# Micro-reversibility and thermalization with collisional baths

## TL;DR

This paper investigates the role of micro-reversibility in thermalization processes involving collisional baths, emphasizing the importance of Liouville's theorem and the full phase space in ensuring proper thermalization and thermodynamic consistency.

## Contribution

It clarifies the conditions under which micro-reversibility and thermalization hold in classical and semi-classical collisional systems, highlighting the necessity of including all phase space variables.

## Key findings

- Liouville's theorem is essential for micro-reversibility in collisional baths.
- The velocity distribution of incident particles is non-Maxwellian when all phase space variables are considered.
- Certain collision rules can violate Liouville's theorem and break the second law of thermodynamics.

## Abstract

Micro-reversibility, that is, the time reversal symmetry exhibited by microscopic dynamics, plays a central role in thermodynamics and statistical mechanics. It is used to prove fundamental results such as Onsager reciprocal relations or fluctuation theorems. From micro-reversibility one can also prove that isolated systems and systems in contact with a thermal bath relax to micro-canonical and canonical ensembles, respectively. However, a number of problems arise when trying to reproduce this proof for classical and quantum collisional baths, consisting of particles from an equilibrium reservoir interacting with a localized system via collisions. In particular, it is not completely clear which distribution for the velocities of the incident particles warrants thermalization. Here, we clarify these issues by showing that Liouville's theorem is a necessary condition for micro-reversibility in classical and semi-classical scenarios. As a consequence, one must take into account all the canonical coordinates and momenta, including the position of the incident particles. Taking into account the position modifies the effective probability distribution of the velocity of the particles that interact with the system, which is no longer Maxwellian. We finally show an example of seemingly plausible collision rules that nonetheless violate the Liouville theorem and allow one to design machines that beat the second law of thermodynamics.