Order-by-order Anisotropic Transport Coefficients of a Magnetised Fluid: a Chapman-Enskog Approach

We derive the first and second-order expressions for the shear, the bulk viscosity, and the thermal conductivity of a relativistic hot boson gas in a magnetic field using the relativistic kinetic theory within the Chapman-Enskog method. The order-by-order off-equilibrium distribution function is obtained in terms of the associate Laguerre polynomial with magnetic field-dependent coefficients using the relativistic Boltzmann-Uehling-Uhlenbeck transport equation. The order-by-order anisotropic transport coefficients are evaluated in powers of the dimensionless ratio of kinetic energy to the fluid temperature for finite magnetic fields. In a magnetic field, the shear viscosity (in all order) splits into five different coefficients. Four of them show a magnetic field dependence as seen in a previous study \cite{Ashutosh1} using the relaxation time approximation for the collision kernel. On the other hand, bulk viscosity, which splits into three components (in all order), is independent of the magnetic field. The thermal conductivity shows a similar splitting but is field-dependent. The difference in the first and second-order results are prominent for the thermal conductivities than the shear viscosity; moreover, the difference in the two results is most evident at low temperatures. The first and second-order results seem to converge rapidly for high temperatures.


Introduction
In linear response theory, the thermodynamic fluxes (such as momentum flow, heat flow, etc.) are assumed to be proportional to the thermodynamic forces ( expressed in terms of the gradients of velocities, temperature, etc.), the proportionality constants are called transport coefficients, a few examples include shear, bulk viscosities, thermal conductivity, etc. The study of these transport coefficients for hot and dense nuclear matter, a.k.a. Quark-Gluon Plasma (QGP) produced at Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) experiments, is one of the primary goals since it was found that the QGP has smallest specific shear viscosity. Also, nuclear collisions at RHIC and LHC experiments produce intense transient electromagnetic fields due to the relativistic velocity of the protons inside the colliding nucleus. The effect of these transient electromagnetic fields on the hot and dense QGP and the subsequent low-temperature phase of hadronic resonance gas is an active area of research . Magnetic fields in the mid-central and peripheral heavy-ion collisions predominantly point perpendicular to the participant plane. This breaks the isotropic symmetry in the momentum space in the transverse plane. While subjected to these directed magnetic fields, the quarks in the QGP phase and the charged hadrons in the HRG phase may give rise to anisotropic transport phenomena [36-38, 51, 52] and associated anisotropic transport coefficients. It is known that some of these new transport coefficients do not contribute to the entropy generation and hence are non-dissipative [39,41]. These anisotropic transport coefficients were studied in Ref. [1] using relativistic Boltzmann equation in the RTA approximation.
Here we use the modified Chapman-Enskog (CE) method [42][43][44][45] for solving the relativistic Boltzmann-Uehling-Uhlenbeck (BUU) transport equation in the presence of magnetic fields and calculate anisotropic transport coefficients. In the CE method, we expand the collisions integral in terms of a complete set of orthogonal polynomials with magnetic fielddependent coefficients. This method can calculate the transport coefficients to a desired degree of accuracy in powers of dimensionless kinetic energy (normalized by temperature) compared to the previous methods; in this approach, the scaled kinetic energy acts as a smallness parameter for the expansion. This paper is organised as follows: In Sec. (2) we discuss the Boltzmann equation and the specific form of the linearised collision integral that is expanded in terms of the orthogonal polynomial. We also give analytical expressions for the magnetic field-dependent first and second-order transport coefficients. In Sec. (3) we discuss the numerical results, including the temperature and magnetic field dependence of anisotropic shear, bulk viscosities, and heat conductivity. Finally, we summarise this work in Sec. (4). Some detailed derivations of the useful formulas are given in App. (A)-(E). Throughout the manuscript we use natural unith = c = k B = 1 and mostly negative metric g µν = diag(+1, −1, −1, −1).

