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 ${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\Omega }}$, 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 (${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\Omega }}$) 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 ${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\Omega }}$ 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.