Revisit to electrical and thermal conductivities, Lorenz number and Knudsen number in thermal QCD in a strong magnetic field

We have explored how the electrical and thermal conductivities in a thermal QCD medium get affected in weak-momentum anisotropy arising due to a strong magnetic field. This study, in turn, facilitates to understand the duration of strong magnetic field, Wiedemann-Franz law, and the Knudsen number. We calculate the conductivities by solving the relativistic Boltzmann transport equation in relaxation-time approximation. The interactions among partons are incorporated within the quasiparticle approach at finite $T$ and strong $B$. We have found that the electrical conductivity decreases with the temperature in a magnetic field-driven anisotropy, which is opposite to its behavior in an expansion-driven anisotropy. Whereas the thermal conductivity increases very slowly with the temperature, unlike its rapid increase in the expansion-driven anisotropy, thus both $\sigma_{\rm el}$ and $\kappa$ may distinguish the origin of anisotropies. The above findings in conductivities are attributed to three factors: {\em Firstly}, the weak-momentum anisotropies are generated either by the strong magnetic field or by the asymptotic expansion. {\em Secondly}, the phase-space factor. {\em Thirdly} the relaxation-time of quarks. We have extracted the time-dependence of initially produced strong magnetic field by $\sigma_{\rm el}$, where the magnetic field expectedly decays slower than in vacuum. However, due to the presence of weak-anisotropy, magnetic field decays relatively faster than in isotropic one. The Knudsen number decreases with the temperature, but the presence of expansion-driven anisotropy reduces its magnitude. However, the strong magnetic field raises its value but, remains less than one. Finally, $\kappa/\sigma_{\rm el}$ in magnetic field, increases with temperature, but with a magnitude larger than in isotropic medium and smaller than in expansion-driven anisotropic medium.


