In-medium viscous coefficients of a hot hadronic gas mixture

# In-medium viscous coefficients of a hot hadronic gas mixture

Utsab Gangopadhyaya Theoretical High Energy Physics Division, Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhannagar Kolkata - 700064, India    Snigdha Ghosh Theoretical High Energy Physics Division, Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhannagar Kolkata - 700064, India    Sukanya Mitra Indian Institute of Technology Gandhinagar, Gandhinagar-382355, Gujarat, India    Sourav Sarkar Theoretical High Energy Physics Division, Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhannagar Kolkata - 700064, India
###### Abstract

We estimate the shear and the bulk viscous coefficients for a hot hadronic gas mixture constituting of pions and nucleons. The viscosities are evaluated in the relativistic kinetic theory approach by solving the transport equation in the relaxation time approximation for binary collisions (, and ). Instead of vacuum cross-sections usually used in the literature we employ in-medium scattering amplitudes in the estimation of the relaxation times. The modified cross-sections for and scattering are obtained using one-loop modified thermal propagators for , and in the scattering amplitudes which are calculated using effective interactions. The resulting suppression of the cross sections at finite temperature and baryon density is observed to significantly affect the and dependence of the viscosities of the system.

## I Introduction

During the last few decades ultrarelativistic heavy ion collision experiments have claimed substantial attention primarily because they provide us with the opportunity to explore strongly interacting matter at high energy densities where we can expect to find coloured degrees of freedom in a deconfined state known as quark-gluon plasma (QGP) in the initial stages followed by a hot hadronic gas mixture expt (). Soon after the collision, this exotic system is believed to approach a state in which the mean value of the macroscopic quantities defining the state of the system becomes considerably larger than their fluctuations. Transport properties have long been employed as probes to understand the characteristics of such a thermodynamic system. The hydrodynamic evolution of the matter created in relativistic heavy ion collision involves different dissipative processes which can be quantified by the transport coefficients. In addition to providing relevant insight about the dynamics of the fluid they also carry information about how far the system is away from equilibrium. Recent results from RHIC have shown clear indications that the produced matter behaves more as a strongly interacting liquid than a weakly interacting gas. In fact, the STAR data on elliptic flow of charged hadrons in Au+Au collision at GeV per nucleon pair could be described (see e.g. Luzum ()) using very small but finite values of shear viscosity over entropy density ratio in their viscous hydrodynamic code. A lower bound on the value of which follows from the uncertainty principle and substantiated using ADS/CFT correspondence KSS () is consistent with the values of shear viscosity extracted from experimental results Gavin1 () and from lattice simulations Nakamura (). Moreover, considering the QCD to hadron gas transition as a cross over, shows a minimum near , the critical temperature, close to the lower bound mentioned Csernai (), whereas the bulk viscosity to entropy density ratio, shows large values around  Karsch ().

Turning out to be an useful signature of the phase transition occurring in the medium created at RHIC and LHC, the estimations of shear () and bulk () viscous coefficients have become a celebrated topic in recent times. These transport coefficients along with their temperature behavior have been studied both below and above the transition temperature . In Thoma (); Gyulassy (); Baym (); Heiselberg (); Hosoya (); AMY () the viscosities have been estimated for interacting QCD matter employing the kinetic theory approach. In JEON (); Carrington (); Basagoiti () shear viscosity has been obtained by evaluating the correlation functions in the linear response theory. The bulk viscosity has been investigated in the same spirit using Kubo formula near QCD critical point in Moore (). In Sasaki (); Lang (); Ghosh () both the shear and bulk viscous coefficients have been estimated employing the Nambu-Jona-Lasinio (NJL) model near chiral phase transition. Recent quasiparticle approaches Greco (); Chandra () and the hydrodynamic simulations in Denicol (); Ryu () have also contributed to the study of the coefficients.

