Zero-th law in structural glasses: an example

We investigate the validity of a zeroth thermodynamic law for nonequilibrium systems. In order to describe the thermodynamics of the glassy systems, it has been introduced an extra parameter, the effective temperature which generalizes the fluctuation-dissipation theorem (FDT) to off-equilibrium systems and supposedly describes thermal fluctuations around the aging state. In particular we analyze two coupled systems of harmonic oscillators with Monte Carlo dynamics. We study in detail two types of dynamics: sequential dynamics, where the coupling between the subsystems comes only from the Hamiltonian; and parallel dynamics where there is another source of coupling: the dynamics. We show how in the first case the effective temperatures of the two interacting subsystems are different asymptotically due to the smallness of the thermal conductivity in the aging regime. This explains why, in structural glasses, different interacting degrees of freedom can stay at different effective temperatures, and never thermalize.