  The received wisdom in statistical mechanics is that isolated systems, when
left to themselves, approach equilibrium. But under what circumstances does an
equilibrium state exist and an approach to equilibrium take place? In this
paper we address these questions from the vantage point of the long-run
fraction of time definition of Boltzmannian equilibrium that we developed in
two recent papers (Werndl and Frigg 2015a, 2015b). After a short summary of
Boltzmannian statistical mechanics and our definition of equilibrium, we state
an existence theorem which provides general criteria for the existence of an
equilibrium state. We first illustrate how the theorem works with a toy
example, which allows us to illustrate the various elements of the theorem in a
simple setting. After commenting on the ergodic programme, we discuss
equilibria in a number of different gas systems: the ideal gas, the dilute gas,
the Kac gas, the stadium gas, the mushroom gas and the multi-mushroom gas. In
the conclusion we briefly summarise the main points and highlight open
questions.
