In-medium thermal conductivity and diffusion coefficients of a hot hadronic gas mixture

# In-medium thermal conductivity and diffusion coefficients of a hot hadronic gas mixture

Utsab Gangopadhyaya    Snigdha Ghosh    Sourav Sarkar Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700 064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400085, India
###### Abstract

The relativistic kinetic theory approach has been employed to study four well-known transport coefficients that characterize heat flow and diffusion for the case of a hot mixture constituting of nucleons and pions. Medium effects on the cross-section for binary collisions (,) have been taken into consideration by incorporating self-energy corrections to modify the propagator of the exchanged baryon in interaction and the and meson propagators for the case of interaction. The temperature dependence of the four coefficients have been investigated for several values of the baryon chemical potential.

## I Introduction

Ultra relativistic heavy ion collisions provide us with the unique possibility to investigate strongly interacting matter in a deconfined state with colored degrees of freedom, known as quark gluon plasma(QGP) Book (). Production and investigation of such matter has been one of the primary goals of the experiments being pursued at the RHIC at BNL and LHC at CERN. Relativistic hydrodynamics is an effective tool which has been widely used to investigate the evolution of the properties of the matter created during these collisions. This is because soon after the collision the system is believed to reach a state where the fluctuation of the thermodynamic quantities describing the system is much smaller compared to the value of the thermodynamic quantities themselves. One of the turning points in these studies has been the observation of a large elliptic flow in 200 AGeV Au+Au collisions at RHIC which led to the conclusion that the system behaves like a near perfect fluid with a small but finite value of shear viscosity to entropy density ratio (see e.g. Luzum ()). Further investigations have revealed that whereas shows a minimum near the QGP-hadron gas crossover Csernai () the bulk viscosity to entropy density ratio shows large values Karsch ().

The fact that transport coefficients not only serve to characterize the nature of matter produced but are useful as signatures of phase transition prompted a lot of activity in the field of non-equilibrium thermodynamics in recent times both above and below the QGP-hadron gas crossover temperature . These coefficients go as inputs to the dissipative hydrodynamic equations for which the second order formulation of Israel and Stewart Israel () is currently most widely used. They are calculated using either linear response theory where they appear in the form of two-point functions or in the kinetic theory approach which is more popular owing to better computational efficiency Jeon (). For QCD matter transport coefficients have been obtained using linear response as in Refs. Defu (); Moore () and Refs. Thoma (); Baym (); Kajantie (); Amy () using kinetic theory.

Quite a substantial amount of recent literature also deals with transport coefficients of hot hadronic matter. We refer to some of them in the following. Shear viscosity has been derived using a hadron resonance gas model in Ref. Noronha (); Wiranata (); Pal (); Kadam (). In the kinetic theory approach viscosities have been estimated using parametrized cross section extracted from empirical data in Refs.  Gavin (); Itakura (); Prakash (); Davesne (), or using lowest order chiral perturbation theory in Refs. Dobado (); Chen (). NJL model has been used in kinetic theory to predict viscous coefficients in Ref. Heckmann (). Again, chiral perturbation theory has been used in Ref. Lu () to calculate the bulk viscosity taking into consideration the number changing inelastic processes, and linear sigma model has been used to study the behavior of bulk viscosity around phase transition in Ref. Dobado2 ().

Compared to the viscosities, the thermal conductivity and diffusion coefficients have received much less attention. This may be due to the absence of a conserved quantum number, the baryon number being insignificantly small for systems produced at RHIC and LHC. However at FAIR energies CBMBook () or in the Beam Energy Scan (BES) program at RHIC, the baryon chemical potential is expected to be significant and baryon number will play a significant role in determining the thermal and diffusion coefficients. A careful analysis also reveals that a system in which the total number of particle is conserved can also sustain thermal conduction or diffusion if the number of particles of individual species is also conserved. Such a scenario is reached as the system expands and cools and reaches chemical freeze out as the collisions become mostly elastic.

