Chapter 19 - Broken Ergodicity

Abstract

Ergodicity is the equivalence of the time and ensemble averages of the properties of a system. A system that is initially ergodic can become nonergodic if its internal relaxation time scale becomes longer than the observation time. The ratio of the internal relaxation time to the external observation time is called the Deborah number. The physics of nonergodic systems was developed by Palmer, who coined the term “broken ergodicity.” Palmer's treatment of broken ergodic systems considers microstates as grouped into components which satisfy the conditions of internal ergodicity and confinement. The condition of internal ergodicity states that the kinetics within a component are fast compared to the observation time, such that local equilibration can be achieved. The condition of confinement states that inter-component transitions are forbidden. Within Palmer's framework of broken ergodicity, the macroscopic properties of a system can be calculated by averages over the individual components. The division of an energy landscape into components upon loss of ergodicity is a partitioning process, which results in a loss of configurational entropy but no change in internal energy or enthalpy. The restoration of ergodicity is a unifying process, which is a spontaneous process resulting in an increase in entropy, consistent with the Second Law. Within Palmer's framework of broken ergodicity, the entropy of any system is zero at absolute zero temperature, consistent with the Third Law. Palmer's assumptions are relaxed in the more general theory of continuously broken ergodicity, which accounts for the gradual loss of ergodicity in a system by considering two sets of probabilities: basin occupation and conditional probabilities. The latter give the probability of the system transitioning to a new basin from a given starting basin after propagating for a specified observation time.