Thermal Relaxation, Electrical Conductivity and Charge Diffusion in a Hot QCD Medium

# Thermal Relaxation, Electrical Conductivity and Charge Diffusion in a Hot QCD Medium

## Abstract

The response of electromagnetic (EM) fields that are produced in non-central heavy-ion collisions to electromagnetically charged quark gluon plasma can be understood in terms of charge transport and charge diffusion in the hot QCD medium. This article presents a perspective on these processes by investigating the temperature behavior of the related transport coefficients, viz. electrical conductivity and the charge diffusion coefficients along with charge susceptibility. In the process of estimating them, thermal relaxation times for quarks and gluons have been determined first. These transport coefficients have been studied by solving the relativistic transport equation in the Chapman-Enskog method. For the analysis, , quark-quark, quark-gluon and gluon-gluon scattering processes are taken into account along with an effective description of hot QCD Equations of state (EOSs) in terms of temperature dependent effective fugacities of quasi-quarks (anti-quarks) and quasi-gluons. Both improved perturbative hot QCD EOSs at high temperature and a lattice QCD EOS are included for the analysis. The hot QCD medium effects entering through the quasi-particle momentum distributions along with an effective coupling, are seen to have significant impact on the temperature behavior of these transport parameters along with the thermal relaxation times for the quasi-gluons and quasi-quarks.

Keywords: Electrical conductivity of hot QCD, Quark-Gluon-Plasma, Heavy-ion collisions, Thermal relaxation times, Charge diffusion, Charge susceptibility

PACS: 12.38.Mh, 13.40.-f, 05.20.Dd, 25.75.-q

## I Introduction

Quantum chromodynamics(QCD)–the underlying theory of strong interaction in nature, predicts a deconfined state of the nuclear matter at high temperature (higher than QCD transition temperature ). Relativistic heavy-ion collision experiments at RHIC, BNL and LHC, CERN have reported the presence of near perfect liquid like hot nuclear matter (1); (2) which turns out to be strongly coupled quark-gluon plasma (QGP). The QGP possess a very tiny value of the shear viscosity to entropy density ratio, (a few times the KSS bound (3)). The has a lower bound near the transition temperature as shown by several studies on QCD matter based on various approaches and the shear viscosity to entropy density ratio for the QGP is found to be lowest among all the known fluids (4); (5). In contrast, the bulk viscosity to entropy density ratio, shows an upper bound with large values (6).

There has been growing interest in understanding the impact of strong electro-magnetic (EM) fields that are produced during the initial stages of the non-central heavy ion collisions (7), while investigating the hadronic observables at the later stages of the collisions. The impact of the EM field will be certainly dependent on the strength of the fields at the later stages as they seen to decay quite rapidly (8). The response of such EM fields (electrical) to the electromagnetically charged QGP can be understood in terms of the electrical conductivity, which characterizes the transport of the conserved charge in the presence of the gradient of a charge chemical potential. There are several recent attempts to understand the in the context of EM field in RHIC (8); (9); (10); (11). The electrical conductivity in the context of charge fluctuations in heavy-ion collisions is investigated in (12). Moreover, there are recent proposals to extract the electrical conductivity from the flow parameters in heavy ion collisions (13).

The other relevant, associated physical process is the charge diffusion, that is being quantified by the charge diffusion coefficient, . The electrical conductivity, and the diffusion coefficient are related by the famous Einstein relation through the charge susceptibility, .

The prime focus of this work is to estimate all these quantities for a hot QCD/QGP medium (their temperature dependence) which is characterized by an effective quasi-particle model. At this juncture, one needs to demarcate between the comoving frames while modeling expanding QGP medium, one with moving charge density and other one with energy density. This requires the introduction of thermal conductivity which characterizes the flow associated with the transport of energy in response to the temperature gradient relative to locally comoving frames with the charge density. There are some recent attempts in the direction  (14); (15), however, our work in this manuscript does not involve any such investigation. As there are two equivalent approaches to estimate the transport coefficients of the hot QCD/QGP medium, viz., the linear response theory where one could relate them to the current-current spectral functions in thermal equilibrium through the Green-Kubo formulae (16), the other one is to solve a linearized transport equation in the presence of electric field along with an appropriate collision term (again linearized) and invoke pertinent equations of motion (for example Maxwell’s equations in the case of electrical conductivity and electrical permittivities). The former approach best suited to lattice QCD estimations of the electrical conductivity and charge diffusion coefficient. There are several attempts from lattice QCD side to obtain the temperature dependence of the conductivity and charge diffusion coefficient (17); (18); (19); (20); (22); (23); (24) along with charge susceptibility (21). The present work follows the latter approach based on the linearized transport equation.

There are a number of estimations of electrical conductivity () by different approaches available in current literatures. In Ref.  (25); (26); (27) the electrical conductivity has been estimated by solving the relativistic transport equation. In Ref.  (28); (29) the has been studied using off-shell parton-hadron string dynamics transport approach for an interacting system. In Ref. (30) as a function of temperature has been estimated using the maximum entropy method (MEM). Electrical conductivity along with diffusion coefficient and charge susceptibility has been estimated employing holographic technique in Ref.  (31). Recently the electrical conductivity has been studied including the momentum space anisotropy also  (32). In the hadronic sector also these quantities have been investigated lately. In Ref. (33) the electrical conductivity has been evaluated for a pion gas where in Ref. (34) the same has been studied for hot hadron gas.

While setting up an appropriate transport equation with a collision term for the determination of and for the QGP medium, one must have a reliable modeling of the equilibrium state of the medium. To that end, quasi-particle descriptions of hot QCD medium play an important role. We employ such a model which is based on mapping the hot QCD medium effects encoded in the equation of state to the non-interacting/weakly interacting quasi-parton degrees of freedom with temperature dependent effective fugacity parameter (35); (36). Further, the model can be understood in terms of renormalization of charges of quasi-partons in the hot QCD medium. This enables us to define an effective coupling constant in hot QCD medium. The effective coupling thus obtained is employed in our analysis.

The manuscript is organized as follows. Section II, offers mathematical formalism to determine thermal relaxation times for quasi-partons (quasi-quarks/antiquarks and quasi-gluons) followed by the analytical estimations of the transport coefficients, , and . Section III, deals with important predictions of the transport coefficients mentioned and related discussions. Finally, in Section IV, the conclusions and outlook of the work are presented.

## Ii Formalism

### ii.1 Quasi-particle description of hot QCD medium

Realization of hot QCD medium effects in terms of effective quasi-particle models has been there since the last few decades. In fact, there are various quasi-particle descriptions viz., effective mass models (45); (46), effective mass models with Polyakov loop (47), NJL (Nambu Jona Lasinio) and PNJL (Polyakov loop extended Nambu Jona Lasinio) based effective models (54), and effective fugacity quasi-particle description of hot QCD (EQPM) (35); (36). The present analysis considers the EQPM (Effective quasiparticle model)for the investigations on the properties of hot and dense medium in RHIC.

There are a number of estimations for different transport coefficients available in current literature which employ various quasiparticle models (37); (38); (39). In Ref.  (37), and have been evaluated for pure gluon plasma employing the effective mass quasi-particle model. On the other hand, in Refs.  (38); (39), and are obtained in gluonic as well as matter sector. Refs.  (40); (41), presented the quasi-particle theory of and and their estimations for the hadronic sector. The thermal conductivity has also been studied, in addition to the viscosity parameters (41), within the effective mass model. In Ref. (42), the ratio of electrical conductivity to shear viscosity has been investigated within the framework of quasiparticle approach as well. However, these model calculations are not able to exactly reproduce the shear and bulk viscosities phenomenologically extracted from the hydrodynamic simulations of the QGP (43); (44), consistently agreeing with different experimental observables measured like the multiplicity, transverse momentum spectra and the integrated flow harmonics of charged hadrons. Nevertheless, these quasiparticle approached could be useful in the equilibrium modeling of the hot QCD/QGP. The predictions based on these model are still useful in the sense of estimating some possible values these transport coefficients from some theoretical models that can considerably describe the interacting system created in heavy ion collisions.