In recent times a substantial amount of interest has been directed towards the analysis of viscous coefficients in the hadronic sector of heavy ion collisions as well. The hadron resonance gas model has been effectively applied in order to extract the value of shear viscosity in Noronha (); Wiranata (); Pal (); Mishra (). In Gavin (); Itakura (); Prakash (); Davesne () the viscosities have been estimated in the kinetic theory approach utilizing parameterized cross sections extracted from empirical data for a hadronic gas mixture, whereas in Purnendu (); Albright () a quasiparticle model has been used to estimate and . The NJL model also has been used to predict the viscous coefficient in the hadronic regime by Buballa () using kinetic theory approach. Scattering amplitudes evaluated using lowest order chiral perturbation theory have been employed in Dobado1 (); Cheng-Nakano () and a unitarized cross section has been used in Dobado2 () using the inverse amplitude method to obtain an estimate of . In LuMoore () the bulk viscosity of a pion gas has been computed including number-changing inelastic processes using chiral perturbation theory. In Dobado3 () the behavior of has been demonstrated around the point of phase transition using the linear sigma model.

The scattering cross-section that appears in the collision integral is the dynamical input in the kinetic theory approach and it is highly suggestive that it contains the effect of the hot and/or dense medium. For the case of a pion gas the consequences of an in-medium cross-section on the temperature dependence of the transport coefficients were extensively discussed in Mitra1 (); Mitra2 (); Mitra3 (). Using effective interactions and the techniques of thermal field theory the scattering amplitudes evaluated with self-energy corrected and meson propagators in the internal lines caused a significant modification in the cross-section and consequently the viscosities Mitra1 (), thermal conductivity Mitra2 () and relaxation times of flows Mitra3 (). In view of the upcoming CBM experiment at FAIR it is thus natural to ask how the presence of a finite baryon density is likely to affect these results. To study such effects one has to include nucleons and consequently the cross-section becomes a significant dynamical input in the study of transport phenomena in addition to the cross-section.

In this work we estimate the shear and bulk viscosities of a hot and dense gas consisting of pions and nucleons. Analogous to the and mesons mediating the interaction we consider scattering to proceed by exchange of the lightest baryon resonance, the , which is close to an ideally elastic resonance, decaying almost entirely into pions and nucleons. We obtain the self-energy at finite temperature and baryon density evaluating several one-loop diagrams with , , and in the internal lines using standard thermal field theoretic methods. The in-medium propagator of the baryon is then used in the scattering amplitudes to obtain the cross-section. The transport equations for the pion and nucleon are solved to obtain the temperature and density dependence of the shear and bulk viscosities.

It is worth emphasizing at this point that whereas vacuum scattering amplitudes have been used in almost all works dealing with transport coefficients of hadronic matter discussed above, the novelty in our approach is the use of in-medium ones. This feature attributes a realistic nature to the evaluation of these quantities and makes the formalism more complete. Moreover, the use of thermal field theoretic methods ensures that scattering and decay of excitations in the medium are included systematically in the evaluation of spectral densities which play the most significant part. Incorporating the in-medium cross-section in addition to the in the coupled transport equations for the hadronic gas mixture we thus aim to provide a more reliable estimate of the viscosities, in particular their dependence on temperature and baryon density, which when used as inputs to the hydrodynamic equations will produce a more realistic scenario of space time evolution of the later stages of heavy ion collisions.

This work is organized in the following manner. Section II deals with the formulation of the shear and bulk viscous coefficients obtained by solving the transport equation, where their collision terms are treated in relaxation time approximation. Section III discusses the in-medium modification of the propagator and its effect on the pion nucleon cross section after briefly recalling the in-medium case. The numerical results and corresponding discussions are presented in Section IV followed by a summary in section V. Some mathematical details are provided in appendices A, B and C.

