The Ehrenfest urn revisited: Playing the game on a realistic fluid model

TL;DR

This paper simulates the Ehrenfest urn model using molecular dynamics of Lennard-Jones fluids, analyzing its Markovian properties and confirming theoretical predictions in a realistic fluid setting.

Contribution

It demonstrates how the Ehrenfest urn process can be realistically modeled through molecular dynamics, validating Markovian behavior and theoretical transition probabilities.

Findings

Waiting time distribution approaches exponential in the limit

Delta z behaves as a Markov chain in the gas phase

Transition probabilities match Ehrenfest theory predictions

Abstract

The Ehrenfest urn process, also known as the dogs and fleas model, is realistically simulated by molecular dynamics of the Lennard-Jones fluid. The key variable is Delta z, i.e. the absolute value of the difference between the number of particles in one half of the simulation box and in the other half. This is a pure-jump stochastic process induced, under coarse graining, by the deterministic time evolution of the atomic coordinates. We discuss the Markov hypothesis by analyzing the statistical properties of the jumps and of the waiting times between jumps. In the limit of a vanishing integration time-step, the distribution of waiting times becomes closer to an exponential and, therefore, the continuous-time jump stochastic process is Markovian. The random variable Delta z behaves as a Markov chain and, in the gas phase, the observed transition probabilities follow the predictions of the Ehrenfest theory.