#### The EQPM

The EQPM employed here, models the hot QCD in terms of effective quasi-partons (quasi-gluons, quasi-quarks/antiquarks). The model is based on the idea of mapping the hot QCD medium effects present in the equations of state (EOSs) either computed within improved perturbative QCD or lattice QCD simulations, into the effective equilibrium distribution functions for the quasi-partons. The EQPM for the QCD EOS at (EOS1) and (EOS2) have been considered here. Additionally, the EQPM for the recent (2+1)-flavor lattice QCD EoS (48) at physical quark masses (LEOS), has been employed for our analysis. There are more recent lattice results with the improved actions and refined lattices (49), for which we need to re-look the model with specific set of lattice data specially to define the effective gluonic degrees of freedom. Therefore, we will stick to the set of lattice data utilized in the model described in Ref. (36).

In either of the cases of above mentioned EOSs, form of the quasi-parton equilibrium distribution functions, (describing the strong interaction effects in terms of effective fugacities ) can be written as.

 fg/q=zg/qexp[−βEp](1∓zg/qexp[−βEp]) (1)

where for the gluons and for the quark degrees of freedom ( denotes the mass of the quarks). and denotes inverse of the temperature. We use the notation for gluonic degrees of freedom, for with number of flavors. As we are working at zero baryon chemical potential, therefore quark and antiquark distribution functions are the same. Since the model is valid in the deconfined phase of QCD (beyond ), therefore, the mass of the light quarks can be neglected as compared to the temperature. As QCD is a gauge theory so for our analysis. Noteworthily, the EOS1 which is fully perturbative, is proposed by Arnold and Zhai (50) and Zhai and Kastening (51). On the other hand, EOS2 which is at is determined by Kajantie et al. (52) while incorporating contributions from non-perturbative scales such as and . Notably, these effective fugacities () are not merely temperature dependent parameters that encode the hot QCD medium effects; they lead to non-trivial dispersion relation both in the gluonic and quark sectors as,

 ωg/q=Ep+T2∂Tln(zg/q), (2)

where denote the quasi-gluon and quasi-quark dispersions (single particle energy) respectively. The second term in the right-hand side of Eq. 2, encodes the effects from collective excitations of the quasi-partons.

The effective fugacities, are not related with any conserved number current in the hot QCD medium. They have been merely introduced to encode the hot QCD medium effects in the EQPM. The physical interpretation of and emerges from the above mentioned non-trivial dispersion relations. The modified part of the energy dispersions in Eq. (2) leads to the trace anomaly (interaction measure) in hot QCD and takes care of the thermodynamic consistency condition. It is straightforward to compute, gluon and quark number densities and all the thermodynamic quantities such as energy density, entropy, enthalpy etc. by realizing hot QCD medium in terms of an effective Grand canonical system (35); (36). Furthermore, these effective fugacities lead to a very simple interpretation of hot QCD medium effects in terms of an effective Virial expansion. Note that scales with , where is the QCD transition temperature.

The number densities, (for gluons), (for quarks, antiquarks) are obtained from Eq. (1) as,

 ng = νg(2π)3∫d3→pfg(→p) (3) = νgT3π2PolyLog[3,zg],
 nq = νq(2π)3∫d3→pfq(→p) (4) = −νqT3π2PolyLog[3,−zq].

The number densities approach to their SB (Stefan Boltzmann) limit only asymptotically (i.e ). On the other hand, the pressure, , Energy density, can be obtained from the relation:

 Pg,q = ∓νg,q∫d3p(2π)3ln(1∓zg,qexp(−βEp)) (5) = ±νg,qT4π2PolyLog[4,±zg,q],
 ϵg,q = νg,q∫d3p(2π)3ωg,qfg,q (6) = ±3νg,qT4π2PolyLog[4,±zg,q] ±T4νg,qπ2T∂Tln(zg,q)PolyLog[3,±zg,q].

The first term in the right-hand side of Eq. (6) is nothing but the , while second term leads to non-vanishing interaction measure in hot QCD. The entropy density and enthalpy can be read off from the expressions of and using well known thermodynamic relations. The energy density and enthalpy density per particle can easily obtained employing results from Eqs.(3-6).

#### Charge renormalization and effective coupling

In contrast to the effective mass models where the effective mass is motivated from the mass renormalization in the hot QCD medium, the EQPM is based on the charge renormalization in high temperature QCD.