Linearised Boltzmann Equation and Transport Coefficients
In this work, we primarily follow the formalism of Ref. [40], which considers Bose enhancement factors in the collision integral of the Boltzmann equation, but here we also consider the presence of finite magnetic fields. In this case, the Boltzmann equation contains a force term due to the magnetic field and has the following form: where, the collision integral In the present work we consider only pion gas, the relevant cross sections for π-π reactions can be found in Ref. [46,47], we will be using a parametrised form of the iso-spin averaged differential cross-section given by, dσ(s) dΩ = 4 q 2 cm 1 9 sin 2 δ 0 0 + 1 3 9 sin 2 δ 1 1 cos 2 θ + 5 9 sin 2 δ 2 0 , where, The widths, Γ σ = 2.06q cm and Γ ρ = 0.095q cm qcm/mπ 1+(qcm/mπ) 2 2 with m σ = 5.8m π and m ρ = 5.53m π . Here √ s = E is the collision energy in the center of mass frame; q cm = E 2 2 − m 2 π . The collision integral vanishes for the local-equilibrium distribution function f 0 , which for our case has the following form , here T is the temperature and U µ is the time-like four velocity, µ is the chemical potential. If the deviation from local equilibrium is small, we can assume the off-equilibrium single particle distribution can be expanded in terms of a small parameter as: To be specific, we use the Knudsen number (the ratio of mean free path to system size) as and use the Chapman-Enskog method to calculate the off-equilibrium part of the distribution function. In equilibrium, the energy momentum tensor is defined as the second moment of f 0 , i.e., T µν 0 = dΓp µ p ν f 0 , the equilibrium energy density and the pressure can be calculated as ε 0 = U µ U ν T µν 0 , and p = − 1 3 ∆ µν T µν ; here ∆ µν = (g µν − U µ U ν ) is the projector operator orthogonal to U µ . From the above definitions, one can identify the parameter T in f 0 as the equilibrium local thermodynamic temperature. When considering an out-ofequilibrium system, identifying the temperature and the fluid four velocities is somewhat ad-hoc and require extra constraints to define them. This is usually done by choosing a specific frame; two widely used frames are Landau-Lifshitz(LL) and Eckart(EK). As per the EK definition, U µ is parallel to the particle four flow N µ = nU µ = dΓp µ f ; hence N µ does not contain any dissipative corrections. Here we notice that the use of the EK frame demands that there must be some conserved charges n, for our study we consider number of pions. With this choice of U µ the energy-momentum tensor can be written as where Π µν is the viscous stress tensor and I µ = (U ν T νρ ∆ µ ρ − h∆ µσ N σ ); h is the enthalpy. The viscous part is further split into the traceless shear part π µν , and non-zero trace bulk part Π as Π µν = π µν + Π∆ µν . In Eckart frame h∆ µσ N σ vanishes in the expression for I µ . While expressing f (x, p) as in Eq.(2.4) µ, T , and U µ are taken to be as parameters; with the further assumptions that n = dΓU µ p µ f = dΓU µ p µ f 0 = n 0 and ε = dΓ (U µ p µ ) 2 f = dΓ (U µ p µ ) 2 f 0 = ε 0 we can assign T and µ as temperature and the chemical potential in the off-equilibrium case. This can be clearly seen from the fact that the number, energy density, and pressure can be expressed as: Here z = m/T , S α n = Σ ∞ k=1 e kµ/T k α K n (kz), where K n (x) is the modified Bessel function of the second kind. With this brief introduction, let us discuss the linearised form of the Boltzmann equation used to calculate the anisotropic transport coefficients in magnetic fields. Introducing in the space-time derivative term of the Boltzmann equation Eq. (2.1), using Eq. (2.4) for the off-equilibrium distribution function we get The book keeping parameter has been introduced in the LHS (also in Eq.(2.4)) to count the order of space derivatives of the macroscopic parameters that determines the off-equilibrium distribution function. Now , matching the coefficients of same power in we get a hirearchy of equations; to the first order we have: where and Here we consider only Bosonic system (appropriate terms in the collision kernel is taken into account) which leads to the BUU equation Eq. (2.9). Now expanding the left hand side of the BUU equation, Where, . (2.15) Using the notation f 0 (x, p) ≡ f 0 p , the right hand side of Eq.(2.9) can be written as : In this work we consider only magnetic field, in this case the Maxwell stress tensor F µν = −Bb µν where b µν = µνρα Bρ B U α . Here B ρ is the magnetic four vector and B = B ρ B ρ is the magnitude of the magnetic field in the rest frame. The last term on the RHS gives : Where we have used the relation µνρα B ρ U α p ν U µ = 0 in deriving the above expression. Our job is to find the expression for φ which will enable us to calculate the transport coefficients discussed in the next few sections.

