Ergodic behavior in supercooled liquids and in glasses

Ergodic behavior in liquids, supercooled liquids, and glasses is examined with a focus on the time scale needed to obtain ergodicity. A measure, d(t), which is based on the time-averaged energies of the individual particles and which is referred to as the ``energy metric,'' is introduced to probe the approach to ergodic behavior. We suggest that d(t) obeys a dynamical scaling law for long but finite times and that it can be used to characterize the degree of stochasticity in measure preserving systems with large numbers of degrees of freedom. Examination of d(t) indicates that the configuration space is explored by a ``diffusive'' process in the space of the dynamical energy variables used in constructing the energy metric. The characteristic diffusion constant associated with this process is argued to be analogous to the well-known maximal Lyapunov exponent which is often used to characterize stochasticity in systems with few degrees of freedom. Based on the long-time behavior of d(t) it is shown that ergodicity is effectively broken in the glassy state. In addition to broken ergodicity, the possibility that a subtle symmetry is broken as the liquid-to-glass transition takes place is examined. It is suggested that a ``discrete'' symmetry, to be referred to as the statistical symmetry, is broken in the glassy phase. This is illustrated by analyzing the distribution of the energy of the particles. Based on this, we expect long-time dynamics and structural relaxation in glasses to be dominated by fluctuations in domains of finite length within which the particles are highly correlated. This is in accord with the ideas of Adams and Gibbs. All of the above arguments are illustrated with the aid of molecular-dynamics simulations of soft-sphere mixtures.