Origin of the Relaxation Time in Dissipative Fluid Dynamics

TL;DR

This paper explains how the relaxation time in dissipative fluid dynamics is determined by the singularity structure of retarded Green's functions, linking microscopic properties to macroscopic relaxation behavior.

Contribution

It establishes a direct connection between the pole structure of Green's functions and the relaxation equations governing dissipative currents.

Findings

Relaxation time corresponds to the inverse of the Green's function pole closest to the origin.

Linearized equations reduce to relaxation-type equations when the pole is a simple imaginary axis pole.

Standard gradient expansion is recovered when the relaxation time approaches zero.

Abstract

We show how the linearized equations of motion of any dissipative current are determined by the analytical structure of the associated retarded Green's function. If the singularity of the Green's function, which is nearest to the origin in the complex-frequency plane, is a simple pole on the imaginary frequency axis, the linearized equations of motion can be reduced to relaxation-type equations for the dissipative currents. The value of the relaxation time is given by the inverse of this pole. We prove that, if the relaxation time is sent to zero, or equivalently, the pole to infinity, the dissipative currents approach the values given by the standard gradient expansion.