  We examine the question of whether the formal expressions of equilibrium
statistical mechanics can be applied to time independent non-dissipative
systems that are not in true thermodynamic equilibrium and are nonergodic. By
assuming the phase space may be divided into time independent, locally ergodic
domains, we argue that within such domains the relative probabilities of
microstates are given by the standard Boltzmann weights. In contrast to
previous energy landscape treatments, that have been developed specifically for
the glass transition, we do not impose an a priori knowledge of the
inter-domain population distribution. Assuming that these domains are robust
with respect to small changes in thermodynamic state variables we derive a
variety of fluctuation formulae for these systems. We verify our theoretical
results using molecular dynamics simulations on a model glass forming system.
Non-equilibrium Transient Fluctuation Relations are derived for the
fluctuations resulting from a sudden finite change to the system's temperature
or pressure and these are shown to be consistent with the simulation results.
The necessary and sufficient conditions for these relations to be valid are
that the domains are internally populated by Boltzmann statistics and that the
domains are robust. The Transient Fluctuation Relations thus provide an
independent quantitative justification for the assumptions used in our
statistical mechanical treatment of these systems.