Introduction
Relativistic heavy-ion experiments at RHIC and LHC create a new state of strongly interacting medium, known as quark gluon plasma (QGP) and are continuing to successfully collect the evidences in the form of elliptic flow [1][2][3], jet quenching [4][5][6], dilepton and photon spectra [7][8][9], anomalous quarkonium suppression [10][11][12] etc. for the existence of QGP. The abovementioned predictions were made for the simplest possible phenomenological setting, i.e. fully central collisions, where the baryon number density is negligible and it is expected that due to the symmetric configuration of the collision, no strong magnetic fields are produced. But only a small portion of heavy-ion collisions are truly head-on, most collisions indeed occur with a finite impact parameter or centrality. As a result, the two highly charged ions impacting with a small offset may produce extremely large magnetic fields reaching between m 2 π (≃ 10 18 Gauss) at RHIC and 15 m 2 π at LHC [13].
However, the naive (classical) estimates for the lifetime of these strong magnetic fields show that they only exist for a small fraction of the lifetime of QGP [14,15]. However, the charge transport properties of QGP have been found to significantly extend their lifetime, thus the study of the transport coefficient, mainly, the electrical conductivity (σ el ) becomes essential. Here our motivations are of two fold, which complement to each other: first we wish to revisit σ el for an isotropic hot QCD medium in absence of any external field to check how long the magnetic field produced in relativistic heavy ion collisions stays appreciably large, i.e., some sort of time-dependence of externally produced magnetic field. However, the issue about the longevity of the magnetic field is not yet settled. So keeping the uncertainties about the exact nature of magnetic field in mind, if the external magnetic field still remains large till the medium is formed, the transport properties of the medium can then be significantly affected and the effect depends on the magnitude of σ el of the medium in presence of strong magnetic field (|q i B| ≫ T 2 , |q i B| ≫ m 2 i , q i (m i ) is the electric charge (mass) of i-th flavor and T , B are the temperature and magnetic field, respectively), whose evaluation is exactly our second motivation. Since σ el is responsible for the production of electric current due to the Lenz's law, its value becomes vital for the strength of Chiral Magnetic Effect [16]. Moreover the electrical field in mass asymmetric collisions has overall a preferred direction, which will eventually generate a charge asymmetric flow and the strength of the flow is given by σ el [17]. Furthermore, σ el is used as a vital input for many phenomenological applications in RHIC, LHC etc., such as the emission rate of soft photons [18], which accounts the raising of the spectra [19,20].
The effects of the magnetic fields on σ el for quark matter have been investigated previously in different models, such as quenched SU(2) lattice gauge theory [21], the dilute instanton-liquid model [22], the nonlinear electromagnetic currents [23,24], axial Hall current [25], real-time formalism using the diagrammatic method [26], effective fugacity approach [27] etc. As we know, the external magnetic field modifies the dispersion relation (E n = p 2 L + 2n|q i B| + m 2 i ) quantum mechanically of the charged particle, where the motion along the longitudinal direction (p L ) (with respect to the magnetic field direction) remains the same as for a free particle and only the motion along the transverse direction (p T ) gets quantized in terms of the Landau levels (n). In strong magnetic field limit (eB >> T 2 as well as eB ≫ m 2 ), only the lowest Landau level will be occupied, i.e. p T ≈ 0, and the particle can only move along the direction of the magnetic field, resulting an anisotropy in the momentum space, i.e. p L ≫ p T . Thus the anisotropic parameter, ξ (= p 2 T 2 p 2 L − 1) comes out to be negative and for a weak-anisotropy (ξ < 1), the distribution function may be approximated by stretching the isotropic one along a certain direction (say, the direction of magnetic field). Thus, to know the effects of strong magnetic field on conductivities in kinetic theory approach, an introduction of anisotropy is automatically needed.
Much earlier than the former one, it was envisaged that the relativistic heavy ion collisions at the initial stage induce a momentum anisotropy in the local rest frame of fireball, due to the asymptotic free expansion of the fireball in the beam direction compared to its transverse direction [28,29]. Unlike the previous one, here p T is greater than p L , hence the anisotropy parameter becomes positive. Therefore, for a weak-anisotropy (ξ < 1), the distribution of partons can be approximated by squeezing an isotropic one along a certain direction and its effects on many phenomenological and theoretical observations have already been made. For example, the leading-order dilepton and photon yields get enhanced due to the anisotropic component [30][31][32][33]. Recently one of us had observed the effect of this kind of anisotropy on the properties of heavy quarkonium bound states [34], the electrical conductivity [35], where the heavy quarkonia are found to dissociate earlier than its counterpart in isotropic one and the electrical conductivity decreases with the increase of anisotropy. Later its relation with the shear viscosity is also explored in [36]. Now we move on to the thermal conductivity (κ), which is related to the efficiency of the heat flow or the energy dissipation in a thermal QCD medium. Our intention is to comment on the validity of the local equilibrium assumed in hydrodynamics in terms of Knudsen number (Ω), which, in turn, is related to the thermal conductivity through the mean free path (λ). Similar to the electrical conductivity, we also wish to explore the effect of strong magnetic field on the thermal conductivity by calculating it in the presence of weak-momentum anisotropy caused by the strong magnetic field. A natural question arises about whether we can distinguish the anisotropies from the abovementioned origins through the transport coefficients and knowing that we can improve the knowledge on the transport properties of the medium. This query may be a worth of investigation.
The electronic contribution of the thermal conductivity and the electrical conductivity are not completely independent rather their ratio is equal to the product of Lorenz number (L) and temperature, widely known as Wiedemann-Franz law. In fact, the ratio, κ/σ el has approximately the same value for different metals at the same temperature. But, it diverges in quasi-one-dimensional metallic phase with decreasing temperature [37], reaching a value much larger than that found in conventional metals, near the insulator-metal transition [38], thermally populated electron-hole plasma in graphene [39] etc. Recently the temperature dependence of the Lorenz number is calculated for the two-flavor quark matter in NJL model [40] and for the strongly interacting QGP medium [41]. In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the Wiedemann-Franz law, which can be explained in terms of independent propagation of charge and heat in a strongly correlated system. However, in this work we intend to observe how the ratio gets affected due to the presence of an ambient strong magnetic field.
In this work, we have evaluated both the conductivities in kinetic theory approach, where the relativistic Boltzmann transport (RBT) equation is employed and is being solved by the relaxation-time approximation (RTA), where, as such there is no scope to incorporate the interaction among the partons 1 . We circumvent the problem by incorporating the interactions among partons through their dispersion relations, known as quasiparticle model (QPM), in their distribution functions. The quasiparticle masses are conveniently obtained from their respective self energies, which, in turn, depends on the temperature and the magnetic field. Thus the presence of magnetic field affects both electrical and thermal conductivities. However, as a base line, we also compute the conductivities with the current quark masses (noninteracting), which give unusually large values, thus motivates us to use the QPM.
In brief, we have observed that the electrical conductivity and the thermal conductivity of the hot QCD medium increase in the presence of strong magnetic field-driven weakmomentum anisotropy, contrary to the decrease of the counterparts in the expansiondriven anisotropic medium. The opposite behavior in two anisotropic mediums may help to distinguish the origin of anisotropy in a thermal medium produced at the initial stage of ultrarelativistic heavy ion collision. From the relative behavior of thermal conductivity and electrical conductivity, we have noticed that the ratio, κ/σ el in a strong magnetic field shows linear enhancement with temperature, whose magnitude and slope are larger than in isotropic medium and smaller than in expansion-driven anisotropic medium, thus describes different Lorenz numbers (κ/(σ el T )). We have also observed that the presence of strong magnetic field makes the Knudsen number slightly larger (but, remains less than one unlike the much larger value in ideal case) than its value in the isotropic medium.
The present work is organized as follows. In section 2, we have formulated the electrical conductivity where we first revisit the electrical conductivity for an isotropic thermal medium in subsection 2.1. Then we proceed to calculate it for the anisotropic thermal mediums due to expansion-induced anisotropy and magnetic field-induced anisotropy in subsection 2.2, and then their results are discussed in the QCD model involving current quark masses. Similarly in section 3 we have determined the thermal conductivity, where we first revisit its form for an isotropic thermal medium in subsection 3.1, which is followed by the calculation of thermal conductivity for the abovementioned anisotropic thermal mediums in subsection 3.2., and then we have discussed their results in the QCD model involving current quark masses. We have studied the applications of aforesaid conductivities in section 4. In section 5, we have discussed the quasiparticle model and calculated the quasiparticle mass in the presence of a strong magnetic field, and then explained the results on electrical conductivity, thermal conductivity, Wiedemann-Franz law and Knudsen number using the quasiparticle model. Finally, we have concluded in section 6.

