# Thermalization of non-stochastic Hamiltonian systems

## TL;DR

This paper proves that non-stochastic Hamiltonian systems can thermalize, establishing that stochasticity is not a necessary condition for relaxation to equilibrium in classical many-body systems.

## Contribution

It demonstrates thermalization in non-stochastic Hamiltonian systems without relying on the thermodynamic limit or specific interaction potentials.

## Key findings

- Thermalization occurs in both stochastic and non-stochastic systems.
- The proof is valid for arbitrary classical Hamiltonian systems.
- Adiabatic invariance links microscopic Hamiltonian structure to thermodynamics.

## Abstract

Ability of dynamical systems to relax to equilibrium has been investigated since the invention of statistical mechanics, which establishes the connection between dynamics of many-body Hamiltonian systems and phenomenological thermodynamics. The key link in this connection is stochasticity, which translates the deterministic evolution of a dynamical system to its probabilistic exploration of the state space. To-date research focuses on determining the conditions of stochasticity for particular systems. Here we propose an alternative agenda and prove thermalization for non-stochastic Hamiltonian systems. This shows that thermalization happens in both stochastic and non-stochastic systems, reducing the need to rely on stochasticity in a "coarse-grained" analysis. The result is valid for an arbitrary classical Hamiltonian system and does not rely on the thermodynamic limit or the particular form of the interaction potential. It utilizes the property of adiabatic invariance, and reveals a deep relation between the structure of the microscopic Hamiltonian and macroscopic thermodynamics.