The Convergence Problem Of Gradient Expansion In The Relaxation Time Approximation

TL;DR

This paper derives a formal solution to the 3+1 D Boltzmann Equation in relaxation time approximation, revealing conditions under which the gradient series converges and challenging the necessity of local thermal equilibrium for hydrodynamic descriptions.

Contribution

It provides a formal integral solution to the Boltzmann Equation and analyzes the convergence of the gradient expansion, offering new insights into hydrodynamic applicability.

Findings

Gradient series can have finite radius of convergence.

Proximity to local thermal equilibrium is not required for hydrodynamics.

Exponential decay of non-hydrodynamic terms in the gradient series.

Abstract

We obtain a formal integral solution to the 3+1 D Boltzmann Equation in relaxation time approximation. The gradient series obtained from this integral solution contains exponentially decaying non-hydrodynamic terms. It is shown that this gradient expansion can have a finite radius of convergence under certain assumptions of analyticity. We then argue that, in the relaxation time model, proximity to local thermal equilibrium is not necessary for the system to be described by hydrodynamic equations.