To investigate how the quasi partonic charges modify in the presence of hot QCD medium, we consider the expression for the Debye mass derived in semi-classical transport theory (55); (56); (57) as,

 m2D = 4παs(T)(−2Nc∫d3p(2π)3∂pfg(→p) (7) + 2Nf∫d3p(2π)3∂pfq(→p)),

where, is the QCD running coupling constant at finite temperature (53).

After performing the momentum integral after substituting the quasi-parton distribution function from Eq. (1) to Eq. (7), we obtain,

 m2D = 4παs(T)T2(2Ncπ2PolyLog[2,zg] (8) − 2Nfπ2PolyLog[2,−zq]).

The Debye mass here reduces to the leading order HTL expression in the limit (ideal EoS: noninteracting of ultra relativistic quarks and gluons),

 m2D(HTL)=αs(T) T2(Nc3+Nf6). (9)

Eq. 8 can be rewritten as,

 m2D = m2D(HTL) ×2Ncπ2PolyLog[2,zg]−2Nfπ2PolyLog[2,−zq]Nc3+Nf6.

We can now define the effective coupling, , so that the . The function reads,

 g(zg,zq)=2Ncπ2PolyLog[2,zg]−2Nfπ2PolyLog[2,−zq]Nc3+Nf6 (11)

Notably, the EQPM employed here has been remarkably useful in understanding the bulk and the transport properties of the QGP in heavy-ion collisions (38); (39); (58); (59); (60).

The behavior of the ratio as a function of temperature () for various EOSs is depicted in Fig. 1. The flavor dependence is also shown in Fig. 1. Clearly the ratio will approach to its value with ideal EOS () which is unity, only asymptotically. The EOS dependence can clearly be visualized from the temperature dependence of the relative coupling in Fig. 1. For example the LEOS result is closest to the running among all the cases. Similiary, other EOS dependent predictions can also be explicated.

There are only two free functions (, and ) in the EQPM employed here which depend on the chosen EOS. In the case of EOS1 and EOS2 employed in the present case, these functions are obtained in (35) and are continuos functions of . On the other hand, for LEOS they are defined in terms of eight parameters obtained in Ref. (36) (See Table I of Ref. (36)). Apart from that effective coupling mentioned above depends on the them and the QCD running coupling constant , that explicitly depends upon how we fix the QCD renormalization scale at finite temperature and up to what order we define . Henceforth, these are the three quantities that need to be supplied throughout the analysis here.

### ii.2 Thermal Relaxation times

In order to estimate the relaxation times of particles due to their mutual interactions we start with the Boltzmann transport equation for an out of equilibrium system that describes the binary elastic process ,

 dfk(x,pk)dt=−C[fk] . (12)

Here is the single particle distribution function for the species in a multicomponent system, that depends upon the particle 4-momentum and 4-space-time coordinates . denotes the collision term that quantifies the rate of change of given in the following manner  (61),

 C[fk]= 12νlN∑l=112ωk∫dΓpldΓp′kdΓp′l(2π)4 × δ4(pk+pl−p′k−p′l)⟨|Mk+l→k+l|2⟩ (13) ×[fk(pk)fl(pl){1±fk(p′k)}{1±fl(p′l)} −fk(p′k)fl(p′l){1±fk(pk)}{1±fl(pl)}] k=1,2,......,N.

The phase space factor is expressed by the notation , as is the energy of the scattered particle which is of species. The overall factor is appearing due to the symmetry in order to compensate for the double counting of final states that occurs by interchanging and . is the degeneracy of particle that belongs to species. It is considered next that the out of equilibrium distribution function of the particle, which is being scattered is given by,

 fk=f0k+δfk=f0k+f0k(1±f0k)ϕk , (14)

where the non-equilibrium part of the distribution function is quantified by the deviation function . The collision term can now be expressed as the distribution deviation over the relaxation time , which is needed by the out of equilibrium distribution function to restore its equilibrium value,

 C[fk]=δfkτk=f0k(1±f0k)ϕkτk . (15)

Putting (14) into the right hand side of (13) by keeping the distribution functions of the particles other than the scattered one vanishingly close to equilibrium and comparing with (15), the relaxation time finally becomes as the inverse of the reaction rate of the respective processes  (62),

 τ−1k≡Γk =νl212ωk∫dΓpldΓp′kdΓp′l(2π)4δ4(pk+pl−p′k−p′l) ×⟨|Mk+l→k+l|2⟩f′0l(1±f′0k)(1±f0l)(1±f0k). (16)

Clearly the distribution function of final state particles are given by primed notation.

