The Boltzmann and Gibbs approaches to statistical mechanics have very
different definitions of equilibrium and entropy. The problems associated with
this are discussed and it is suggested that they can be resolved, to produce a
version of statistical mechanics incorporating both approaches, by redefining
equilibrium not as a binary property (being/not being in equilibrium) but as a
continuous property (degrees of equilibrium) measured by the Boltzmann entropy
and by introducing the idea of thermodynamic-like behaviour for the Boltzmann
entropy. The Kac ring model is used as an example to test the proposals
