Ergodic theory, interpretations of probability and the foundations of statistical mechanics

The traditional use of ergodic theory in the foundations of equilibrium
statistical mechanics is that it provides a link between thermodynamic
observables and microcanonical probabilities. First of all, the ergodic
theorem demonstrates the equality of microcanonical phase averages and
infinite time averages (albeit for a special class of systems, and up to a
measure zero set of exceptions). Secondly, one argues that actual
measurements of thermodynamic quantities yield time averaged quantities,
since measurements take a long time. The combination of these two points
is held to be an explanation why calculating microcanonical phase averages
is a successful algorithm for predicting the values of thermodynamic
observables. It is also well known that this account is problematic.
This survey intends to show that ergodic theory nevertheless may have
important roles to play, and it explores three other uses of ergodic theory.
Particular attention is paid, firstly, to the relevance of specific interpretations
of probability, and secondly, to the way in which the concern with
systems in thermal equilibrium is translated into probabilistic language.
With respect to the latter point, it is argued that equilibrium should not be
represented as a stationary probability distribution as is standardly done;
instead, a weaker definition is presented.