Simplifying utilizing the -function we finally obtain in the center of momentum frame of particle interaction as,

 τ−1k=Γk= νl∫d3→pl(2π)3d(cosθ)dσd(cosθ)f′0l(1±f′0k)(1±f0l)(1±f0k) , (17)

where is the scattering angle in the center of momentum frame and is the interaction cross section for the respective scattering processes. Now in terms of the Mandelstam variables and the expression for can be reduced simply as,

 τ−1k=Γk=νl∫d3→pl(2π)3dtdσdtf′0l(1±f′0k)(1±f0l)(1±f0k) . (18)

The differential cross section relates the scattering amplitudes as . The quark-gluon scattering amplitudes for processes are taken from  (63), that are averaged over the spin and color degrees of freedom of the initial states and summed over the final states.

Now in order to take into account the small-angle scattering scenario that results into divergent contributions from -channel diagrams of QCD interactions, a transport weigh factor have been introduced in the interaction rate  (64); (65). Furthermore considering the momentum transfer is not too large we can make following assumptions, and  (64) to finally obtain,

 τ−1k=Γk=νl∫d3→pl(2π)3f0l(1±f0l)∫dtdσdt2tus2 . (19)

In the integration involving -channel diagrams from where the infrared logarithmic singularity appears, the limit of integration is restricted from to in order to avoid those divergent results using the cut-off as infrared regulator. Here with being the coupling constant of strong interaction as already mentioned in section-A.

Now in the QGP medium the quark and gluon interaction rates result from the following interactions respectively,

 Γg=Γgg+Γgq ,      Γq=Γqg+Γqq , (20)

where , and are the interaction rates between gluon-gluon, gluon-quark, quark-gluon and quark-quark respectively.

Finally after pursuing the angular integration in (19) we are left with the thermal relaxation times of the quark and gluon components in a QGP system in the following way,

 τ−1g= {νg∫d3→pl(2π)3f0g(1+f0g)} × [9g416π⟨s⟩gg{ln⟨s⟩ggk2−1.267}] + {νq∫d3→pl(2π)3f0q(1−f0q)} × [g44π⟨s⟩gq{ln⟨s⟩gqk2−1.287}], (21)
 τ−1q= {νg∫d3→pl(2π)3f0g(1+f0g)} × [g44π⟨s⟩qg{ln⟨s⟩qgk2−1.287}] + {νq∫d3→pl(2π)3f0q(1−f0q)} × [g49π⟨s⟩qq{ln⟨s⟩qqk2−1.417}], (22)

where is the thermal average value of with . Clearly in order to account for a hot QCD medium the quasiparticle effects must be invoked in the expressions of these thermal relaxation times obtained far. As discussed in section-A, the distribution functions of quarks and gluons and the coupling will carry the quasiparticle descriptions accordingly. Since the cut-off parameter also depends upon and the thermal average of includes , they will reflect the hot QCD equation of state effect as well. Following the definition of equilibrium distribution function of quarks and gluons from Eq.(1), within the quasiparticle framework, the thermal averages of gluon and quark momenta respectively are obtained as,

 ⟨pg⟩=3TPolyLog[4,zg]PolyLog[3,zg] , (23) ⟨pq⟩=3TPolyLog[4,−zq]PolyLog[3,−zq] . (24)

### ii.3 Electrical conductivity

In this work, we have adopted the kinetic theory approach for evaluating the analytical expression of electrical conductivity, based on solving the relativistic transport equation for a charged QGP system.

