Modelling stochastic fluctuations in relativistic kinetic theory

TL;DR

This paper develops a Lorentz-covariant stochastic framework for relativistic kinetic theory, demonstrating causality, stability, and the impact of system size on Boltzmann's molecular chaos, with applications to ultrarelativistic gases and collision kernels.

Contribution

It introduces a covariant stochastic theory for relativistic kinetic systems, ensuring causality and stability, and analyzes fluctuations and correlations in various relativistic scenarios.

Findings

The stochastic theory is causal and covariantly stable.

Boltzmann's molecular chaos breaks down in small systems with N<5.

Ultrarelativistic gases' transient hydrodynamics captures 80% of equilibrium fluctuations.

Abstract

Using the information current, we develop a Lorentz-covariant framework for modeling equilibrium fluctuations in relativistic kinetic theory in the grand-canonical ensemble. The resulting stochastic theory is proven to be causal and covariantly stable, and its predictions do not depend on the choice of spacetime foliation used to define the grand-canonical probabilities. As expected, in a box containing $N{>}5$ particles, Boltzmann's molecular chaos postulate is broken with (almost exact) probability $N^{-1/2}$, leading to a breakdown of the Boltzmann equation in small systems. We also verify that, in ultrarelativistic gases, transient hydrodynamics already accounts for at least 80% of the equilibrium fluctuations of the stress-energy tensor at a given time. Finally, we compute the correlators at non-equal times for two selected collision kernels: That of a chemically active diluted solution, and that of ultrarelativistic scalar particles self-interacting via a quartic potential. For the former, we compute the density-density correlators analytically in real space, and dehydrodynamization of the stochastic theory is proven to occur whenever the mean free path diverges at high energy.