Electrical conductivity
Transport coefficients such as electrical conductivity and thermal conductivity of a hot QCD system can be determined using different models and approaches namely relativistic Boltzmann transport equation [36,[42][43][44], the Chapman-Enskog approximation [41,45], the correlator technique using Green-Kubo formula [22,46,47] and lattice simulation [48][49][50][51]. However, we will employ the relativistic Boltzmann transport equation with the relaxation-time approximation to calculate the electrical conductivity for both isotropic and anisotropic hot QCD mediums in subsections 2.1 and 2.2, respectively.

Electrical conductivity for an isotropic thermal medium
When an isotropic and hot medium of quarks, antiquarks and gluons in thermal equilibrium is disturbed infinitesimally by an electric field, an electric current J µ is induced as where the summation is over three flavors (u, d and s) and q i , g i and δf q i (δfq i ) are the electric charge, degeneracy factor and infinitesimal change in the distribution function for the quark (antiquark) of ith flavor, respectively. In our calculations we will be using δf q i = δfq i = δf i for zero chemical potential. According to Ohm's law, the longitudinal component of the spatial part of four-current is directly proportional to the external electric field and the proportionality factor is known as the electrical conductivity, The infinitesimal change in quark distribution function is defined as is the equilibrium distribution function in the isotropic medium for ith flavor, with ω i = p 2 + m 2 i . It is possible to obtain δf i from the relativistic Boltzmann transport equation (RBTE) [52], where F µν denotes the electromagnetic field strength tensor and the collision term, C[f i (x, p)] is given in the relaxation-time approximation as where u ν is the four-velocity of fluid in the local rest frame and the relaxation-time (τ i ) for ith flavor in a thermal medium is given [53] by To take into account the effect of the electric field, we use only µ = i and ν = 0 and vice versa components of the electromagnetic field strength tensor, i.e. F 0i = −E and F i0 = E in our calculation, thus the RBT eq. (4) takes the following form, which gives the solution, δf i , Now substituting the value of δf i in eq. (1) for zero chemical potential, we obtain the electrical conductivity for the thermal isotropic medium, which can now be used to show how the magnetic field varies with time in the isotropic thermal conducting medium. According to electrodynamics, the magnetic field created due to the spatial variation of the electric field, rapidly changes over time. However for an medium with substantial value of electrical conductivity, the momentary magnetic field would induce an electric current which ultimately would help to enhance the lifetime of the strong magnetic field.

Electrical conductivity for an anisotropic thermal medium
Here we will mainly discuss two types of momentum anisotropies, which may arise in the very early stages of ultrarelativistic heavy ion collisions. The first one is due to the preferential flow in the longitudinal direction compared to the transverse direction and the second one is due to the creation of strong magnetic field. We will first revisit the former one.

Expansion-induced anisotropy
At early times, the QGP created in the heavy ion collision experiences larger longitudinal expansion than the radial expansion and this develops a local momentum anisotropy. For the weak-momentum anisotropy (ξ < 1) in a particular direction (say n), the distribution function is written [54] as which can be expanded in a Taylor series, and upto O(ξ) it takes the following form, The anisotropic parameter (ξ) is generically defined in terms of the transverse and longitudinal components of momentum as where p L = p · n, p T = p − n · (p · n), p ≡ (p sin θ cos φ, p sin θ sin φ, p cos θ), n = (sin α, 0, cos α), α is the angle between z-axis and direction of anisotropy, (p · n) 2 = p 2 c(α, θ, φ) = p 2 (sin 2 α sin 2 θ cos 2 φ + cos 2 α cos 2 θ + sin(2α) sin θ cos θ cos φ). For p T ≫ p L , ξ takes positive value, which explains the removal of particles with a large momentum component along n direction due to the faster longitudinal expansion than the transverse expansion [28].
Now we are going to observe how the weak-momentum anisotropy affects the electrical conductivity of the thermal medium. Thus, after solving the RBTE (4) for the anisotropic distribution function, we get δf i as which is then substituted in eq. (1) to yield the expression of electrical conductivity, where the first term in R.H.S. is the electrical conductivity for an isotropic medium. So in terms of σ iso el , σ aniso el,ex is written as