Diffusion coefficients arise in the treatment of a multicomponent gas in addition to thermal conductivity DegrootBook (). For the pion-nucleon system under study there arise two thermal coefficients (thermal conductivity and Dufour coefficient) and two diffusion coefficients (diffusion and thermal diffusion coefficient). We have not encountered an elaborate study of the temperature and density dependence of these coefficients in the literature. Additionally we have introduced medium effects in the dynamics to obtain a more realistic description. Note that in the kinetic theory approach the dynamic input appears in the collision integral in the form of the scattering cross section. In most of the analyses employing kinetic theory to evaluate transport coefficients vacuum cross section have been employed. Since the system under study is produced during the later stages of heavy ion collisions and is presumably at a high temperature and/or density we have incorporated medium effects in the cross-section and this is a novel feature in our approach. For the case of a pion gas the medium effects on the cross-section were incorporated by means of self-energy corrections on the intermediate and meson propagators appearing in the scattering matrix elements. A significant modification of the cross-section due to medium effects resulted in a noticeable consequences on the temperature dependence of the viscosities Sukanya1 (), thermal conductivity Sukanya2 () and the relaxation time of flows Sukanya3 () for a hot pion gas. For the case of a hadronic gas mixture of nucleons and pions the in-medium cross-section obtained using a modified propagator Snigdha () resulted in significant changes in the viscous coefficients Utsab ().

In the present work we obtain the temperature and (baryon) density dependence of the thermal and diffusion coefficients of a hadronic gas mixture of pions and nucleons introducing in-medium cross-sections in the kinetic theory approach. In the next section the expressions for the coefficients are discussed followed by a section on the evaluation of in-medium cross-section using thermal field theoretic methods. Results are followed by appendices containing calculational details.

## Ii Formalism

We begin with a brief description of the hydrodynamic equations. The energy momentum tensor can be decomposed into reversible and irreversible contributions, given by

 T(0)μν = enUμUν−PΔμν T(1)μν = [(Iμq+hΔμσNσ)Uν+(Iνq+hΔνσNσ)Uμ]+Πμν , (1)

where , and are the pressure, energy density per particle and particle density respectively; is the hydrodynamic four velocity and is the projection operator. In the above equations and are the heat flow and the total particle four-flow density respectively and contains the shear and bulk part of the irreversible momentum flow. Throughout this paper the convention of metric that has been used is .

It follows from from Eq. (1) that the heat transfer can be expressed as

 Iμq=T(1)μνUν−hΔμσNσ , (2)

since and . In linear hydrodynamic theory that satisfies the second law of thermodynamics, the relation between the thermodynamic forces and the reduced heat flow and particle flow for a binary mixture at mechanical equilibrium (i.e. ) is given by Degroot1 (); Degroot2 (),

 ¯Iμ = Iμ−2∑k=1hkNμk ¯Iμ = λ▽μT+D′Tnx1T(∂μ1∂x1)P,T▽μx1 (3) Iμk = Nμk−xkNμ Iμk = DFn▽μx1+DTnx1x2▽μT (4)

where , , and are the Thermal, Dufour, Thermal diffusion and Diffusion coefficients respectively. In the above equations, , are the concentrations of first and the second species respectively, and being the particle density of the individual species.

In the transport theory approach the transfer of momentum and energy is due to collisions between the constituent particles. The diffusion flow and the irreversible part of the energy momentum tensor is non-trivial when the system is not in local equilibrium. For slight deviation from equilibrium the local distribution function can be 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) . (5)

Here is the equilibrium (Bose-Einstein or Fermi-Dirac) distribution function and represents the slight deviation from equilibrium of the particle species. The sign in the expression of denotes the Bose enhancement or Pauli blocking. The quantity parametrizes the deviation from equilibrium. Thus the heat flow and the diffusion flow are given by

 ¯Iμq = 2∑k=1∫dΓk(pνkUν−hk)pμkδfk , (6) Iμ1 = 2∑k=1∫dΓk(δ1k−xk)pμkδfk (7)

where and, . On the right side of Eq.(3) and Eq.(4) we find space derivatives of temperature and the concentration of the first species. The right hand side of Eq. (6) and Eq.(7) which refers to the same quantity as expressed in Eq.(3) and Eq.(4) involves integral over the particle three momentum of both the species. In order that these two sets of equations conform with each other must be a linear combination of the space derivative of temperature and the concentration of first species with proper coefficients Utsab (),

 ϕk=−B(k)qμ▽μTT−B(k1)μ1x2(∂μ1∂x1)P,T▽μx1 . (8)

Substituting the above expression for in Eq.(6) and Eq.(7), with the help of Eq.(5) and then comparing the coefficients of space derivate of temperature and concentration of the first species we get

