A systematic expansion for relativistic causal hydrodynamics

TL;DR

This paper derives a relativistic causal diffusion equation from the Boltzmann equation using a systematic expansion, highlighting the importance of next-to-leading order terms and connecting microscopic correlation functions to hydrodynamics.

Contribution

It provides a systematic derivation of Kelly's relativistic causal diffusion equation from microscopic theory, extending the Green-Kubo relation and the f-sum rule.

Findings

NLO term is crucial in certain dynamical regimes

Derivation from correlation functions confirms the microscopic basis of Kelly's equation

Method can be generalized to full hydrodynamic equations

Abstract

A systematic expansion of the Boltzmann equation for the diffusion of dilute tracers, in powers of the Knudsen number, carried out to next-to-leading order (NLO), gives the relativistic causal diffusion equation (Kelly's equation). Using dimensionless combinations of dynamical quantities, we show when the small NLO term plays a crucially important role. We proceed to show that a derivation of Kelly's equation from a microscopic theory of the correlation function of the number density of diffusers is possible. The correlator fulfils the Green-Kubo relation for the diffusion constant, as well as an f-sum which goes beyond a purely phenomenological causal theory. We argue that the construction generalizes to the full hydrodynamic equations.