A study of the Boltzmann and Gibbs entropies in the context of a stochastic toy model

In this article we reconsider a stochastic toy model of thermal contact, first introduced in Onorato et al (2017 Eur. J. Phys. 38 045102), showing its educational potential for clarifying some current issues in the foundations of thermodynamics. The toy model can be realized in practice using dice and coins, and can be seen as representing thermal coupling of two subsystems with energy bounded from above. The system is used as a playground for studying the different behaviours of the Boltzmann and Gibbs temperatures and entropies in the approach to steady state. The process that models thermal contact between the two subsystems can be proved to be an ergodic, reversible Markov chain; thus the dynamics produces an equilibrium distribution in which the weight of each state is proportional to its multiplicity in terms of microstates. Each one of the two subsystems, taken separately, is formally equivalent to an Ising spin system in the non-interacting limit. The model is intended for educational purposes, and the level of readership of the article is aimed at advanced undergraduates.