Shear Viscosity
In this section we derive the expression for shear viscous coefficients. Using Eq. (2.9), Eq. (2.11), and Eq. (2.17) we have Here we only consider the traceless symmetric part of the velocity gradient, and F µ = qBb µν p ν . Since magnetic field breaks the isotropy of the system the most general form of φ for shear viscosity only can be written as Ref. [1]: where, The expansion coefficient X n (B, τ ) are expressed in terms of the orthogonal Laguerre polynomial of order 5 2 and degree m as Ref. [40], C (n)µναβ are the projection tensors, whose properties have been discussed in details in App. (A) and in Ref. [1,41]. The particular form of φ was chosen so that we have a similar tensorial structure on both sides of Eq. (2.18). Once φ is known, we can evaluate the shear stress tensor as: µν αβ . For a system with anisotropy, the fourth-rank shear-viscous coefficient η µνψθ is defined as: where η n are the unknown coefficients to be determined by using appropriate tensor contractions defined in App. (A). The final expressions are: ; for brevity, henceforth, we do not explicitly write the B dependence in c One can calculate the transport coefficients to a desired degree by truncating the series Eq. (2.22) to the desired order. The expression for the first-order shear viscosity are as follows: The definitions of various symbols in the above set of equations are given in App. (B). We also report expression for the second-order shear viscosities and detailed calculations in App. (B) due to length constraints.

Bulk Viscosity
The bulk viscosity near the crossover temperature (from the hadronic phase to the QGP phase) is conjectured to show a peak; hence for the temperature range achieved in high energy heavy-ion collisions, bulk viscosity is an important transport coefficient to study along with the shear viscosity. The bulk viscosity is associated with the non-zero trace part of the dissipative correction to the energy-momentum tensor. In terms of φ, the bulk stress can be written as: We can further express ζ ηδ in terms of the projection basis as ζ || C ηδ . Now, the explicit expression of φ for the bulk viscosity in presence of the magnetic field is (2.37) Now comparing the coefficients of linearly independent C ηδ (n) on both sides of Eq. (2.37) we get, With the help of the following matching conditions dΓ To evaluate A (n) we consider the BUU equation for the thermodynamic force ∇ α U α The last term in the above equation vanishes because where we have used the fact that F µν U ν = 0. Now contracting the Boltzmann equation with P (n)ηδ (defined in App. A) and making use of the properties mentioned therein we get, and A 2 = 0, where α j is a function of z, S α n , γ and thermodynamic quantities (see Eqs. (C.15)) of App. (C) for details). Now expanding A (n) 's using Laguerre polynomial Restricting the expansion of a n after first term we get the first-order expression for bulk viscosities, We see that ζ's are independent of B which corroborates the finding of Ref. [1,18]. Similarly we calculate the second-order expressions for ζ's by truncating the series after second-order, The detailed calculation, along with the explicit expressions for various terms is given in App. (C).

Thermal Conductivity
Another quantity of interest is the thermal conductivity of hot and dense nuclear matter. The expression for the reduced heat flow is where h = (ε + p) /n is the enthalpy per particle of the system. In linear response theory the reduced heat flow is proportional to the temperature-gradient and fluid acceleration , when expressed using the kinetic theory definition, gives: The functional form of φ in this case was chosen considering the linearised BUU equation for heat flow The explicit expression for φ in this case is, Substituting it in Eq. (2.52), using λ µν = 2 n=0 λ n C µβ (n) and equating the coefficients of the thermodynamic force for heat flow we have: m (τ ) we find the scalar unknowns λ (n) using the properties of the projection operators (given in App. (A)) as:  .52), which leads to the following set of constraints: Here we have taken into consideration the fact that b

Temperature and field dependence of the shear viscosity
We intend to study the dependency of anisotropic parameters on the magnetic field orderby-order. First, we show the result for the shear viscosity in Fig.(1). Panel (a),(b), (c) of Fig.(1) shows the temperature dependence of η (n) (n = 0-4) for B = m 2 π , 5m 2 π , and 10m 2 π respectively. The bold lines correspond to the first-order results, whereas the dashed lines represent the second-order results. The following conclusions are made: (i) all five shear viscous coefficients monotonically increase with temperature irrespective of the magnetic fields, (ii) η 0 (the parallel component) has the largest value in the temperature range considered here; the difference between the first-order and second-order results only visible for η 0 at low temperature (≈ 80 MeV), we do not see any visible difference between the first and second-order results for other η's irrespective of the magnetic field. In panel (d) of

Temperature and field dependence of the bulk viscosity
We find that the bulk viscosity does not explicitly depend on the magnetic fields, and the order-by-order results are similar to Ref. [40] ; hence we do not discuss it in detail. However, it is interesting to note that the magnitude of ζ differs significantly order-by-order at low temperatures (∼ 100 MeV) compared to the shear viscosity.

