Mind the gap: Boltzmannian versus Gibbsian equilibrium

There are two main theoretical frameworks in statistical mechanics, one associated with Boltzmann and the other with Gibbs. Despite their well-known differences, there is a prevailing view that equilibrium values calculated in both frameworks coincide. We show that this is wrong. There are important cases where the Boltzmannian and Gibbsian equilibrium concepts yield different outcomes. Furthermore, the conditions under which equilibria exists are different for Gibbsian and Boltzmannian statistical mechanics. There are, however, special circumstances under which it is true that the equilibrium values coincide. We prove a new theorem providing sufficient conditions for this to be the case.