Ballistic particle transport and Drude weight in gases

TL;DR

This paper demonstrates that particle transport in non-relativistic gases is inherently ballistic, characterized by a Drude weight directly proportional to the gas density, and provides a method to compute this in integrable quantum systems.

Contribution

It establishes a universal relation between the Drude weight and gas density in any dimension and ensemble, and derives explicit formulas for the Lieb-Liniger Bose gas using generalized thermodynamics.

Findings

Particle current in non-relativistic gases is always ballistic.

The Drude weight equals twice pi times the gas density.

The derived formulas agree with generalized hydrodynamics predictions.

Abstract

Owing to the fact that the particle current operator in non-relativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight $\Delta$. The Drude weight can be calculated within linear response theory. It is given by the formula $\Delta = 2 \pi D$, where $D$ is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current auto-correlation function.