosh ξ + 4 (cid:16) I (1)31 βθ + I (1)30 ˙ β (cid:17) τ eq sinh ξ. (F1)1Using the recurrence relation (B14) we can write down I (1)20 = I (0)10 = n , I (1)31 = I (0)21 = − P , I (1)30 = I (0)20 = ε . (F2)Substituting the above values for I (1)20 , I (1)31 , I (1)30 and the values of ˙ ξ and ˙ β from Eqs. (73) and (74) into Eq. (F1), wecan show that the right-hand side of Eq. (F1) vanishes.
2. Proving u µ δT µν = 0 Projecting Eq. (D23) along u µ we obtain u µ δT µν = − τ eq sinh ξ (cid:104) I (1)30 ˙ ξu ν + I (1)31 ∇ ν ξ (cid:105) + 4 τ eq cosh ξ (cid:104) I (1)40 ˙ βu ν + I (1)41 ( β ˙ u ν + ∇ ν β + βθu ν ) (cid:105) . (F3)Using Eq. (81), the above equation can be written as u µ δT µν = − τ eq (cid:104) I (1)30 ˙ ξ sinh ξ − I (1)40 ˙ β cosh ξ − I (1)41 βθ cosh ξ (cid:105) u ν − τ eq (cid:20) I (1)31 sinh ξ − I (1)41 cosh ξ n tanh ξε + P (cid:21) ∇ ν ξ. (F4)Using the recurrence relations (B14) and (B15) we can write I (1)30 = I (0)20 , I (1)40 = I (0)30 = n , I (1)41 = I (0)31 = − β (cid:16) I (0)20 − I (0)21 (cid:17) . (F5)Using the above relations along with the values of ˙ ξ and ˙ β from Eqs. (73) and (74), we see that the first square bracketterm on the right-hand side of Eq. (F4) vanishes; see Eq. (71) for details. Using the relations (F2) and (F5), it canalso be shown that the second square bracket term in Eq. (F4) is zero.
3. Proving u µ δS λ,µν = 0 Projecting Eq. (D41) along u λ we obtain u λ δS λ,µν = 4 s τ eq (cid:34) − sinh ξ (cid:26) I (0)10 ˙ ξ ω µν + 2 m (cid:16) I (1)41 (cid:16) u [ µ ω ν ] α ∇ α ξ + u α ω [ να ∇ µ ] ξ (cid:17) + ˙ ξ (cid:16) I (1)40 u α u [ µ ω ν ] α + I (1)41 ∆ α [ µ ω ν ] α (cid:17)(cid:17)(cid:27) + cosh ξ (cid:26)(cid:16) I (1)31 βθω µν + I (1)30 ˙ βω µν (cid:17) + 2 m ˙ βI (1)50 u α u [ µ ω ν ] α + 2 m I (1)52 (cid:16) βθ ∆ α [ µ ω ν ] α + β ( ∇ [ µ u α + ∇ α u [ µ ) ω ν ] α (cid:17) + 2 m I (1)51 (cid:16) βθu α u [ µ ω ν ] α + ( β ˙ u α + ∇ α β ) u [ µ ω ν ] α + (cid:16) β ˙ u [ µ + ∇ [ µ β (cid:17) ω ν ] α u α + ˙ β ∆ α [ µ ω ν ] α (cid:17) − (cid:18) I (1)20 − m I (1)41 (cid:19) ˙ ω µν + 2 (cid:16) m I (1)20 − I (1)41 (cid:17) (cid:16) I (1)40 − I (1)41 (cid:17) u α u [ µ ˙ ω ν ] α − m I (1)41 (cid:16) u [ µ ∇ α + u α ∇ [ µ (cid:17) ω ν ] α (cid:27)(cid:35) . (F6)2Using Eq. (E2), the above equation can further be written as u λ δS λ,µν = 4 s τ eq (cid:34) − sinh ξ I (0)10 ˙ ξω µν − ξm (cid:16) I (1)41 (cid:16) u [ µ ω ν ] α ∇ α ξ + u α ω [ να ∇ µ ] ξ (cid:17) + ˙ ξ (cid:16) I (1)40 u α u [ µ ω ν ] α + I (1)41 ∆ α [ µ ω ν ] α (cid:17) (cid:17) + cosh ξ (cid:18) I (1)31 βθω µν + I (1)30 ˙ βω µν + 2 m ˙ βI (1)50 u α