Temperature and field dependence of the thermal conductivity
As was discussed earlier, as per our tensorial decomposition of thermal conductivity tensor λ µν , we have three scalar coefficients λ 0,1,2 . In panel (a),(b),(c) Fig. (2) we show the temperature dependence of λ n 's (n = 0-2) for B = m 2 π , B = 5m 2 π , and B = 10m 2 π respectively. The solid lines in all three panels correspond to the first-order, and the dashed lines correspond to the second-order results. As can be seen from Fig. (2) λ 0 (orange lines) is independent of magnetic fields (it is the same as the heat coefficient for zero magnetic fields); whereas λ 1,2 (black and green lines) increases with magnetic fields. It is also interesting to note that [λ 2 ] 1 , and [λ 2 ] 2 significantly differs at low temperatures (∼ 100 MeV) and the difference increases with larger magnetic fields.

Summary and Conclusion
In this work, we explored the magnetic field dependence of the anisotropic transport coefficients shear, bulk viscosity, and thermal conductivity order-by-order (up to secondorder) using the relativistic Boltzmann-Uehling-Uhlenbeck transport equation. The offequilibrium distribution function is obtained from the Chapman-Enskog method by expanding in terms of the associate Laguerre polynomial with magnetic field-dependent coefficients. For the current work, we only consider pions (bosons) while calculating the transport coefficients in the temperature range of 80-200 MeV. The magnetic field varies from zero to 10m 2 π . Five shear viscous coefficients (some of which are non-dissipative) monotonically increase as a function of temperature; a visible difference between first and second-order results was seen for η 0 (also known as the longitudinal viscosity) at low temperature ∼ 80-100 MeV. Other shear viscosity coefficients η 1-4 show no visible difference between first and second-order cases in the temperature range considered here; however, these coefficients are sensitive to the magnitude of magnetic fields. On the other hand, bulk viscous coefficients show no explicit dependence on the external magnetic field; however, they show greater sensitivity for first and second-order cases compared to the shear viscosity in the low-temperature regime. The thermal conductivity turned out to be sensitive to both the magnetic field and the expansion order. The difference between the first-and second-order results is prominent in the low-temperature regime and is also sensitive to the external magnetic fields.
At this point, we would like to discuss a few relevant facts. Firstly, our approach must work in the low field limit as we consider the local equilibrium distribution function is the same as the zero-field case and the magnetic field enters only in the correction δf . Secondly, we assume that the electric field vanishes, or in other words, our result is applicable for large magnetic Reynolds numbers only (a.k.a. the ideal-MHD limit). Moreover, the expansion of the collision integral in terms of the Laguerre polynomial is not the unique choice; other choices are possible (see Ref. [48][49][50]). While calculating the collision integral, we consider the pion cross-section for vacuum; hence one can improve upon this point by taking the pionic cross-section in the medium. Last but not least similar calculations can be carried out in the future for strong-field limits.

A Projection Tensors
In this appendix we give details about the projection tensor used to construct the Second C µν (n) and fourth-rank tensor C µναβ (n) . C µν (n) 's are constructed from the second-rank basis projection tenor P (m) µν which are defined as, µν have the following properties, We define C µν (n) 's in terms of these basis as: The basis for the fourth-rank tensor C µναβ (n) 's are constructed from the P (m) µν 's as Using the second rank projection tensors we define the fourth rank projection tensors as, νµαβ .
An important property of P Using these basis fourth-rank projection tensor, we construct C µναβ (n) 's as: µν αβ .
One can further verify that

B Detail Derivation of the Shear Viscosity
In this appendix we give the details of the results presented in Sec. (2.1). We start with the right hand side of the BUU equation (2.18) and denote the two terms as T 1 , and T 2 , where and, The first term of the above equation vanishes because, where we have used Eq. (2.22) for X n . Thus, We further simply the above equation and express in terms of p µ p ν using the relation − 1 2 b µκ νδ + b µδ νκ + b νκ µδ + b νδ µκ = 1 2 C µνκδ (2) + C µνκδ (4) , and using the properties of the basis projection tensors, .

(B.4)
Our goal is to find the unknown coefficients X n , for that purpose we use the above form of the collision integral in the BUU equation Eq.(2.18) and contract both sides with P ηδαβ (n) ; after that we integrate with a weight function L 5 2 j (τ p ) p α p β and get Eqs. (2.28)-(2.30). Various terms in these equations are defined as, Where γ n with only subscript corresponds to the one mentioned in Ref. [40], c nj corresponds to the quantity c nj in the above mentioned paper, and In Eq. (2.29) and Eq. (2.30) we also get terms like Eqs. (B.6),(B.7),(B.8) for C (2) , C (4) , and ξ (2) nj and ξ (4) nj but on careful observation one see that these terms vanishes because ∆ αβ ∆ ηδ + ∆ αη ∆ βδ + ∆ αδ ∆ βη = 0. Once we evaluate the collision integral as described above we further proceed to calculate the shear viscous coefficients from the definition: Now using the definition Eq. (2.24) in the above expression and equating the coefficient of ∂ λ U κ we have: λκψθ . If we keep terms up to first-order in X , we have the first-order results given in Eq. (2.31)- (2.35). Similarly by keeping terms up to second in the coefficient X n we get,  The determinant D in the above expressions is: 1 l 00 l 01 + 4ξ 10 + 2γ

