Weighted Time Averages and Weak Convergence to Equilibrium in Quantum Integrable Systems

TL;DR

This paper introduces a weighted time-averaging method for quantum integrable systems, proving convergence to equilibrium states despite quasiperiodic dynamics, and validates the approach with a three-spin model.

Contribution

It provides a novel weighted averaging technique that ensures convergence to equilibrium in quantum integrable systems with pure point spectra.

Findings

Weighted time averages converge to equilibrium states.

The method applies to systems with pure point spectra.

Validated with a three-spin quantum integrable model.

Abstract

This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as $t\to\infty$. To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.