u [ µ ω ν ] α + 2 m I (1)52 (cid:16) βθ ∆ α [ µ ω ν ] α + β ( ∇ [ µ u α + ∇ α u [ µ ) ω ν ] α (cid:17) + 2 m I (1)51 (cid:16) βθu α u [ µ ω ν ] α + ( β ˙ u α + ∇ α β ) u [ µ ω ν ] α + (cid:16) β ˙ u [ µ + ∇ [ µ β (cid:17) ω ν ] α u α + ˙ β ∆ α [ µ ω ν ] α (cid:17) + (cid:16) I (0)10 θ ω µν + I (0)10 ˙ ξ ω µν tanh ξ − ˙ β I (0)20 ω µν (cid:17) + 2 m (cid:40) tanh ξ (cid:16) I (0)30 − I (0)31 (cid:17) ˙ ξu α u [ µ ω ν ] α + tanh ξ I (0)31 (cid:16) ∇ λ ξ + ˙ ξu λ (cid:17) (cid:16) u [ µ ω ν ] λ − ω µν u λ + u α g λ [ µ ω ν ] α (cid:17) − (cid:16) ˙ β (cid:16) I (0)40 − I (0)41 (cid:17) u α u [ µ ω ν ] α + (cid:16) ∇ λ β + ˙ βu λ (cid:17) I (0)41 u [ µ ω ν ] λ − ˙ βI (0)41 ω µν + (cid:16) ∇ λ β + ˙ βu λ (cid:17) I (0)41 u α g λ [ µ ω ν ] α (cid:17) + (cid:16) I (0)30 − I (0)31 (cid:17) θ u α u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) ˙ u α u [ µ ω ν ] α + (cid:16) I (0)30 − I (0)31 (cid:17) u α ˙ u [ µ ω ν ] α + I (0)31 (cid:16) ω [ νλ ∇ λ u µ ] + u [ µ ∇ λ ω ν ] λ − θ ω µν + ω [ να ∇ µ ] u α + u α ∇ [ µ ω ν ] α (cid:17)(cid:41) − m I (1)41 (cid:16) u [ µ ∇ α + u α ∇ [ µ (cid:17) ω ν ] α (cid:19)(cid:35) . (F7)Now using Eq. (81) we rewrite the above equation as u λ δS λ,µν = 4 s τ eq (cid:34) ω µν (cid:18) − I (0)10 ˙ ξ sinh ξ + 2 m I (1)41 ˙ ξ sinh ξ + I (1)30 ˙ β cosh ξ + I (1)31 β θ cosh ξ − m I (1)51 cosh ξ ˙ β − m I (1)52 β θ cosh ξ + I (0)10 ˙ ξ sinh ξ − I (0)20 ˙ β cosh ξ + I (0)10 θ cosh ξ − m I (0)31 ˙ ξ sinh ξ + 2 m I (0)41 ˙ β cosh ξ − m I (0)31 θ cosh ξ (cid:19) + 2 m ( ∇ α ξ ) u [ µ ω ν ] α (cid:18) − I (1)41 sinh ξ + n tanh ξε + P I (0)41 cosh ξ + I (0)31 sinh ξ − n tanh ξε + P I (0)41 cosh ξ (cid:19) + 2 m (cid:16) ∇ [ µ ξ (cid:17) u α ω ν ] α (cid:18) − I (1)41 sinh ξ + n tanh ξε + P I (1)51 cosh ξ + I (0)31 sinh ξ − n tanh ξε + P I (0)41 cosh ξ (cid:19) + 2 m u α u [ µ ω ν ] α (cid:18) − I (1)40 ˙ ξ sinh ξ + I (1)41 ˙ ξ sinh ξ + I (1)50 ˙ β cosh ξ + I (1)51 β θ cosh ξ − I (1)51 ˙ β cosh ξ − I (1)52 β θ cosh ξ + (cid:16) I (0)30 − I (0)31 (cid:17) ˙ ξ sinh ξ + 2 I (0)31 ˙ ξ sinh ξ − (cid:16) I (0)40 − I (0)41 (cid:17) ˙ β cosh ξ − I (0)41 ˙ β cosh ξ + (cid:16) I (0)30 − I (0)31 (cid:17) θ cosh ξ (cid:19) + 2 m (cid:16) ∇ [ µ u α + ∇ α u [ µ (cid:17) ω ν ] α (cid:18) I (1)52 β cosh ξ + I (0)31 cosh ξ (cid:19) + 2 m (cid:16) u [ µ ∇ α + u α ∇ [ µ (cid:17) ω ν ] α (cid:18) I (0)31 cosh ξ − I (1)41 cosh ξ (cid:19)(cid:35) . (F8)From this equation it can be clearly seen that the coefficient of all the tensor objects on the right-hand side cancelsout. Thus, we confirm that u λ δS λ,µν = 0 . (F9) Appendix G: Calculation of ν α , π αβ and Π By contracting Eq. (D16) with ∆ αµ , the following expression for particle diffusion current can be obtained ν α = ∆ αµ δ N µ = − τ eq ( ∇ α ξ ) cosh ξI (1)21 + 4 τ eq I (1)31 ( β ˙ u α + ∇ α β ) sinh ξ. (G1)3Using Eq. (81), the above equation can be cast in the following simpler form ν α = − τ eq ( ∇ α ξ ) (cid:34) cosh ξI (1)21 − (cid:18) n tanh ξε + P (cid:19) I (1)31 sinh ξ (cid:35) . (G2)Contracting Eq. (D23) with ∆ αβµν yields π αβ = ∆ αβµν δT µν = ∆ αβµν (cid:34) − τ eq sinh ξ (cid:16) ∇ ρ ξ + u ρ ˙ ξ (cid:17) (cid:16) I (1)30 u µ u ν u ρ + I (1)31 (∆ µν u ρ + ∆ ρµ u ν + ∆ νρ u µ ) (cid:17) + 4 τ eq cosh ξ (cid:16) β ∇ ρ u λ + βu ρ ˙ u λ + u λ ∇ ρ β + u λ u ρ ˙ β (cid:17) (cid:26) I (1)40 u λ u µ u ν u ρ + I (1)41 (cid:0) ∆ µλ u ν u ρ + ∆ νλ u µ u ρ + ∆ µν u λ u ρ + ∆ λρ u µ u ν + ∆ µρ u λ u ν + ∆ νρ u λ u µ (cid:1) + I (1)42 (cid:0) ∆ µλ ∆ νρ + ∆ µρ ∆ νλ + ∆ λρ ∆ µν (cid:1)(cid:27)(cid:35) . (G3)Doing simple algebraic manipulations where Eqs. (C1)–(C6) are used, we find π αβ = 8 τ eq cosh ξβI (1)42 σ αβ , (G4)where σ αβ = (cid:0) ∇ β u α + ∇ α u β − ∆ αβ ∇ λ u λ (cid:1) is the shear flow tensor. Thus, the bulk pressure Π can be expressedby the formulaΠ = −
13 ∆ µν δT µν = −
13 ∆ µν (cid:34) − τ eq sinh ξ (cid:16) ∇ ρ ξ + u ρ ˙ ξ (cid:17) (cid:110) I (1)30 u µ u ν u ρ + I (1)31 (∆ µν u ρ + ∆ ρµ u ν + ∆ νρ u µ ) (cid:111) + 4 τ eq cosh ξ (cid:16) β ∇ ρ u λ + βu ρ ˙ u λ + u λ ∇ ρ β + u λ u ρ ˙ β (cid:17) (cid:26) I (1)40 u λ u µ u ν u ρ + I (1)41 (cid:0) ∆ µλ u ν u ρ + ∆ νλ u µ u ρ + ∆ µν u λ u ρ + ∆ λρ u µ u ν + ∆ µρ u λ u ν + ∆ νρ u λ u µ (cid:1) + I (1)42 (cid:0) ∆ µλ ∆ νρ + ∆ µρ ∆ νλ + ∆ λρ ∆ µν (cid:1) (cid:27)(cid:35) . (G5)Using the relations defined in Eq. (C1) we obtainΠ = 4 τ eq (cid:20) I (1)31 ˙ ξ sinh ξ − cosh ξ (cid:18) I (1)41 ˙ β + 53 I (1)42 β ∇ λ u λ (cid:19)(cid:21) . (G6)Now using the recurrence relation (B14) we can write I (1)41 = I (0)31 = − β (cid:16) I (0)20 − I (0)21 (cid:17) = − β ( ε + P ) , (G7) I (1)31 = I (0)21 = − P = − n β . (G8)Substituting I (1)41 and I (1)31 from the above equations and the convective derivatives ˙ ξ and ˙ β from Eqs. (73) and (74)into Eq. (G6) the following result for the bulk pressure can be obtainedΠ = − τ eq (cid:34) n (cid:0) cosh ξ sinh ξ (cid:0) ε ( P + ε ) − n T (cid:0) P (cid:0) z + 3 (cid:1) + 3 ε (cid:1)(cid:1)(cid:1) β (cid:0) ε sinh ξ − n T cosh ξ ( P ( z + 3) + 3 ε ) (cid:1) − cosh ξβ (cid:32) n ( P + ε ) (cid:0) P cosh ξ + ε (cid:1) n T cosh ξ ( P ( z + 3) + 3 ε ) − ε sinh ξ (cid:33) + 5 β I (1)42 (cid:35) θ. (G9) [1] STAR
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