# Entropy and the second law for driven, or quenched, thermally isolated systems

## TL;DR

This paper investigates the behavior of entropy in isolated classical systems under external driving or quenches, demonstrating that while extensive entropy generally does not decrease for large systems, small systems can exhibit negative mean entropy changes.

## Contribution

It introduces a refined micro-canonical entropy framework and demonstrates that the second law's non-decreasing entropy principle holds in the large system limit but can be violated in finite systems.

## Key findings

- Extensive entropy change remains non-negative in large systems.
- Finite systems can exhibit negative mean entropy change under small driving.
- Results are validated using quenched N-dimensional harmonic oscillators.

## Abstract

The entropy of a thermally isolated system should not decrease after a quench or external driving. For a classical system following Hamiltonian dynamics, we show how this statement emerges for a large system in the sense that the extensive part of the entropy change does not become negative. However, for any finite system and small driving, the mean entropy change can well be negative. We derive these results using as micro-canonical entropy a variant recently introduced by Swendsen and co-workers called "canonical". This canonical entropy is the one of a canonical ensemble with the corresponding mean energy. As we show by refining the micro-canonical Crooks relation, the same results hold true for the two more conventional choices of micro-canonical entropy given either by the area of a constant energy shell, the Boltzmann entropy, or the volume underneath it, the Gibbs volume entropy. These results are exemplified with quenched $N$-dimensional harmonic oscillators.