Thermalisation as Diffusion in Hilbert Space

TL;DR

This paper presents a microscopic theory of thermalisation in quantum systems, modeling interactions as random variables, and predicts relaxation timescales and non-Markovian effects validated by numerical tests.

Contribution

It introduces a diffusion-based framework for thermalisation beyond Markovian assumptions, accounting for interaction-induced level broadening and non-Markovian dynamics.

Findings

Good agreement with numerical simulations for heavy-tailed couplings

Predicts thermalisation timescales from level broadening distributions

Generalizes global balance to non-Markovian regimes

Abstract

We develop a microscopic theory of thermalisation for a thermometer coupled to a many-body bath beyond standard Markovian and Fermi-golden-rule assumptions. By modeling interaction matrix elements in the non-interacting basis as independent random variables, we derive a diffusion-propagator expression for the reduced dynamics and show that relaxation is controlled by the distribution of interaction-induced level broadenings. The theory predicts a thermalisation timescale set by the inverse typical broadening and yields a non-Markovian generalization of global balance. Exact-diagonalization tests for heavy-tailed L{\'e}vy couplings, an all-to-all transverse-field Ising model, and the one-dimensional Imbrie model show good agreement with these predictions.