## Ii Formalism

In order to describe the hot and dense matter created in heavy ion collisions we have two Lorentz-covariant dynamical frameworks at our disposal; covariant transport theory and relativistic hydrodynamics. While the evolution of macroscopic parameters related to transport processes involving thermodynamic forces and fluxes can be studied using hydrodynamics, the microscopic collisions between constituents giving rise to dissipative phenomena is described by the transport theory. In linear hydrodynamic theory that satisfies the second law of thermodynamics, the relation between the thermodynamic forces and the corresponding fluxes is given by Weinberg2 (); Degroot (),

 Tμν=enuμuν−PΔμν+Πμν Πμν=2η⟨∂μuν⟩+ζ(∂⋅u)Δμν (1)

where , and are the pressure, energy density per particle and particle density respectively. and are the coefficients of shear and bulk viscosities, is the hydrodynamic four velocity, is the projection operator and denotes symmetric traceless combination. Throughout this paper the convention of metric that has been used is .

At the microscopic level energy and momentum are carried by the constituent particles and their exchange occurs due to the flow and collision of the particles. Viscous forces appear when the system is away from local equilibrium and work to bring the system back to equilibrium. The correspondence between kinetic theory and viscous hydrodynamics can thus be established by considering small deviation from equilibrium for which the distribution function is expressed as,

 fk(x,p)=f(0)k(x,p)+δfk(x,p),    δfk(x,p)=f(0)k(x,p)[1±f(0)k(x,p)]ϕk(x,p) (2)

where the equilibrium distribution function as a function of position and momentum is given by

 f(0)k(x,p)=[expp⋅u(x)−μk(x)T(x)−σ]−1 (3)

with , and representing the local temperature, flow velocity and chemical potential respectively. For bosonic (fermionic) degree of freedom . The quantity parametrizes the deviation of the distribution function of the particle specie from equilibrium. The sign in the expression of denotes Bose enhancement or Pauli blocking. The viscous part of the energy momentum tensor is then given by

 Πμν=N∑k=1∫dΓkΔμσΔντpσkpτk δfk. (4)

where is the number of particle species and with . On the right hand side of eq. (1) the thermodynamic forces appear with different tensorial ranks involving the space derivative of the hydrodynamic four velocity. The right hand side of eq. (4) involves integration over the particles’ three momenta and in order that it conforms to the form of as expressed in eq. (1), must be a linear combination of the thermodynamic forces with proper coefficients and appropriate tensorial ranks. Consequently is expressed as.

 ϕk=Ak∂⋅u−Cμνk⟨∂μuν⟩ , (5)

where and are the unknown coefficients needed to be determined. It is convenient to decompose into a traceless part and a remainder as

 Πμν=˚Πμν+ΠΔμν (6)

where the viscous pressure is defined as one third of the trace of the viscous pressure tensor,

 Π=N∑k=013∫dΓkΔστpσkpτk δfk. (7)

So the traceless part of viscous trace tensor comes out to be,

 ˚Πμν = Πμν−ΠΔμν (8) = N∑k=0∫dΓk{ΔμσΔντ−13ΔστΔμν}pσkpτk δfk.

Substituting the expression for from (2) and (5) in eqs. (7) and (8) and comparing the coefficients with the same tensorial ranks in (1) the expressions for shear and bulk viscosity turn out respectively as

 η=−N∑k=1110∫dΓk⟨pkμpkν⟩f(0)k(1±f(0)k)Cμνk (9)

and

 ζ=N∑k=113∫dΓkΔμνpμkpνk f(0)k(1±f(0)k) Ak . (10)

In order to obtain the shear and bulk viscous coefficients we need to find and for which we turn to the relativistic transport equation for a multi-particle system. This is given by

 pμk∂μfk(x,p)=N∑l=1gl1+δklCkl[fk], (11)

where in the degeneracy of particle and the collision integral on the right hand side for binary elastic collisions is

 Ckl[fk] = ∫dΓpl dΓp′k dΓp′l[fk(x,p′k)fl(x,p′l){1±fk(x,pk)}{1±fl(x,pl)} (12) −fk(x,pk)fl(x,pl){1±fk(x,p′k)}{1±fl(x,p′l)}] Wkl .

