Stability,equilibrium and metastability in statistical mechanics

We survey a body of work, containing some new material, concerning the characterisation of equilibrium and metastable states of large assemblies of particles in terms of a variety of stability conditions. The theory is formulated in the thermodynamic limit and is based on the premise that the former states are those that are stable against all dynamical and thermodynamical perturbations, whereas the latter ones are endowed with only limited stability, sufficing to guarantee their long lifetimes and good thermodynamical behaviour. The Kubo-Martin-Schwinger (KMS) fluctuation-dissipation conditions play a central role in the developments stemming from this viewpoint, since it turns out that these conditions represent stability against localised disturbances of both the dynamical and thermo-dynamical kinds. Consequently, the stability arguments invoked here lead us to the following principal conclusions: (1) The equilibrium states are those that minimise the free energy density of the system and also satisfy the KMS conditions. This substantiates Gibbs's hypothesis that these states correspond to the standard ensembles. (2) Metastable states are of two kinds, that we term “ideal” and “normal”. Those of the former type satisfy the KMS conditions but minimise only the restriction of the free energy density to some reduced state space: those of the latter type are characterised by a still lower grade of stability. (3) The conditions on the forces under which ideal metastable states can exist are very restrictive, and thus the normal ones generally correspond to those observed in nature.