Life-span of magnetic field
Earlier people had thought that the magnetic field generated in the heavy ion collision decays instantly. However in the presence of transport coefficient such as electrical conductivity, the lifetime of magnetic field may be elongated. To reaffirm this, we are going to see the variation of magnetic field using the value of electrical conductivity that we have calculated above for both isotropic and anisotropic mediums.
Thus for a charged particle moving in x-direction, a magnetic field will be produced in the perpendicular direction of the particle trajectory, say z-direction. According to the Maxwell's equations, the magnetic field created along z-direction is expressed, as a function of time and electrical conductivity [55] for an isotropic medium as However for an anisotropic medium, the expression for eB is not available, so we assumed the same expression by replacing σ iso el → σ aniso el,ex , For the sake of comparison, the magnetic field produced in vacuum [55] is given by,  where b and γ denote the impact parameter and the Lorentz factor of heavy ion collision, respectively. In equations (16) and (17), the electrical conductivity is taken as a function of time through the cooling law, T 3 ∝ t −1 , where initial time and temperature are set at 0.2 fm and 390 MeV, respectively. From figures 1 and 2, which are plotted at x = 0 for the centre of mass energies 200 GeV and 2.76 TeV, respectively, we see that the magnetic field in the isotropic conducting medium decays very slowly as compared to the vacuum. At initial time, the fluctuation of magnetic field in a thermal medium is quite high, however after certain time, it gradually stabilizes.
However for a conducting medium in the presence of weak-momentum anisotropy (ξ = 0.6), we have investigated (from figure 3) that the lifetime of existence of a nearly stable magnetic field in the anisotropic thermal medium is slightly less than in the isotropic thermal medium, whereas at initial time, this difference in the variations of magnetic field in two mediums is less illustrious.
As we can see from figures 1, 2 and 3, the decay of magnetic field with time is very slow in conducting medium and it nearly remains strong. So, it is plausible to explore the effect of strong magnetic field-induced anisotropy on the thermal medium.

Magnetic field-induced anisotropy
In the presence of an extremely strong magnetic field, the quarks are confined to only lowest Landau levels (LLLs), because they could not be excited to the higher Landau levels (HLLs) due to very high energy gap ∼ O( √ eB). Thus the motion of quark is restricted to only one spatial dimension (along the direction of magnetic field) unlike the gluons who still move in three spatial dimensions.
The distribution function for quark at finite temperature and strong magnetic field is given by where ξ is the anisotropic parameter which characterizes the distribution of particles in a strong magnetic field, p ′ = (0, 0, p 3 ) and n = (sin α, 0, cos α). In the strong magnetic field regime, the anisotropy is mainly produced by the magnetic field, so the direction of anisotropy coincides with the direction of magnetic field (z-direction). Thus one can set α = 0, which yields (p ′ · n) 2 = p 2 3 . From the definition of ξ in eq. (12), it is evident that, ξ will approach negative value for a medium embedded in a magnetic field with a very large strength, because in this case the momentum along the direction of anisotropy dominates over the momentum perpendicular to the direction of anisotropy. i.e. there will be more number of particles with large longitudinal component of momentum along the direction of anisotropy than along the transverse direction of anisotropy (p T ≪ p L ).
For very small ξ, the Taylor series expansion of eq. (19) where ξ-independent part of the quark distribution function is given by with ω i = p 2 the electrical conductivity can be obtained from the third component of current in Ohm's law, Due to dimensional reduction in the presence of a strong magnetic field, the density of states in two spatial directions perpendicular to the direction of magnetic field can be written in terms of |q i B| and as a result, the (integration) phase factor gets modified [57,58] as The infinitesimal perturbation around the equilibrium distribution function i.e.
due to the action of external magnetic field is obtained from the relativistic Boltzmann transport equation in RTA, in conjunction with the strong magnetic field limit, where τ B i denotes the relaxation time for quark in the presence of strong magnetic field. In the LLL approximation, 1 ⇋ 2 (g ⇋ qq) scattering process is dominant over 2 ⇋ 2 (gq ⇋ gq) scattering process [59,60]. In the dominant process (1 ⇋ 2), for the quark momentum ∼ O(T ), the momentum-dependent relaxation time is calculated [60] as where C 2 is the Casimir factor and the primed notations are used for antiquark. Now solving the RBTE (25), we obtain the value of δf i as which, after substituting in eq. (22) for zero chemical potential, we get the electrical conductivity of the anisotropic thermal QCD medium in the presence of a strong magnetic field, where the first term in R.H.S. is the ξ-independent term. Decomposing into ξ = 0 and ξ = 0 terms, eq. (28) takes the following form, Before analysing the results on the electrical conductivity in the presence of anisotropies arising either due to the expansion or due to the strong magnetic field, we wish to understand first how the distribution function in an isotropic medium gets affected in the presence of anisotropies because, in kinetic theory approach, the conductivities are mainly affected by the distribution function embodying the effects of anisotropy, the phase-space factor and the relaxation time. Therefore, we must understand how the ratios, f ex aniso /f iso , f B aniso /f iso depend on the temperature at low and high momenta or vice-versa, which are numerically plotted in figures 4 and 5, respectively. The observations in the above figures can be readily understood by estimating an order of estimate from equations (3), (10) and (19), for u quark. For weak-momentum anisotropy (ξ ≪ 1), the ratios are: f ex aniso /f iso ∼ e −c p T and f B aniso /f iso ∼ e +c ′ p T , in both low and high momentum limits, with the constant, c < c ′ < 1. The crucial negative and positive signs in exponentials arise due to the positive and negative anisotropic parameter, in expansion-driven and magnetic field-driven cases, respectively.
Let us start with the variation of f ex aniso /f iso with T in low momentum regime (figure 4a). As T increases, p/T decreases, resulting an increase in f ex aniso /f iso due to the lesser Boltzmann damping and an obvious decrease in f B aniso /f iso . The slower and relative faster variations are due to the smaller value of c with respect to c ′ . For higher momentum the variations (in figure 4b) as well as the magnitudes of the ratios are more pronounced. The variations of the ratios with momentum at a fixed temperature (in figures 5a and 5b) are much easier to understand because the variable (p) in the exponential is proportional to p/T , hence the variations become just opposite to the variation with temperature in figure 4.
The above observations on the distribution functions facilitate to understand the results on the electrical conductivity for a thermal QCD medium with three flavors (u, d and s) with their current masses in figure 6. For the isotropic medium (denoted by solid line), σ el increases with the increase of temperature, whereas due to insertion of weakmomentum anisotropy (labelled as dotted line), σ el decreases a little because the ratio f ex aniso /f iso is always less than 1 for the entire range of temperature (as in figure 4a). On the other hand, the relative magnitude of σ el in magnetic field-driven anisotropic medium (labelled as dashed-dotted line) becomes very large due to relatively large ratio, f B aniso /f iso (as in figure 4). However σ el increases with T , albeit the ratio, f B aniso /f iso decreases with temperature (as in figure 4). The decrease in f B aniso /f iso at high temperature becomes much slower and approaches towards unity, but the phase-space factor (∼ |q i B|) and the relaxation time in SMF limit (|q i B| ≫ T 2 ) together compensate the minimal decrease in f B aniso and gives an overall increasing trend in σ el in the presence of strong magnetic field (seen in figure 6).
Such large value of σ el in the presence of strong magnetic field arises due to the large relaxation-time (τ B ), because it is inversely proportional to the square of the mass. The similar results are also recently found in [26], where σ el is calculated in the diagrammatic method in the strong magnetic field regime and its large value is due to the smaller value of current quark masses. This motivates us to recalculate it with quasiparticle masses in subsection 5.1.

