# Equilibration of energy in slow-fast systems

## TL;DR

This paper demonstrates that in slow-fast systems, the violation of ergodicity in the fast subsystem can facilitate the entire system reaching statistical equilibrium, especially when the fast dynamics have multiple ergodic components.

## Contribution

It introduces a principle showing ergodicity violation in fast dynamics promotes equilibration in slow-fast systems, supported by analysis of springy billiards.

## Key findings

- Equilibration always occurs in the studied systems.
- Positive exponential rate of equilibration requires multiple ergodic components.
- Fast subsystem ergodicity violation aids in driving the whole system to equilibrium.

## Abstract

Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. On the other hand, it is known that in slow-fast systems ergodicity of the fast sub- system impedes the equilibration of the whole system due to the presence of adiabatic invariants. Here, we show that the violation of ergodicity in the fast dynamics effectively drives the whole system to equilibrium. To demonstrate this principle we investigate dynamics of the so-called springy billiards. These consist of a point particle of a small mass which bounces elastically in a billiard where one of the walls can move - the wall is of a finite mass and is attached to a spring. We propose a random process model for the slow wall dynamics and perform numerical experiments with the springy billiards themselves and the model. The experiments show that for such systems equilibration is always achieved; yet, in the adiabatic limit, the system equilibrates with a positive exponential rate only when the fast particle dynamics has more than one ergodic component for certain wall positions.