Statistical mechanics of time independent non-dissipative nonequilibrium states

Amorphous solids are typically nonergodic and thus a more general formulation of statistical mechanics, with a clear link to thermodynamics, is required. We present a rigorous development of the nonergodic statistical mechanics and the resulting thermodynamics for a canonical ensemble, where the 6N dimensional phase space contains a set of distinct nonoverlapping domains. An ensemble member which is initially in one domain is assumed to remain there for a time long enough that the distribution within the domain is Boltzmann weighted. The number of ensemble members in each domain is arbitrary. The lack of an a priori specification of the number of members in each domain is a key differences between the work presented here and existing energy landscape treatments of the glass transition. Another important difference is that the derivation starts with the phase space distribution function rather than an equilibrium expression for the free energy. The utility of this newly derived statistical mechanics is demonstrated by deriving an expression for the heat capacity of the ensemble. Computer simulations on a model glass former are used to provide a demonstration of the validity of this result which is different to the predictions of standard equilibrium statistical mechanics.