Transient oscillations in a macroscopic effective theory of the Boltzmann equation

TL;DR

This paper develops a new transient effective theory for the relativistic Boltzmann equation that captures oscillatory behavior of non-hydrodynamic modes in momentum-anisotropic systems, revealing phenomena previously observed only in strongly coupled theories.

Contribution

It introduces a systematic approach to include non-hydrodynamic modes in the Boltzmann equation, demonstrating transient oscillations in weakly coupled systems with momentum anisotropy.

Findings

Transient oscillations occur in weakly coupled Boltzmann systems.

Non-perturbative treatment of momentum anisotropy is crucial.

Oscillations resemble phenomena seen in strongly coupled theories.

Abstract

A new transient effective theory of the relativistic Boltzmann equation is derived for locally momentum-anisotropic systems. In the expansion of the distribution function around a local "quasi-equilibrium" state a non-hydrodynamic dynamical degree of freedom is introduced at leading order that breaks local momentum isotropy. By replacing the deviation of the distribution function from this quasi-equilibrium state in terms of moments of the leading-order distribution and applying a systematic power counting scheme that orders the non-hydrodynamic modes by their microscopic time scales, a closed set of equations for the dynamical degrees of freedom is obtained. Truncating this set at the level of the slowest non-hydroynamic mode we find that it exhibits transient oscillatory behavior -- a phenomenon previously found only in strongly coupled theories, where it appears to be generic. In weakly coupled systems described by the Boltzmann equation, these transient oscillations depend on the breaking of local momentum isotropy being treated non-perturbatively at leading order in the expansion of the distribution function.