# Irreversibility and typicality: A simple analytical result for the Ehrenfest model

## TL;DR

This paper uses the Ehrenfest model to analytically demonstrate that macroscopic irreversibility is a typical property of stochastic processes, showing most trajectories behave irreversibly and align with ensemble averages, confirmed by simulations and proofs.

## Contribution

It provides a simple analytical framework clarifying the typicality of irreversibility in the Ehrenfest model, supported by rigorous proofs and numerical validation.

## Key findings

- Most trajectories exhibit irreversible behavior
- Trajectories stay close to ensemble averages
- Rigorous proof of typicality in the thermodynamic limit

## Abstract

With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a non-ambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining "very close" to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit.