Gibbs variational principles and Boltzmann irreversible theorem

Abstract

We analyze the Gibbs variational principles associated with the probability distributions of (i) an isolated system and (ii) a system at constant temperature. We give an example of using the Gibbs inequality to obtain the free energy and analyze the phase diagram of an Ising model with competing interactions. We also review the Boltzmann irreversible theorem, and show how it is connected to the Gibbs variational principles. This connection is established by using the Kolmogorov equation for the evolution of the probability distribution, which predicts a monotonic increase of entropy for an isolated system, and a decrease of the free energy for a system in contact with a thermal reservoir.