 λ = (LqqT) (9) D′T = (Lq1nx2x1T) (10) D = (L11nx2)(∂μ1∂x1)P,T (11) DT = (L1qnx2x1T) , (12)

where,

 Lqq = −132∑k=1∫dΓk(pk⋅U−hk)pσkΔασB(k)qαf(0)k[1±f(0)k] (13) Lq1 = −132∑k=1∫dΓk(pk⋅U−hk)pσkΔασB(k1)αf(0)k[1±f(0)k] (14) L11 = −132∑k=1∫dΓk(δk1−xk)pσkΔασB(k1)αf(0)k[1±f(0)k] (15) L1q = −132∑k=1∫dΓk(δk1−xk)pσkΔασB(k)qαf(0)k[1±f(0)k] . (16)

To get the different coefficients we need to find , , and which can be obtained by solving the relativistic transport equation for two particle system,

 pμk∂μfk(x,p)=2∑l=1(gl1+δkl)Ckl[fk] . (17)

Here is the degeneracy of species of particle and the collision integral for binary elastic collisions is given by

 Ckl[f] = ∫∫∫dΓpldΓp′kdΓp′l[fk(x,p′k)fl(x,p′l){1+fk(x,pk)}{1+fl(x,pl)} (18)

with the interaction rate , where is the square of the c.m. energy. The collision term on the right hand side of Eq.(17) in the relaxation time approximation (RTA) Utsab (); Prakash () is expressed as the deviation of the distribution function over the thermal relaxation time which is actually a measure of the time scale for restoration of the out of equilibrium distribution to its local equilibrium value. We thus have

 2∑l=1(gl1+δkl)Ckl[fk]≃−(δfkτk)=−⎡⎣fk−f(0)kτk⎤⎦ . (19)

The relaxation time is taken as the inverse of the reaction rate of the particle the explicit expression for which will be given in the next section. Note that there are other ways of obtaining the relaxation time. Transport relaxation rates can be used besides other possible parametrizations. Moreover, though RTA offers a simple and reasonably accurate way to handle the collision kernel, for better precision the 9-dimensional collision integral given by Eq. (18) needs to be considered along with more advanced methods of simplification.

Expanding the distribution function using Chapman expansion in terms of the Knudsen number and restricting ourself to the first order, the transport equation assumes the form Utsab (); Degroot2 ().

 pμkUμDf(0)k+pμk▽μf(0)k=−⎛⎝f(1)kτk⎞⎠ , (20)

where , and . The covariant derivative on the left hand side has been split into a time-like part and a space-like part. Using the conservation equations the time derivative of the different thermodynamic parameters are converted into space derivatives of temperature and fluid velocity. The left hand side of Eq.(20) then becomes,

 f(0)k(1±f(0)k)TEk[Qk(∂νuν)−⟨pμkpνk⟩⟨∂μuν⟩+(pσkUσ)pμk(▽μTT−▽μPnh)+Tpμk▽μ(μkT)] . (21)

Here where , and ; being the enthalpy per particle belonging to species. The details of the calculation along with the expressions of ’s and ’s will be given in Appendix A. Since we intend to extract the thermal and diffusion coefficients the terms including the space derivative of the hydrodynamic velocity are ignored. Using the Gibbs-Duhem relation and (21), Eq. (20) can be written for a two component system as

 f(0)k(1+f(0)k)TEk[{pk⋅U−h+(δk1−x1)T2(∂∂T(μ1T)P,x1−∂∂T(μ2T)P,x1)}pμk(▽μTT) +(δk1−x1)x2(∂μ1∂x1)P,Tpμk▽μx1]=−(δfkτk) . (22)

The details of the above calculation are provided in Appendix B. Substituting the expression for from Eq. (5) in Eq.(22) using the expression for from Eq.(8) and comparing the coefficients of the thermodynamic forces we obtain

 B(k)qν△μν = τkEkT[pk⋅U−h+(δk1−x1)T2(∂∂T(μ1T)P,x1−∂∂T(μ2T)P,x1)]△μνpkν (23) B(k1)ν△μν = τkEkT(δk1−x1)△μνpkν . (24)

Using the above expressions in Eq.(13) to Eq.(16) we finally get

 Lqq = 13T2∑k=1gk∫d3pk(2π)3(→p2kE2k)(pk⋅U−hk)[pk.U−h+(δk1−x1)T2β]τkf(0)k(1±f(0)k) (25) Lq1 = 13T2∑k=1gk∫d3pk(2π)3(→p2kE2k)(pk⋅U−hk)(δk1−x1)τkf(0)k(1±f(0)k) (26) L11 = 13T2∑k=1gk∫d3pk(2π)3(→p2kE2k)(δk1−x1)2τkf(0)k(1±f(0)k) (27) L1q = 13T2∑k=1gk∫d3pk(2π)3(→p2kE2k)[pk⋅U−h+(δk1−x1)T2β](δk1−x1)τkf(0)k(1±f(0)k) (28)

where .The expressions for , and needed to calculate the transport coefficients have been derived in Appendix C.