Thermal conductivity
This section is devoted to the determination of the thermal conductivity of a hot QCD medium using the relativistic Boltzmann transport equation. In non-relativistic case, the heat equation is obtained by the validity of the first and second laws of thermodynamics, where the flow of heat is proportional to the temperature gradient and the proportionality factor is called the thermal conductivity. This implies that when two bodies at different temperatures are set in thermal contact, heat flows from the hotter body to the colder body. The heat does not flow directly, but it diffuses, depending on the internal structure of the medium it travels through. Similarly for a relativistic QCD system also, the behavior of heat flow depends on the features of the medium. Thermal conductivity of a particular medium helps to describe the heat flow in that medium and it may leave significant effects on the hydrodynamic evolution of the systems with nonzero baryon chemical potential. To see how the heat flow gets affected, we have calculated thermal conductivity for both isotropic and anisotropic hot QCD mediums in subsections 3.1 and 3.2, respectively.

Thermal conductivity for an isotropic thermal medium
Heat flow four-vector is defined as the difference between the energy diffusion and the enthalpy diffusion, where the projection operator is defined as ∆ µα = g µα −u µ u α , h is the enthalpy per particle which in terms of energy density, pressure and particle number density is represented as h = (ε+P )/n, T αβ denotes the energy-momentum tensor and N α is the particle flow fourvector. N α and T αβ are also known as the first and second moments of the distribution function with the following expressions [61], It is also possible to obtain the particle number density from eq. (31), the energy density and the pressure from eq. (32) as n = N α u α , ε = u α T αβ u β and P = −∆ αβ T αβ /3, respectively. From equations (30), (31) and (32), one can find that in the rest frame of the heat bath or fluid, heat flow four-vector is orthogonal to the fluid four-velocity, i.e.
Thus in the rest frame of the fluid, the heat flow is purely spatial and this component of heat flow due to the action of external disturbances can be written, in terms of the nonequilibrium part of the distribution function as In order to define the thermal conductivity for a system, the number of particles in that system must be conserved and therefore it requires the associated chemical potential to be nonzero. In the beginning of the universe and also in the initial stage of ultrarelativistic heavy ion collision, the value of chemical potential (µ) is small but nonzero. In the Navier-Stokes equation, the heat flow is related to the thermal potential (U = µ/T ) as [61] where the coefficient κ is known as the thermal conductivity and ∇ µ = ∆ µν ∂ ν = ∂ µ − u µ u ν ∂ ν is the four-gradient, which, in the rest frame of the heat bath, i.e. in the local rest frame, is replaced by ∂ j (or ∂/∂x j ). Thus, in the local rest frame, the spatial component of the heat flow is written as The thermal conductivity (κ) can be determined by comparing equations (34) and (36), so we need to first find δf i . In the local rest frame, the flow velocity and temperature depend on the spatial and temporal coordinates, so the distribution function can be expanded in terms of the gradients of flow velocity and temperature. Thus, the relativistic Boltzmann transport equation (4) takes the following form, where f i = f iso i + δf i and p 0 = ω i − µ i , which, for very small value of µ i , can be approximated as p 0 ≈ ω i . After dropping out the infinitesimal correction to the local equilibrium distribution function (δf i ) from the left hand side of eq. (37) and then using the following partial derivatives, ∂f iso we solve eq. (37) to get the disturbance, Substituting ∂ j µ T = − h T 2 ∂ j T − T nh ∂ j P and using ∂ 0 u ν = ∇ ν P/(nh) from the energymomentum conservation, we get the final expression for δf i , after simplifying, as After substituting the δf i expression in eq. (34) and then comparing it with eq. (36), we get the thermal conductivity for the isotropic medium,

