Continuously broken ergodicity

A system that is initially ergodic can become nonergodic, i.e., display "broken ergodicity," if the relaxation time scale of the system becomes longer than the observation time over which properties are measured. The phenomenon of broken ergodicity is of vital importance to the study of many condensed matter systems. While previous modeling efforts have focused on systems with a sudden, discontinuous loss of ergodicity, they cannot be applied to study a gradual transition between ergodic and nonergodic behavior. This transition range, where the observation time scale is comparable to that of the structural relaxation process, is especially pertinent for the study of glass transition range behavior, as ergodicity breaking is an inherently continuous process for normal laboratory glass formation. In this paper, we present a general statistical mechanical framework for modeling systems with continuously broken ergodicity. Our approach enables the direct computation of entropy loss upon ergodicity breaking, accounting for actual transition rates between microstates and observation over a specified time interval. In contrast to previous modeling efforts for discontinuously broken ergodicity, we make no assumptions about phase space partitioning or confinement. We present a hierarchical master equation technique for implementing our approach and apply it to two simple one-dimensional landscapes. Finally, we demonstrate the compliance of our approach with the second and third laws of thermodynamics.