## Iii Dynamical Inputs

We now focus on the calculation of the relaxation times. Inter-particle collisions are responsible for transport phenomena and so dynamical information in the transport equation enter through the scattering cross section. For the case at hand spices 1 & 2 denote the nucleon and the pion respectively. The relaxation times for nucleons and pions are coupled and involve the , and cross sections.They are respectively given by

 τ−1N = τ−1Nπ+τ−1NN τ−1π = τ−1ππ+τ−1πN (29)

where

 τ−1kl(pk) = (νl1+δkl)12Ek∫∫∫dΓpldΓp′kdΓp′l(2π)4δ4(pk+pl−p′k−p′l) (30) × |Mk+l→k+l|2×f0l(1±f′0k)(1±f′0l)(1±f0k) ,

being the amplitude for binary elastic scattering processes. A popular approach is to use phenomenological amplitudes designed to reproduce experimental data of elastic scatterings. However, the system under consideration is produced in the later stages of HICs and in presumably at a high temperature and/or baryon density where the vacuum amplitude could be modified by many body effects. A realistic treatment thus suggests the use of in-medium amplitudes. Here we adopt a dynamical approach where effective interactions are used to evaluate the hadronic scattering amplitudes. Considering resonant scattering in the -channel, the propagators in this channel are replaced by effective ones obtained by a Dyson-Schwinger sum of one-loop self energy diagrams.

Let us first consider elastic scattering. Considering with , the scattering proceeds via exchange of the baryon. The propagation of the is modified in the hot/dense medium. This is quantified through the self energy evaluated using standard methods of thermal field theory. The spin averaged expressions for the real and imaginary parts of self energy are given by,

 Re ΠΔ(q) = (31) ⎛⎝~npj+ωkiNΔij(k0i=q0−ωpj)(q0−ωpj)2−ω2ki⎞⎠−⎛⎝~npj−ωkiNΔij(k0i=q0+ωpj)(q0+ωpj)2−ω2ki⎞⎠⎤⎦

and

 Im ΠΔ(q) = −πϵ(q0)∑i∈{π,ρ} ∑j∈{N,Δ}∫d3ki(2π)314ωkiωpj× (32) [NΔij(k0i=ωki){(1+nki+−~npj+)δ(q0−ωki−ωpj)+(−nki+−~npj−)δ(q0−ωki+ωpj)}

In Eqs. (31) and (32), the distribution functions for mesons and baryons are given by and respectively with . The expressions for may be found in Ref Snigdha (). For and , the contribution to the self energy have been further folded with the vacuum spectral function of and respectively, details of which can be found in Ref. Snigdha (). The real part shifts the pole (which in the present case turns out to be very small) and the imaginary part modifies the width. The first term in Eq. (32) is the contribution from decay and formation of baryon weighted by thermal factors; the fourth term is due to scattering processes in the medium which leads to the absorption of the ; the second and third terms do not contribute for physical time-like momenta of defined in terms of and .

The decay width of is defined as,

 ΓΔ(T,μπ,μN)=−Im ΠΔ(q0=mΔ,→q=→0)mΔ . (33)

Note that in heavy ion collisions pions get out of chemical equilibrium at 170 MeV and a corresponding chemical potential starts building up with decrease in temperature. The kinetics of the gas is then dominated by elastic collisions. We take the temperature dependent pion chemical potential from Ref. Hirano () which reproduces the slope of the transverse momentum spectra of identified hadrons observed in experiments. Here, by fixing the ratio of entropy and number density to the value at chemical freeze-out where , one can go down in temperature up to the kinetic freeze-out by increasing the pion chemical potential. This provides the temperature dependence leading to whose value starts from zero at chemical freezeout and rises to a maximum at kinetic freezeout. The temperature dependence is parametrized as

 μπ(T)=a+bT+cT2+dT3

with , , , and , in MeV.

