Thermalization is typical in large classical and quantum harmonic systems

TL;DR

This paper provides an analytical criterion demonstrating that large harmonic systems, both classical and quantum, typically thermalize for most initial conditions, offering a new perspective beyond chaos and ergodicity.

Contribution

The authors derive a general criterion for thermalization in large harmonic systems applicable to both classical and quantum models, independent of chaos or ergodicity.

Findings

Thermalization is typical for large systems with random initial conditions.

Non-typical initial conditions can lead to energy equipartition without thermalization.

The results bridge classical and quantum theories of thermalization in integrable models.

Abstract

We establish an analytical criterion for dynamical thermalization within harmonic systems, applicable to both classical and quantum models. Specifically, we prove that thermalization of various observables, such as particle energies in physically relevant random quadratic Hamiltonians, is typical for large systems ($N \gg 1$) with initial conditions drawn from the microcanonical distribution. Moreover, we show that thermalization can also arise from non-typical initial conditions, where only a finite fraction of the normal modes is excited. A different choice of initial conditions, such as all the initial energy localized in a single particle, instead leads to energy equipartition without thermalization. Since the models we consider are integrable, our findings provide a general dynamical basis for an approach to thermalization that bypasses chaos and ergodicity, focusing instead on the physical requirement that thermodynamic observables depend on a large number of normal modes, and build a bridge between the classical and quantum theories of thermalization.