The dynamical input which goes into the determination of the distribution function appears in the interaction rate .

The distribution function of the particles may be expanded in a series, in powers of the non-uniformity parameter (or Knudsen number, which acts as a book-keeping factor) as, and substituted in the multi-particle transport equation. Restricting to the first order the transport equation becomes Degroot (),

 pμuμDf(0)k+pμ∇μf(0)k=N∑l=1gl1+δklCkl[f(1)k] (13)

where the derivative on the left hand side is separated into a time-like and a space-like part using where and . Using the conservation equations the time derivative is replaced with space derivatives of the fluid four-velocity so that the left hand side of eq. (13) becomes

 1Tf(0)k(1±f(0)k)[Qk∂⋅u−⟨pμkpνk⟩⟨∂μuν⟩] (14)

where , , and with is the enthalpy per particle belonging to species. The details of the calculation along with the expressions of ’s and ’s are given in Appendix-B. The bulk and shear viscous forces are represented by the first and second terms respectively in (14) where we have ignored terms related to thermal conductivity and diffusion.

The term on the right hand side of (11) is now simplified by assuming that all particles involved in the scattering process except the particle with momentum is in equilibrium. This is the so-called relaxation time approximation. In the collision integral given by eq. (12) we thus use and in place of and respectively to get

 N∑l=1gl1+δklCkl[fk]=−δfkτkEk (15)

where the relaxation time is given by

 [τk(pk)]−1= N∑l=1[τkl(pk)]−1 (16)

with

 [τkl(pk)]−1=gl1+δklcsh(ϵk/2)Ek∫dωldω′kdω′lWkl (17)

where , and the function
if represents a fermion and if is a boson.

The transport equation (11) in the relaxation time approximation thus takes the form

 1TEk[Qk∂⋅u−⟨pμkpνk⟩⟨∂μuν⟩]=−ϕkτk (18)

where use has been made of (2). Substituting the expression of from eq. (5) in eq. (18) and comparing the coefficients of independent thermodynamic forces on both sides, we obtain the coefficients and in terms of ,

 Ak= −τkTEkQk , (19) Cμνk= −τkTEk⟨pμkpνk⟩ . (20)

From eq. (9) and (10), using the conditions mentioned in Appendix-C, we finally arrive at the expressions of shear and bulk viscosity,

 η= 115TN∑k=1∫d3pk(2π)3τkE2k|→pk|4f(0)k(1±f(0)k) , (21) ζ= 1TN∑k=1∫d3pk(2π)3τkE2k{Q2k}f(0)k(1±f(0)k) . (22)

## Iii Dynamical Inputs

We now specialize to the case of a pion nucleon gas mixture so that the index . The relaxation times of pions and nucleons are thus respectively given by

 τ−1π = τ−1ππ+τ−1πN τ−1N = τ−1πN+τ−1NN . (23)

Here we aim to demonstrate the effects of the in-medium cross-sections on the viscosities. Correspondingly, we will first obtain the and cross-sections in the medium. The case of has been discussed extensively in Mitra1 (). We however recall the main results for completeness. We assume the scattering to proceed via and meson exchange. Using the interaction with and , the matrix elements in the isoscalar and isovector channels are given by

 MI=0 = 2g2ρ[s−ut−m2ρ+s−tu−m2ρ] + g2σm2σ[3s−m2σ+Σσ+1t−m2σ+1u−m2σ] MI=1 = g2ρ[2(t−u)s−m2ρ+Σρ+t−su−m2ρ−u−st−m2ρ] (24) + g2σm2σ[1t−m2σ−1u−m2σ] .