In Fig. 1 the decay width is plotted as a function of temperature for = 80 MeV and = 500 MeV. It increases as can be understood from Eq. (32) where the imaginary part of the self energy increases due to the increase of thermal distribution functions with the increase of . This physically indicates to the enhancement of scattering and decay processes HadronBook (). The effective propagator of the obtained from Eqs. (31) and (32) when used in the -channel diagram for results in a form

 ¯¯¯¯¯¯¯¯¯¯¯¯|M|2=13(fπNΔmπ)2[A|s−m2Δ−ΠΔ|2+B(u−m2Δ)2+C(u−m2Δ)|s−m2Δ−ΠΔ|] (34)

where the factors , and contain trace over Dirac matrices, vertex form factors etc. and can be read off from Snigdha (). The cross-section is then given by . The significant broadening of the width in the medium is reflected in the suppression of the cross section as can be seen in Fig. 2(a). It is to be noted that the vacuum cross section (obtained using the vacuum width of the in the propagator) is in very good agreement with the experimental data.

The cross section in the medium is obtained analogously. In this case, the scattering is assumed to be proceed via and meson exchange. Using with and  Serot:1984ey (), the spin averaged self energies of and are given by

 Re Πρ,σ(q) = ∑i∫d3kπ(2π)312ωkπωpiP[(nkπ+ωpiNπi,σ(k0π=ωkπ)(q0−ωkπ)2−ω2pi)+(nkπ−ωpiNπi,σ(k0π=−ωkπ)(q0+ωkπ)2−ω2pi) (35) +(npi+ωkπNπi,σ(k0π=q0−ωpi)(q0−ωpi)2−ω2kπ)+(npi−ωkπNπi,σ(k0π=q0+ωpi)(q0+ωpi)2−ω2kπ)]

and

 Im Πρ,σ(q) = −πϵ(q0)∑i∫d3kπ(2π)314ωkπωpi× (37) [Nπi,σ(k0π=ωkπ){(1+nkπ++npi+)δ(q0−ωkπ−ωpi)+(−nkπ++npi−)δ(q0−ωkπ+ωpi)} +Nπi,σ(k0π=−ωkπ){(−1−nkπ−−npi−)δ(q0+ωkπ+ωpi)+(nkπ−−npi+)δ(q0+ωkπ−ωpi)}],

where for the self energy and for self energy. The expressions for can be found in Ref. Ghosh:2009bt (). The in-medium decay width of meson, given by

 Γρ(T,μπ)=−Im Πρ(q0=mρ,→q=→0)mρ , (38)

has been plotted against at = 80 MeV in Fig. 1, which shows similar behavior as implying an enhancement of decay and scattering of with the increase of temperature. The in-medium cross section obtained from the matrix elements of scattering using the medium modified and propagators Sukanya2 () is shown in Fig. 2(b). The vacuum cross section being in agreement with the experimental data, we find significant suppression in cross section at high temperature sue to the broadening of and width.

## Iv Results

We first discuss the behavior of the relaxation times of pions and nucleons making up the hadron gas mixture as a function of temperature and density. These contain the dynamical information of the medium embedded in the binary collisions leading to various transport phenomena. Both and cross-sections contribute in determining the individual relaxation times of the pion and nucleon as seen from Eq. (29). In Fig. 3(a) the temperature dependence of the pion and nucleon relaxation times are shown. The curves with vacuum cross-section are of similar order of magnitude as given in Prakash (). Both the cross-sections decrease in the medium as discussed in the previous section leading to an effective increase of the relaxation times in the medium. Though at lower temperatures near kinetic freeze-out the magnitude of the nucleon relaxation time is about a factor of two lower than that of the pions, they are found to even out at higher temperatures. Shown in Fig. 3(b) is the dependence on the nucleon chemical potential. Here too the relaxation times decrease with the increase in chemical potential but the rate is much slower. The larger gap between the results with vacuum and medium cross-sections shows the dominant role played by temperature especially in the later stage of the evolution.

