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.