where the corresponding -channel propagators have been replaced by effective ones obtained by a Dyson-Schwinger sum of one-loop self-energy diagrams in vacuum. The cross-section is given by where the isospin averaged amplitude is defined as . As seen in fig. 1 the cross-section agrees fairly well with the experimental values up to about 1 GeV. In the medium the vacuum self energies and are replaced with in-medium ones evaluated using thermal field theory Sourav_RT (); Bellac (). For the meson only the loop graph is evaluated in the medium whereas in case of the meson in addition to the loop diagram, , , self-energy diagrams are included Sabya (). The imaginary part of the self-energy is given by

 ImΣ(q0,→q)=−π∫d3k(2π)34ωπωh× [L1(1+n+(ωπ)+n+(ωh))δ(q0−ωπ−ωh) +L2(n−(ωπ)−n+(ωh))δ(q0+ωπ−ωh)] (25)

where is the Bose distribution function with arguments and . The terms and arise from factors coming from the vertex etc, details of which can be found in Sabya (). The angular integration is done using the -functions which define the kinematic domains for occurrence of scattering and decay processes leading to loss or gain of (or ) mesons in the medium. The term with arises from the unitary cut and corresponds to formation and decay in the medium weighted by Bose enhancement factors and the second term corresponds to the so-called Landau cut contribution arising from resonant scattering in the medium. To account for the substantial and branching ratios of some of the unstable particles in the loop the self-energy function is convoluted with their spectral functions Sabya (). The increase of the imaginary part for 130 and 160 MeV is manifested in a suppression of the magnitude of the cross-section as seen in fig. 1.

The case of scattering is treated analogously. It is taken to proceed via the exchange of the -baryon which is the lightest baryon resonance. Despite being relatively broad, it is well separated from other resonances. We use the well-known interaction with to evaluate the scattering matrix elements. Averaging over isospin, the squared invariant amplitude for the process is given by

 ¯¯¯¯¯¯¯¯¯¯¯¯|M|2 = 13(fπNΔmπ)4⎡⎣F4(k,p)Ts∣∣s−m2Δ−Π∣∣2+F4(k,p′)Tu(u−m2Δ)2 (26) +2F2(k,p)F2(k,p′)Tm(s−m2Δ−ReΠ)3(u−m2Δ)∣∣s−m2Δ−Π∣∣2⎤⎦

where , and stand for

 Ts = Tr[(p′+mN)Ds(p+mN)γ0D†uγ0] (27) Tu = Tr[(p′+mN)Du(p+mN)γ0D†uγ0] (28) Tm = Tr[(p′+mN)Ds(p+mN)γ0D†uγ0] (29)

in which

 Ds = kαk′βOβνΣμν(qs)Oμα (30) Du = k′αkβOβνΣμν(qu)Oμα (31)

and

 Σαβ(q)=(q+mq)[−gαβ+13m2qqαqβ+13γαγβ+13mq(γαqβ−γβqα)] .

At each vertex we consider the form factor Snigdha ()

 F(p,k)=Λ2Λ2+(p⋅kmp)2−k2 (32)

in which and denote the momenta of the fermion and boson respectively. The cut-off is taken as MeV Snigdha (). As seen from fig. 2 the one-loop self-energy in vacuum produces a good description to the experimental scattering cross-section. At finite temperature we consider additional contributions coming from on-shell particles in the medium by evaluating , , and self-energies using the real time method. The expression for the spin averaged imaginary part of self-energy is given by

 ImΠ= −π∫d3k(2π)314ωkωp (33) [N1(1+n+(ωk)−~n+(ωp))δ(q0−ωk−ωp) +N2(n−(ωk)+~n+(ωp))δ(q0+ωk−ωp)]

where the distribution function for the fermions is given by . The expressions for and for the different loops may be found in Snigdha (). As before the first term is the contribution from decay and formation of the baryon weighted by thermal factors. The second term is a result of scattering processes in the medium leading to the absorption of the . These processes contribute significantly to the imaginary part which is reflected as a suppression of the cross-section. The suppression increases with increasing temperature and chemical potential as seen in fig. 2.

