Density dynamics from current auto-correlations at finite time- and length-scales

TL;DR

This paper links the broadening of inhomogeneous density distributions in quantum systems to current auto-correlation functions at finite times, providing a method to estimate diffusion constants and confirming previous results in spin chains.

Contribution

It introduces a generalized relation connecting density variance growth to current auto-correlations at finite times, applicable beyond diffusive regimes.

Findings

Derived a generalized Einstein relation for quantum systems.

Validated the method by confirming the magnetization diffusion constant in a spin chain.

Provided a practical approach to estimate transport coefficients from finite-time auto-correlations.

Abstract

We consider the increase of the spatial variance of some inhomogeneous, non-equilibrium density (particles, energy, etc.) in a periodic quantum system of condensed matter-type. This is done for a certain class of initial quantum states which is supported by static linear response and typicality arguments. We directly relate the broadening to some current auto-correlation function at finite times. Our result is not limited to diffusive behavior, however, in that case it yields a generalized Einstein relation. These findings facilitate the approximation of diffusion constants/conductivities on the basis of current auto-correlation functions at finite times for finite systems. Pursuing this, we quantitatively confirm the magnetization diffusion constant in a spin chain which was recently found from non-equilibrium bath scenarios.