Thermal conductivity for an anisotropic thermal medium
In this subsection we will consider an anisotropic QCD medium, where the particle distribution is anisotropic in the momentum space and may be generated at the early stages of the ultrarelativistic heavy ion collisions. In this process we will first observe the effects due to the weak-momentum anisotropy on the thermal conductivity of hot QCD medium caused by the initial asymptotic expansion and then by the strong magnetic field as well.

Expansion-induced anisotropy
Using the Taylor series expansion of the anisotropic distribution function (f aniso ex,i ) up to the first order in ξ, the following partial derivatives have been calculated as ∂f aniso ∂f aniso ex,i ∂f aniso ex,i which are then substituted in eq. (37) to obtain δf i , Now substituting the value of δf i in eq. (34), we find the thermal conductivity in an expansion-driven anisotropic thermal QCD medium, where the first expression in R.H.S. is the thermal conductivity for the isotropic thermal QCD medium. Thus one can write κ aniso ex in terms of κ iso as We are now going to see how the thermal conductivity of the hot QCD medium gets modified due to the anisotropy developed by the strong magnetic field.

Magnetic field-induced anisotropy
The strong magnetic field restricts the dynamics of quarks to one spatial dimension i.e. along the direction of magnetic field. So there will be no conduction of heat by the quarks in the transverse directions. In this strong magnetic field scenario, eq. (34) is thus modified into Similarly eq. (36) takes the following form, where h B = (ε + P )/n represents the enthalpy per particle in a strong magnetic field. For the charged particles in the SMF limit, the particle number density (n) is obtained from the following particle flow four-vector, similarly the energy density (ε) and the pressure (P ) are obtained from the following energy-momentum tensor, Now in terms of the gradients of flow velocity and temperature, the RBTE (25) in the presence of a strong magnetic field can be written as where the variables,p µ = (p 0 , 0, 0, p 3 ) andx µ = (x 0 , 0, 0, x 3 ) are suited to the strong magnetic field calculation. Using the following partial derivatives, ∂f aniso ∂f aniso in eq. (54), we obtain δf i as which is then substituted in eq. (50) to give the value of thermal conductivity in a strong magnetic field, where the first expression in R.H.S. is the ξ = 0 part of the thermal conductivity. Thus κ aniso B can be rewritten in terms of ξ-dependent and independent parts as Figure 7 depicts how the thermal conductivity varies with temperature for the isotropic medium and for the anisotropic mediums due to expansion-driven anisotropy and strong magnetic field-driven anisotropy. We have observed that κ for the isotropic medium increases with the increase in temperature. Similar increasing behavior of κ is also noticed for the expansion-driven anisotropic medium, however its magnitude decreases. If the origin of anisotropy is strong magnetic field, then the magnitude of κ jumps to a higher value, but its increase with temperature becomes relatively faster than in isotropic medium and expansion-driven anisotropic medium as well. The above observations on the thermal conductivity could similarly be attributed to the behaviors of respective distribution functions, the phase-space factor and the relaxation time scale, where the last two factors are severely affected by the strong magnetic field only. This again validates the use of quasiparticle masses for the thermal conductivity in subsection 5.2.

Applications
This section is devoted to study how the above behaviors observed in the electrical and thermal conductivities will help to understand some specific properties of medium. In subsection 4.1, we will observe how the interplay between the conductivities through the Wiedemann-Franz law gets modified in a thermal QCD medium in the presence of anisotropies arising due to different causes. In subsection 4.2, we will calculate the Knudsen number to have a say whether the thermal QCD medium is still in local equilibrium even in the presence of different anisotropies discussed hereinabove.

