Analog of Planck's formula and effective temperature in classical statistical mechanics far from equilibrium

Abstract

We study the statistical mechanics very far from equilibrium for a classical system of harmonic oscillators colliding with point particles (mimicking a heat reservoir), for negligible initial energies of the oscillators. It is known that for high frequencies the times of relaxation to equilibrium are extremely long, so that one meets with situations of quasiequilibrium very far from equilibrium similar to those of glassy systems. Using recent results from the theory of dynamical systems, we deduce a functional relation between energy variance and mean energy that was introduced by Einstein phenomenologically in connection with Planck's formula. It is then discussed how this leads to an analog of Planck's formula. This requires using Einstein's relation between specific heat and energy variance to define an effective temperature in a context of quasiequilibrium far from equilibrium, as is familiar for glassy systems.