C Detail Derivation of the Bulk Viscosity
The order-by-order calculation of the bulk viscosity is similar to the shear viscosity. The kinetic theory definition of the bulk viscous stress is given by, where C ηδ To calculate the ζ || , ζ ⊥ , ζ × we contract both sides of the Eq.(C.2) with µν P n ηδ and get, From the above set of equations it is evident that the term associated with Z 2 must vanish. Now integrating Eq. (C.9) with weight L 1 2 j (τ ) we have, The above equation can be rewritten as, where, 0 (τ ) = 3 2 − τ , dΓ p f 0 1 + f 0 Qφ = 0, one can show α 0 = α 1 = 0 and a 0j = a j0 = a 1j = a j1 = 0. If we limit the series to a finite l, l+1 n=0 a n(l) Finally truncating the series upto first-order we arrive at the expressions for [ζ n ] 1 s: Similarly the second-order result is calculated and they are ,

D Detail Derivation of the Thermal Conductivity
In this case the φ in δf has the following form: The expression for the reduced heat flow is gven by, where h is the enthalpy per particle of the system. Using φ in the kinetic theory definition we have Like for the other cases, the thermal conductivity tensor can be decomposed as λ µν = 2 n=0 λ n C µν (n) ; hence we have As usual, we obtain λ 0 by contracting both sides of the above equation with P βθ (0) , Contracting Eq. (D.4) with P βθ (1) , and P βθ (2) gives Solving the above two equations for λ 1 and λ 2 we have To derive the final expressions for λ's as given in Eqs. (2.54),(2.55),(2.56) we note the fact that the following relation holds for the integral appeared in the above equation The first term disappears on contraction with C (2)αµ µ σ , and I 2 is Now we observe that C (2)αµ µα = 0 and C (1)αµ µα = −2; hence we get the final expressions Eqs. (2.54),(2.55),(2.56). Further, we expand the coefficients Y j as Substituting the above expansion in Eq.(2.54) we get, where the expression for β n , (D.14) n .

(D.15)
To derive the unknown coefficients b (1) n we use the BUU equation, Now, the last term on the right hand side of the above equation can be further simplified as, Here the β n 's are function of z and the full form an be found in Ref. [40]. For n = 1, 2 we have the following expression for β n From the above definition, for n = 1, 2 we have Where the coefficients b (m) n (m = n = 1, 2) can be found from

E Other Useful Formulas
For all the calculations presented in this manuscript we have further made use of the following simplification of the collision term. Starting with Eqs. (C. 15),(C.16) we transform the incoming and outgoing four-momentum of the colliding particles p µ , k µ , p µ , k µ to the following relative and global momentum: Further using the polar-coordinate and rapidity ψ we express P α = P (cosh ψ, sinh ψ sinθ cosφ, sinh ψ sinθ sinφ, sinh ψ cosθ). For convenience we use the following orthonormal i.e.,(e α (i) e α(j) = δ ij ) space-like four vectors in subsequent calculations: e α (i) are also orthogonal to the P α , i.e., e α (i) P α = 0 for (i = 1, 2, 3). In the centre of mass frame, P α reads P α = (P, 0), and g α = (0, g ), and e α (i) = (0, e i ) have only space components. Therefore we can write any three vector g in terms of the basis vectors as: g = g (sin θ cos ϕ e 1 + sin θ sin ϕ e 2 + cos θ e 3 ) . (E.4) Here θ, φ are the angle subtended by g with the e i . In the permanent local rest frame we can define a four-velocity U α = (cosh χ, sinh χē 3 ) so that P 0 = P α U α (see Ref. [42,43] for more details). Let us define Θ the scattering angle in the center of mass frame g · g = g 2 cos Θ and ψ such as sinh ψ = g/mc so that P = 2mc cosh ψ. P 0 = 2mc cosh ψ cosh χ, g 0 = mc sinh ψ sinh χ cos θ, g 0 = mc sinh ψ sinh χ cos θ , (E.5) where cos θ = cos θ cos Θ − sin θ sin Θ cos φ . Using these transformed variables we can write: f Further, on evaluation of the terms a mn , b mn , c mn (for bulk, thermal conductivity, and shear respectively) we get Ref. [40]. Finally C mn 's are evaluated in term of the χ integrals (defined below):