Microscopic and probabilistic approach to thermal steady state based on a dice and coin toy model

In this article we present an educational approach to thermal equilibrium which was tested on a group of 13 undergraduate students at the University of Trento. The approach is based on a stochastic toy model, in which bodies in thermal contact are represented by rows of squares on a cardboard table, which exchange coins placed on the squares based on the roll of two dice. The discussion of several physical principles, such as the exponential approach to equilibrium, the determination of the equilibrium temperature, and the interpretation of the equilibrium state as the most probable macrostate, proceeds through a continual comparison between the outcomes obtained with the toy model and the results of a real experiment on the thermal contact of two masses of water at different temperatures. At the end of the sequence, a re-analysis of the experimental results in view of both the Boltzmann and Clausius definitions of entropy reveals some limits of the toy model, but also allows for a critical discussion of the concepts of temperature and entropy. In order to provide the reader with a feeling of how the sequence was received by students, and how it helped them understand the topics introduced, we discuss some excerpts from their answers to a conceptual item given at the end of the sequence.