Order-by-order anisotropic transport coefficients of a magnetised fluid: a Chapman-Enskog approach

Abstract

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 [1] 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.

Transport Coefficients of Relativistic Matter: A Detailed Formalism with a Gross Knowledge of Their Magnitude

“…By taking constant relaxation time and tuning it from τ c = 0.41 fm (blue dash line) to τ c = 3.94 fm (red solid line), we can cover the theoretical data of Marty et al [211], Singha et al [216], and Plumari et al [250]. This gives the impression that the relaxation time of QGP lies somewhere between 0.4 to 4.0 fm in the momentum and temperature-independent relaxation time model of the Next, we can also write down the final expression of η in the hadronic phase by using Equation (205) for multicomponent baryonic and mesonic mixtures in the limit µ = 0. Final expressions of η for the hadronic (H) phase with all the baryons (B) and mesons (M) can be written as:…”

“…for bosons, and a = −1 for fermions. The BTE with the C[ f ] given in Equation ( 40) is often called the Relativistic Uheling-Uhlenbeck Equation(UUE) [36,205]. The BTE with C[ f ] given in Equation (39) or Equation (40) is notoriously complicated to solve due to the non-linear nature of its collision term, which makes it a non-linear partial integro-differential equation.…”

“…In off-central HIC, the spectators moving with a relativistic speed produce electromagnetic fields [123][124][125][126][127][128][129][130][131][132], which affect the medium constituents of the QGP created [128,; apart from that, there can also be a significant magnitude of the electromagnetic field produced by the medium constituents of the QGP itself [146,194,195]. Inspired by this, the study of the properties of QGP under electromagnetic fields has gained considerable attention, and researchers working on HIC phenomenology have carried out extensive work on the subject, which spans from deriving magnetohydrodynamics equations from BTE to the calculation of the multi-component structure of transport coefficients of QGP [196][197][198][199][200][201][202][203][204][205]. A detailed review on the topic of magnetohydrodynamics can be found in Refs.…”