## Iv Results

In this section we show how the medium modified and cross-sections discussed above are reflected in the relaxation times and consequently in the viscosities of the system. We will show results for the temperature range 100 to 160 MeV which is typical of a hadron gas produced in the later stages of heavy ion collisions between kinetic and chemical freeze-out. Accordingly, we consider a non-zero value of the pion chemical potential Bebbie () in addition to the chemical potential for nucleons.

We plot in fig. 3 the relaxation time of pions in the hadron gas consisting of pions and nucleons as a function of temperature for 200 and 500 MeV and 80 MeV Hirano (). The order of magnitude of along with its decreasing trend with increasing temperature is consistent with Prakash (). As can be seen from (23) it’s magnitude is decided by both and cross-sections. As discussed above, both of these decrease in the medium causing the in-medium relaxation time of pions to effectively increase.

Similar features are observed in fig. 4 where the relaxation time of nucleons is plotted as a function of . The medium effect arising from the reduced cross-section because of the additional scattering and decay processes in this case cause an increase in the nucleon relaxation time.

We now look into the behavior of the viscous coefficients for different values of parameters evaluated numerically. The shear viscous coefficient as a function of temperature is depicted in fig. 5. The lower set of curves are the ones evaluated using the vacuum cross sections, while the upper set is evaluated taking into consideration the medium effect on the and cross sections. The different curves in each set correspond to different values of the nucleon chemical potential while the pion chemical potential is taken to be 80 MeV. Both in vacuum and medium the shear viscosity appears to increase with increasing and is the result of interplay of various factors. This was already noted in Itakura () where by means of a simplified estimate of the viscosity of the mixture this feature could be understood as resulting from an enhancement of the nucleon component with increasing . Moreover, a considerable change in the value of the shear viscous coefficient is seen due to the introduction of the medium effect. The decrease of the in-medium cross-section with increasing and results in an increase in the relaxation time and hence the viscosity as shown by the upper set of curves.

We also find a significant influence of the medium on the kinematic shear viscosity (i.e,; is the entropy density of the system). Fig. 6 depicts the variation of kinematic viscosity with temperature for nucleon chemical potential MeV and MeV represented by lines with and without symbols respectively. The lower curve in each set corresponds to vacuum cross section while the higher one corresponds to the one calculated taking into account the medium effect for the same nucleon chemical potential. The monotonous decrease in with may be attributed to the increase in the entropy density with  Itakura (). The decrease however, respects the lower bound KSS () around the transition temperature.

Now we turn to the bulk viscous coefficient whose temperature dependence is shown in fig. 7. Proceeding from the bottom of the diagram each pair of graphs corresponds to a different value of nucleon chemical potential. The lower curve for each pair corresponds to the one where vacuum cross section has been used to calculate the relaxation time while the upper curve corresponds to the one where the medium effects have been taken into consideration. Although the magnitude of bulk viscosity appears to be much smaller that that of shear viscosity the effects of the thermal medium are quite visible in this case too. In fig. 8 the variation of the kinematic bulk viscosity (i.e, ) has been studied where the lines correspond to the same combination of parameters as in fig. 6. The dependence with and in vacuum and medium show similar features as the shear viscosity. Significant difference is observed between the magnitude of viscosities calculated with vacuum and medium cross-sections.

## V Summary

In this work we evaluate the shear and bulk viscosities of a hadronic gas mixture consisting of pions and nucleons. We aim to study the difference caused by the use of in-medium cross-sections in the transport equations. Analogous to the scattering amplitudes which have been modified through self-energy corrections of the exchanged and mesons, the interaction at finite temperature and baryon density has been obtained by means of loop corrections to the propagator using the techniques of thermal field theory. Because of a significant contribution to the imaginary part basically coming from (resonant) scatterings of the exchanged baryon in the medium there is a significant suppression of the cross-section around the peak position. This causes the relaxation times to increase with respect to their vacuum values which in turn bring in a quantitative change in the viscosities of the system. With such input in the viscous hydrodynamic simulations, the space-time evolution of the later stages of heavy ion collisions might be observably affected.