Before proceeding for the solution of transport equation, we introduce here some of the thermodynamic quantities needed for developing the required framework. We start with particle 4-flow for the species of particle in a multicomponent system  (66),

 Nμk(x)=∫d3→pk(2π)3p0kpμkfk(x,pk) . (25)

Next the total particle 4-flow and the energy momentum tensor of the system are defined respectively as the following,

 Nμ(x)= N∑k=1Nμk(x)=N∑k=1∫d3→pk(2π)3p0kpμkfk(x,pk) , (26) Tμν(x)= N∑k=1∫d3→pk(2π)3p0kpμkpνkfk(x,p) . (27)

With the help of the above quantities we define the diffusion flow of the component as  (66),

 Iμk=Nμk−xkNμ , (28)

where is the particle fraction corresponding to species, and are the particle number density for species and total particle number density of the multicomponent system respectively, which are related by . We can readily notice , i,e. sum of the diffusion flows vanishes.

The total electric current density of such a system is given by  (67),

 Jμ(x)=N∑k=1qkIμk=N−1∑k=1(qk−qN)Iμk , (29)

where is the electric charge associated with the species.

A realistic description of non-equilibrium phenomena in relativistic systems must take reactive processes into account which incorporates all kinds of inelastic collisions beside elastic ones. In such a case the system must include a number of conserved quantum numbers and the diffusion flow in such situations will become,

 Iμa= N∑k=1qakIk,                [a=1,2,..........N′] (30) = N∑k=1qak{Nμk−xkNμ} . (31)

Here stands for the index of conserved quantum number and is the conserved quantum number associated with component. Following the prescription we are able to define the particle number density of the independent components as, .

After defining these basic thermodynamic quantities let us present the relativistic transport equation (12) in covariant form with the force term present in it  (25),

 pμk∂μfk+qkFαβpβ∂fk∂pαk=−C[fk] . (32)

Here is the electromagnetic field tensor with electric field , in the absence of any magnetic field in the medium. We identify as hydrodynamic 4-velocity. Throughout this paper we will use the metric system .

Now using the Chapman-Enskog (CE) method the transport equation is linearized around a local equilibrium distribution function and finally the CE hierarchy reduces the left hand side of the transport equation in terms of . The collision term is simplified using (15) giving rise to,

 pμk∂μf0k+1Tf0k(1±f0k)qkEμpμk=−ωkτkf0k(1±f0k)ϕk . (33)

To proceed further the distribution functions of constituent particles is needed to be provided in covariant notations. In a comoving frame and involving the quasiparticle description discussed in section-A, it can be given in the following way,

 f0k(x,pk)=zkexp[−pμkuμT+μkT]1∓zkexp[−pμkuμT+μkT] , (34)

where we have introduced as the energy per particle of the species and is the chemical potential for the same. Within the quasiparticle framework, for quarks and gluons is defined by Eq.(2).

In order to retrieve the transport equation in terms of the thermodynamic forces, the first term on the left hand side of Eq.(33) is needed to be reduced by decomposing the derivative over the distribution function into a time-like and a space-like part as , with the covariant time derivative and the spatial gradient , expressed in terms of hydrodynamic 4-velocity and projection operator . Whence the spatial gradients over velocity, temperature and chemical potentials directly link with the viscous flow, heat flow and the diffusion flow of the fluid respectively, the time derivatives are needed to be eliminated using a number of thermodynamic identities so that they contribute in the expressions of the thermodynamic forces as well. The time derivative over particle number density and the time derivative over energy per particle however follow the equation of continuity and equation of energy as in the case of a system without the influence of electric field,

 Dnk= −nk∂⋅u , (35) N∑k=1xkDωk= −∑Nk=1Pk∑Nk=1nk∂⋅u , (36)

where is the partial pressure attributed to species. But the equation of motion in the presence of the electric field will be different from the one without electric field. In a multicomponent system in the presence of an electric field the equation of motion takes the follow form,

 Duμ=∇μP∑Nk=1nkhk+∑Nk=1qknk∑Nk=1hknkEμ . (37)

Clearly even the pressure gradient is zero, the Lorentz force acting on the particle produces non-zero acceleration. By utilizing these identities and retaining the thermodynamic forces involving thermal and diffusion terms only, (shear and bulk viscous part not considered in this work), the transport equation becomes,

 1T[pνk{(pk.u)−hk}Xqk+pνkN′−1∑a=1(qak−xa)Xaν]=−ωkτkϕk , (38)

where and are the thermal and diffusion forces respectively given by,

 Xqμ= [∇μTT−∇μPnh]+[−1hN∑k=1xkqkEμ] , (39) Xkμ= [(∇μμa)P,T−hknh∇μP]+ (40) [qk−qN−hk−hNhN∑l=1xlql]Eμ .

