Ergodic hypothesis in classical statistical mechanics

Abstract

An updated discussion on physical and mathematical aspects of the ergodic hypothesis in classical equilibrium statistical mechanics is presented. Then a practical attitude for the justification of the microcanonical ensemble is indicated. It is also remarked that the difficulty in proving the ergodic hypothesis should be expected. Keywords: ergodic hypothesis, statistical mechanics, microcanonical ensemble.Apresenta-se uma discussão atual sobre aspectos físicos e matemáticos da hipótese ergódica em mecânica estatística de equilíbrio. Então indica-se uma eventual postura para se justificar o ensemble microcanônico. Observa-se, também, que a dificuldade em se demonstrar a hipótese ergódica deveria ser esperada. Palavras-chave: hipótese ergódica, mecânica estatística, ensemble microcanônico.

Strain Mechanisms in Lead-Free Ferroelectrics for Actuators

“…415,416 This result can be qualitatively explained by enhanced elastic shear occurring in the yz or xz planes due to the transition from the T to the O phase rather than in the transition from the O to the R phase. In other words, as reorients between [100] and [110] directions, higher shear components are encountered than when reorients between [110] and [111] directions. Although enhanced elastic shear softening was predicted for monodomain single crystals, this effect was also shown experimentally in polycrystalline materials.…”

“…104,105 This state is characterized by distinct historydependent functional properties 55,86 , as a result of the multiple metastable states of canonical relaxors in their non-ergodic state. 100 Non-canonical relaxors feature an analogous high temperature paraelectric and ergodic state when compared to canonical relaxors, as displayed in Figure 2.10 (b). Upon further decrease in temperature below , however, they have a spontaneous transition into a ferroelectric state.…”

“…In other words, in an ergodic system an arbitrary function can be defined in a certain mathematical space within the phase space of the whole system, in which the characteristic property´s time average becomes indistinguishable from the ensemble over all accessible points. 99,100 Expressing a higher degree of ergodicity can be understood as increasing the fraction of phase space in which its property time average characteristic becomes indistinguishable from the ensemble average for the distribution of all accessible points in the phase space of the system. Expressing a degree of non-ergodicity within a phase space is therefore not proper but a consequence of the degree of ergodicity.…”

“…106 It should be noted that due to the history-dependent properties in the non-ergodic state of canonical relaxors 100 , isothermal (red) and isofield (blue) representations differ. 106,107 An electric field induced phase transition that originates from an ergodic state will be reversible, whereas from a non-ergodic state (below ) will be irreversible.…”

“…This behavior was attributed to an anomalous softening reflected by an increased 11 . 417 Therefore, it can be suggested that an analogy may exist when the shifts through shear from the [100] direction to another direction corresponding to the M phase. In fact, Sabat et al 401 showed that the 55 is more than two times higher than all of the other elastic compliance components near the T to M phase boundary.…”