Wiedemann-Franz law
According to the Wiedemann-Franz law, the ratio of charged particle contribution of thermal conductivity to electrical conductivity is proportional to the temperature, where the proportionality factor L is called the Lorenz number. This law is perfectly satisfied by the matter which are good thermal and electrical conductors, such as metals. However for different cases, the deviation of the Wiedemann-Franz law has been observed, such as for the thermally populated electron-hole plasma in graphene, describing the signature of a Dirac fluid [39], for the two-flavor quark matter in the Nambu-Jona-Lasinio (NJL) model [40] and for the strongly interacting QGP medium [41]. In this work we intend to see how the Lorenz number for the thermal QCD matter varies by observing the ratio (κ/σ el ) as a function of temperature in the presence of expansion-driven and strong magnetic-field driven anisotropies in figure 8.
In the isotropic medium, the ratio is found to increase linearly with temperature. When the isotropic medium is subjected to an expansion-driven anisotropy, κ/σ el shows almost same increasing behavior with temperature like in isotropic case, but its magnitude and the slope of the linear increase get enhanced. If the origin of anisotropy is strong magnetic field, then both the magnitude and the slope of the linear increase of the ratio with the temperature become smaller than the former descriptions. Thus in two different types of anisotropies we have observed nearly opposite behavior of κ/σ el , which can also be understood from the opposite behavior in electrical and thermal conductivities for the two aforesaid anisotropic mediums. This observation thus implies different Lorenz numbers (κ/(σ el T )) at the same temperature, thus violates the Wiedemann-Franz law.

Knudsen number
The Knudsen number is required to be small for small deviation from the equilibrium in the hydrodynamic regime. The Knudsen number (Ω) is defined as where λ denotes the mean free path and L is the characteristic length scale. One can calculate the mean free path by using the thermal conductivity (κ) of the medium, where v is the relative speed and C V is the specific heat. Therefore the Knudsen number can be recast in terms of the thermal conductivity as where we take v ≃ 1, L = 3 fm, and C V is evaluated from the energy density.
In an isotropic medium, the Knudsen number decreases with the increase of temperature, which explains that the mean free path becomes much smaller than the characteristic length scale of the system. As a result the medium approaches the equilibrium faster. When the medium exhibits a weak-momentum anisotropy due to the asymptotic expansion initially, the Knudsen number does not deviate considerably from its value in the isotropic medium (seen in the left panel of figure 9). However if the origin of anisotropy is the strong magnetic field (eB = 15 m 2 π ), a significant deviation from the isotropic one can be seen, where the Knudsen number has a larger magnitude (denoted as dashed-dotted line in the right panel of figure 9), which defies physical interpretation and has fortunately been cured in the quasiparticle model (seen in figure 15).

Quasiparticle description of hot QCD matter
Till now we, in fact, have not incorporated any interactions among quarks and gluons in a thermal QCD medium either in the presence or absence of strong magnetic field. As a matter of fact, the magnitude and the variation of the electrical conductivity, thermal conductivity and Knudsen number become unrealistic. Hence we must resort to the quasiparticle description of particles, known as QPM, where different flavors acquire the medium generated masses, in addition to their current masses. The thermal mass is generated due to the interaction of quark with the other particles of the medium, thus the quasiparticle model properly describes the collective properties of the medium. Earlier this model was explained in different approaches such as the Nambu-Jona-Lasinio and PNJL based quasiparticle models [62][63][64][65], quasiparticle model based on Gribov-Zwanziger quantization [66,67] and thermodynamically consistent quasiparticle models [68,69]. However, for our calculation, the effective mass (squared) of i-th flavor in a pure thermal medium is taken from [70], where m i0 is the current mass and m iT is the thermal mass of i-th flavor, which is given [71,72] by where g ′ is the running coupling that runs with the temperature of the medium (we have taken its one-loop result).
Now, for a thermal medium in the presence of a strong magnetic field, the above effective mass (squared) can be generalized as where m iT B is the mass of i-th flavor at finite temperature and strong magnetic field.
Let us revisit first, how the medium generated mass can be calculated in the thermal medium only. The medium generated mass of the particle can be determined from the pole of effective propagator, which is defined as where the quark self-energy, Σ(p) is of the order of gT . Now, in additional presence of strong magnetic field along the z-direction (|q i B| ≫ T 2 ), the transverse motion of quark ceases to exist (p ⊥ ≈ 0), as a result, the effective quark propagator becomes a function of the parallel component of the quark momentum only, where γ µ p µ = γ 0 p 0 − γ 3 p z with the notations of p µ ≡ (p 0 , 0, 0, p z ), γ µ ≡ (γ 0 , 0, 0, γ 3 ) and g µν = diag(1, 0, 0, −1) 2 . The following notations are also required to be known, The quark self-energy, Σ(p ) will now give the mass correction (m iT B ), due to thermal QCD medium in the ambience of strong magnetic field. In terms of the quark (S(k)) and gluon (D µν (p − k)) propagators, the quark self-energy is written by the Feynman rules, where 4/3 is the Casimir factor and g is the running coupling that runs mainly with the magnetic field [73,74], because magnetic field is the largest energy scale in the strong magnetic field regime. The quark propagator gets modified in the strong magnetic field [56,75,76] as The two k z integrations, I 1 kz and I 2 kz are respectively performed [78] to get which are then substituted in eq. (77) to get where m 2 iT B is introduced as the mass (squared) at finite temperature and strong magnetic field with the following value, which depends on both temperature and magnetic field.
In the quasiparticle description of particles, the distribution function now contains the effective masses of the particles. Therefore, the distribution functions in isotropic medium as well as in expansion-driven anisotropic medium depend only on temperature, whereas the distribution function in magnetic field-driven anisotropic medium depends on both temperature and magnetic field. So, from figures 10 and 11, we noticed that the behaviors of ratios (f ex aniso /f iso and f B aniso /f iso ) get flipped in comparison to their respective behavior in ideal case (as in figures 4 and 5). As the transport coefficients such as electrical conductivity and thermal conductivity are expressed in terms of the distribution function at finite temperature and/or magnetic field, so the knowledge about the behavior of distribution function in the QPM description is useful in understanding the transport properties of the hot QCD medium.
In the coming subsections we are going to discuss the results on the electrical conductivity, thermal conductivity and their applications using the quasiparticle model with three flavors (u, d and s).