The detail of the computation in offered in Appendix-A.

We identify and as the enthalpy per particle for species and for total system respectively and . Here and are the particle fraction and chemical potential associated with quantum number respectively. Clearly in the expressions of thermal and diffusion driving forces, terms proportional to electric field give rise to electrical conductivity. Now in order to be a solution of this equation the deviation function must be a linear combination of the thermodynamic forces,

 ϕk=BkμXμq+1TN′−1∑a=1BμakXaμ , (41)

with, and where and .

Putting (41) into the right hand side of (38) and comparing both sides of (38) (noting thermodynamic forces are independent) we finally obtain,

 Bμk=⟨Πμk⟩ωk−hk{−ωkτk} ,    Bμak=⟨Πμk⟩qak−xa{−ωkτkT} , (42)

from which the complete structure of can be obtained. Now going back to equation (31) we notice for equilibrium distribution function the clearly vanishes, while with it gives a finite diffusion flow as following,

 Iμa=N∑k=1(qak−xa)∫d3→pk(2π)3p0kpμkf0k(1±f0k)ϕk . (43)

Putting the value of from (41) with the help of Eqs. (42) into (43) we get the linear law of diffusion flow,

 Iμa=laqXμq+N′−1∑b=1labXμb ,a=1,....,(N′−1) . (44)

where the coefficients are now expressed in terms of the relaxation time ,

 laq = N∑k=1(qak−xa)1T∫d3→pk(2π)3f0k(1±f0k)τk (45) ×(ωk−hk) ,
 lab=N∑k=1(qak−xa)(qbk−xb)1T∫d3→pk(2π)3f0k(1±f0k)τk. (46)

Now substituting the expression of diffusion flow into Eq. (29), and pertaining the terms proportional to electric field only we finally reach the expression for the electric current density,

 Jμ = N−1∑k=1(qk−qN)[N−1∑l=1lkl{ql−qN−hl−hNhN∑n=1xnqn} (47) −lkqhN∑n=1xnqn]Eμ .

We also know the current density relates with the electric field by the linear relation via the electrical conductivity as,

 Jμ=σelEμ . (48)

By comparing (47) and (48) we finally obtain the detailed expression of electrical conductivity in the following manner,

 σel= N−1∑k=1(qk−qN)[N−1∑l=1lkl{ql−qN− (49) hl−hNhN∑n=1xnqn}−lkqhN∑n=1xnqn] .

Now for a quark-gluon system the expression of the electric conductivity boils down to,

 σel=q2ql11hg−l1qxqh. (50)

The subscript and stands for quarks and gluons respectively. So finally we are left with the coefficients as,

 l1q= 1T[−xqτg∫d3pg(2π)3f0g(1+f0g)(ωg−hg) (51) +xgτq∫d3pq(2π)3f0q(1−f0q)(ωq−hq)],
 l11= 1T[x2qτg∫d3pg(2π)3f0g(1+f0g) (52) +x2gτq∫d3pq(2π)3f0q(1−f0q)] .

The is simply the square of the fractional quark charges taking sum over quark degeneracies. For up, down and strange quarks the fractions quark charge is taken to be , and respectively.

### ii.4 Charge Diffusion

We recall Eq. (44), where the diffusion flow is linearly expressed in terms of thermal and diffusion driving forces respectively. The diffusion driving force does not include the terms containing because diffusion flow vanishes for those values of . Since presently we are dealing with a quark-gluon plasma which incorporates binary elastic collisions that conserve particle numbers, in such case the distinction between the independent particle fractions and the particle fraction of separate components vanishes. So in present situation the diffusion flow rather follow the relation , as mentioned earlier. In such scenario the original diffusion driving forces (not containing the electric field) conjugate to (N-1) independent diffusion flows is given by  (66),

 Xμk= [