## Appendix-A

The quantities like energy density, pressure, entropy etc. of the system consisting of pions and nucleons are expressed in terms of and , where and . They appear as

 nπ = gπ∫dΓpπEpπf(0)π(pπ)=gπ2π2z2πT3S12(zπ), Pπ = gπ∫dΓpπ→p2π3f(0)π(pπ)=gπ2π2z2πT4S22(zπ), nπeπ = gπ∫dΓpπE2pπf(0)π(pπ)=gπ2π2z2πT4[zπS13(zπ)−S22(zπ)] nπhπ = nπzπTS13(zπ)S12(zπ) , (34)

where , and . Using the formula the function has been expanded, hence the integrals was expanded to a sum of integral which can be compactly expressed as , denoting the modified Bessel function of order given by

 Kn(x)=2nn!(2n)! xn∫∞x dτ(τ2−x2)n−12e−τ (35)

or

 Kn(x)=2nn!(2n−1)(2n)! xn∫∞x τdτ(τ2−x2)n−32e−τ . (36)

In the corresponding expressions for ,, etc, the will be replaced by defined as .

## Appendix-B

The left hand side of the linearized transport equation for each species

 pμ∂μf(0)k=pμuμDf(0)k+pμ∇μf(0)k=−δfkτkEk. (37)

is to be expressed in terms of thermodynamic forces. In order to do this the derivative on the left is expressed in terms of the derivatives of the thermodynamic parameters getting

 (pk⋅u)[pk⋅uT2DT+D(μkT)−pμkTDuμ]+pμ[pk⋅uT2∇μT+∇μ(μkT)−pνkT∇μuν]=−δfkτkEk . (38)

Note and which are the time derivative and the space derivative respectively in the local rest frame. The thermodynamic forces do not contain terms like and . Rather, it contains space derivatives of temperature, chemical potential and the thermodynamic velocity . In order to express the time derivative in terms of space derivative of the thermodynamic parameters we make use of the conservation equations which are given up to first order by

 ∂μNμk=0 , Dnk=−nk∂μuμ . (39)

and

 ∂νTμν(0)=0 ,                         uμ∂νT(0)μν=0 De=−Pn∂μuμ ,        ∑knkDek=−(∑kPk)∂μuμ (40)

where and ; . The quantities and are the heat flow and the viscous part of the energy momentum tensor respectively. Note that we have used Eckart’s definition of flow velocity of the fluid.

Expanding the equations in terms of derivative of temperature and chemical potential over temperature we get,

 ∂nπ∂TDT+∂nπ∂(μπ/T)D(μπT)+0⋅D(μNT)=−nπ∂μuμ ∂nπ∂TDT+0⋅D(μπT)+∂nN∂(μN/T)D(μNT)=−nN∂μuμ (41) [nπ∂eπ∂T+nN∂eN∂T]DT+nπ∂eπ∂(μπ/T)D(μπT)+nN∂eN∂(μN/T)D(μNT)=−P∂μuμ (42)

where is the total pressure. Using the relations from Appendix A we get

 ∂eπ∂T = 4zπS13S12+zπS22S03(S12)2−S22S12+z2π[S02S12−S13S03(S12)2] ∂eπ∂(μπ/T) = −T[1−S22S02(S12)2]+Tzπ[S03S12−S13S02(S12)2] . ∂nπ∂T = 4π(2π)2T2[−z2πS12+z3πS03] ∂nπ∂(μπ/T) = 4π(2π)3 z2πT3S02 . (43)

Putting in eq. (42) and solving for , and we get111To get the expression for the derivative of and , and are to be replaced by and respectively.

 T−1DT = (1−γ′)∂μuμ (44) TD(μπT) = T[(γ′′π−1)^hπ−γ′′′π]∂μuμ (45) TD(μNT) = T[(γ′′N−1)^hN−γ′′′N]∂μuμ (46)

where,

 +gN[z3N(4S02