Model for the derivation of the Gibbs distribution

We propose a simple stochastic model, as a pedagogical aid, to illustrate how the equilibrium Gibbs distribution comes about. (N+1) distinguishable systems with identical non-degenerate energy spectra interact pairwise; the energy levels of all systems are equally spaced; a quantum of energy is exchanged at random whenever two systems interact. Probability distributions are derived, without assumptions, for various bath sizes (that is, for various values ofN); they only become Gibbsian asN→∞. The underlying conditions that bring about the Gibbs distribution become quite transparent.