Kubo formulas for relativistic fluids in strong magnetic fields

Abstract

Magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction, as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev's non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of the equilibrium theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium (imaginary-time) infrared Green's functions defined with respect to the equilibrium statistical operator.

Relativistic magnetohydrodynamics

“…Papers [25,26] studied transport coefficients for relativistic fluids subject to an external magnetic field. While this does not correspond to MHD in the sense described in this paper (we define MHD as a theory in which magnetic field or its auxiliary is a dynamical degree of freedom), a fluid in external field is a fundamental building block for MHD.…”

“…Some of our results differ from those in refs. [25,26]: the analysis of thermodynamics, the number of transport coefficients, constraints on transport coefficients imposed by the positivity of entropy production, and some of the Kubo formulas. The details are given in appendix B.…”

“…Rewriting our constraints (3.19) in terms of the HSR coefficients, we find On the other hand, the constraints coming from the second law in ref. [25] state that all the dissipative HSR transport coefficients must be positive. We find that the constraints on dissipative transport coefficients (B.2) are in fact weaker.…”

“…Denoting the transport coefficients in ref. [25] with the subscript HSR, the relations to our transport coefficients are as follows: assuming the convention 0123 = 1. This lists eleven transport coefficients compared to ten HSR coefficients, hence under this mapping the eleven transport coefficients are not independent.…”

“…Ref. [25], abbreviated below as HSR, studied relativistic hydrodynamics of parity-invariant fluids in external non-dynamical magnetic field. HSR enumerated the transport coefficients, giving a relativistic version of the classification in the book [30], §13, and derived the Kubo formulas for transport coefficients in an operator formalism.…”