We now turn to the transport coefficients as defined by Eqs. (13)-(16). The species and appearing in these equations denote nucleon and pion respectively. Fig. 4(a) depicts the behavior of the thermal conductivity as a function of temperature where the pion chemical potential is taken as MeV which is its value at kinetic freezeout. The nucleon chemical potential is taken as 500 MeV. The coefficient of thermal conductivity is found to go down with the increase in temperature. The introduction of medium effect not only causes significant change in value of the thermal conductivity but also changes its behavior with temperature though it changes only slightly with the rise in temperature after 130 MeV. Fig. 4(b) shows the variation of thermal conductivity with the nucleonic chemical potential at temperature 160 MeV. In this case also we find a significant change in its value with the introduction of medium effect. Note that the increase in average energy with temperature is in fact canceled by the factor in the expression of and hence the coefficient of thermal conductivity follows the trait of relaxation time. Increase in density (chemical potential) is found to have very little effect on its behavior.

The variation of the Dufour and the Thermal diffusion coefficients with temperature are presented in Fig. 5(a) and Fig. 6(a) respectively. Here we find that the two coefficients are approximately of similar magnitude (in fact, the values would have been identical if the Chapman-Enskog formalism was used to derive the values of these coefficients physica40 (); DegrootBook ()). The introduction of medium effect has very little effect on the value of these coefficients though there is a slight increase in their magnitudes. The coefficients are found to fall sharply with the increase in temperature; the presence of in their expression causes the rate of decrease to be sharper than that of the relaxation time. Fig. 5(b) and Fig. 6(b) show the behavior of the Dufour coefficient and the thermal diffusion as a function of the nucleon chemical potential at 160 MeV. The trend follows that of the relaxation time.

In Fig. 7(a) we see that the diffusion coefficient decreases with temperature. The introduction of medium effects in the relaxation time as expected causes a surge in its value. Its variation with the nucleon chemical potential is given in Fig. 7(b). It behaves quite differently from the relaxation time of collision and its value is found to depend strongly on the density of nucleons (which goes up with the increase of nucleon chemical potential). We plot for two values of in this case to show its different behavior at lower and higher values resulting from the interplay of the two factors. At 160 MeV temperature the value of the diffusion coefficient is found to be quite steady up to 450 MeV after which it decreases slowly. In contrast, at lower temperatures (100 MeV) the value of diffusion coefficient goes up with increase in chemical potential.

## V Summary

In the present work we have estimated the temperature and (baryon) density dependence of the thermal conductivity in addition to associated thermal and diffusion coefficients for the case of a hot hadronic gas mixture composed of nucleons and pions in the kinetic theory approach. The transport equation is solved in the Chapman-Enskog approximation and the collision term is handled in the relaxation time approximation. In order to account for the hot and dense environment we have incorporated the in-medium cross-sections for and scattering obtained using effective Lagrangians in the framework of thermal field theory. It is observed that the temperature dependence of thermal conductivity is significantly affected by the medium in comparison with the other coefficients. It may have observable consequences on the evolution of the late stages of relativistic heavy ion collisions when included in hydrodynamic simulations.

## Acknowledgement

Snigdha Ghosh acknowledges Center for Nuclear Theory, Variable Energy Cyclotron Centre and Department of Atomic Energy, Government of India for support.

## Appendix A

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

 pμkUμDf(0)k+pμk▽μf(0)k=−δfkτk , (39)

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. It is to be noted that, and , i.e. the time derivative and the space derivative respectively in the local rest frame. So the above equations is written as,

 f(0)k(1±f(0)k)TEk{(pk⋅U)[pk⋅UT2DT+D(μkT)−pμkTDUμ] (40)

The thermodynamic forces do not contain terms like and ; it contains the space derivative of temperature, chemical potential and the thermodynamic velocity . In order to express the time derivative as spatial derivatives of the thermodynamic parameters we make use of the conservation equations (which are correct up to first order),

 ∂μNμk = 0 Dnk = −nk∂μUμ ∂νTμν(0) = 0 Uμ∂νT(0)μν = 0 De = −Pn∂μuμ ∑knkDek = −(∑kPk)∂μUμ

where, , and . The quantities and are the heat flow and the viscous part of the energy momentum tensor respectively. 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 ∂nN∂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)=−PδμUμ

where is the total pressure. The different thermodynamic quantities like energy density, pressure, entropy etc. of the system consisting of pions and nucleons are expressed as follows,

 nπ = gπ∫dΓπEπf(0)π(pπ)=g2π2z2πT3S12(zπ), (42) Pπ = gπ∫dΓπ→p2π3f(0)π(pπ)=g2π2z2πT4S22(zπ), (43) nπeπ = gπ∫dΓπE2πf(0)π(pπ)=gπ2π