Electrical conductivity
With the quasiparticle description as input, we have now recomputed the electrical conductivity for isotropic and anisotropic mediums by substituting the effective quark masses from eq. (66) in equations (9), (15), and from eq. (68) in eq. (29). We have replotted σ el as a function of temperature in figure 12 and found that there is an overall decrease in σ el . Interestingly, for a magnetic field-driven weak-momentum anisotropy (denoted by dashed-dotted line), the magnitude of σ el now becomes smaller, which is at par with the counterparts in isotropic and expansion-driven anisotropic mediums. However, σ el for this magnetic field-driven anisotropic medium, now decreases with the temperature, which is opposite to its variation in the expansion-driven anisotropy. The above observations on σ el fully resonate with the distributions seen in figures 10 and 11. We are now convinced that the quasiparticle description of particles tames the unusually large value of σ el in the strong magnetic field.

Thermal conductivity
We have also calculated the thermal conductivity for isotropic and anisotropic mediums with the quasiparticle description by substituting the effective quark masses from eq. (66) in equations (43), (49), and from eq. (68) in eq. (61). Figure 13 plots the variation of κ with temperature for the isotropic medium, expansion-and strong magnetic fielddriven anisotropic mediums with the quasiparticle description. The effects of quasiparticle description on the thermal conductivity can again be understood through the distribution functions with quasiparticle masses in figures 10 and 11. For the isotropic as well as expansion-driven anisotropic mediums, κ is found to increase with temperature as in ideal case. The only noticeable finding is that, although the magnitude of κ for the strong magnetic field-driven anisotropic medium is still larger than in isotropic medium but it has now become smaller and comparable with the value in isotropic medium at very large temperature.

Wiedemann-Franz law
Wiedemann-Franz law makes us understand the relation between the charge transport and the heat transport in a system. Here we have revisited the law in quasiparticle description of particles, unlike using the ideal description of particles earlier in previous subsection 4.1. From figure 14 we found that the magnitude of the ratio, κ/σ el for isotropic and expansion-driven anisotropic mediums now becomes smaller whereas for the magnetic field-driven anisotropic medium it becomes larger as compared to their respective values in the ideal case (figure 8).

Knudsen number
We have seen earlier that for a strong magnetic field-driven anisotropic medium, the Knudsen number (Ω) in the ideal case (seen in figure 9) was very large. As a result, the thermal medium in the presence of strong magnetic field deviates much away from its equilibrium which is, however, not desirable. This is exactly circumvented here in quasiparticle description in figure 15, where we have found that Ω has now been decreased drastically in the presence of strong magnetic field at per with the estimates for B = 0 cases. However, there is an overall decrease of Knudsen number for all cases. Thus in the quasiparticle description, the probability of finding the system to be in local equilibrium is higher, due to the smaller value of Knudsen number.

Conclusions
In this work, we have studied the effect of strong magnetic field-driven anisotropy on the transport coefficients such as electrical conductivity and thermal conductivity of the hot QCD matter and compared them with their behavior in the expansion-driven anisotropy. In order to find these conductivities we have solved the relativistic Boltzmann transport equation in relaxation-time approximation. First we revisited the formulation of electrical and thermal conductivities for the isotropic thermal medium and then calculated these for the expansion-induced anisotropic thermal medium. Using the value of electrical conductivity we have then observed the variation of magnetic field with time and this explains that the lifetime of magnetic field becomes larger for an electrically conducting medium as compared to the vacuum, hence the strong magnetic field is expected to affect the charge transport and heat transport in the QCD medium and this motivated us to derive the aforesaid conductivities for a thermal medium in the presence of strong magnetic field-induced anisotropy. We have observed that both the electrical and thermal conductivities have noticeably larger values in the presence of strong magnetic field-driven anisotropy as compared to their respective values in the isotropic medium, however if the anisotropy is induced due to asymptotic expansion, then the values of the conductivities are seen to get marginally lowered than their values in the isotropic medium. So, in the two different types of anisotropic mediums, we noticed nearly opposite behavior of conductivities. However the large values of conductivities in a strong magnetic field are avoided using the quasiparticle masses. Next, as applications of electrical conductivity and thermal conductivity, we have studied the Wiedemann-Franz law to see the relative behavior of these conductivities, where this law is found to be violated in the presence of strong magnetic field. Then we have calculated the Knudsen number to observe whether the system is still in equilibrium in the presence of weak-momentum anisotropy which may be caused by either sources. We found that, in the quasiparticle description, the Knudsen number becomes less than one, thus the medium may remain in local equilibrium.