Ergodic convergence properties of supercooled liquids and glasses

The dynamical properties of two measures to probe ergodic behavior in Hamiltonian systems with a large number of degrees of freedom are investigated. The measures, namely, the energy metric d(t) and the fluctuation metric {Ω}(t) are both based on the time-averaged energies of the individual particles comprising the system. The energy metric d(t) is obtained by following the dynamics of the system using two independent configurations, whereas {Ω}(t) is expressed in terms of the properties of a single trajectory. Both measures obey a dynamical scaling law for long but finite times. The scaling law for d(t), which was previously established numerically, and for {Ω}(t) is derived for systems that are effectively ergodic. The scaling function suggests that the configuration space is explored by a ``diffusive'' process in the space of the variables used to construct d(t) and {Ω}(t). Furthermore, a single parameter, namely DE and D_Ω, characterizes the time scales needed for effective ergodicity to be obtained. These ideas are used to study the behavior of supercooled and glassy states of soft-sphere binary alloys using microcanonical molecular dynamics. The scaling forms for d(t) and {Ω}(t) are explicitly demonstrated. The temperature dependence of the numerically computed diffusion constants (DE and D_Ω) reveals that the nature of phase-space exploration changes both near the fluid-solid transition and near the liquid-to-glass transition. The changes in DE and D_Ω with temperature are well described by a Vogel-Fulcher equation close to the glass transition temperature. In addition, the distribution of the time-averaged individual particle energies P(e;t), moments of which are related to {Ω}(t), is shown to be broad in the glassy state. We argue that the time dependence of P(e;t) can be used to probe local structural relaxations in glasses.