On the quantum-statistical ergodic and H -theorems

Abstract

Metrical indecomposability is a condition for the validity of the classical ergodic theorem. Its counterpart in the quantum-statistical theorem is believed to be non-degeneracy of the energy eigenvalues. It is shown here, however, that the coarse-grained quantum-statistical theorems (due to von Neumann and to Pauli & Fierz) remain valid in the presence of energy degeneracy. This requires, therefore, a reassessment of the significance of these theorems; this is discussed, and it is concluded that the von Neumann approach is unsatisfactory in that it fails to yield a microscopic criterion for either ergodicity or validity of the H -theorem.

Generalization of von Neumann’s Approach to Thermalization

“…This is the viewpoint adopted, e.g., in Refs. [35,42] but now formulated within our present generalization of von Neumann's original approach (see also [31]). …”

“…[35,36,41,42]. In other words, it is not right to say that von Neumann's approach is inadequate to investigate thermalization, since the same applies to the approach from Refs.…”

“…In other words, it is not right to say that von Neumann's approach is inadequate to investigate thermalization, since the same applies to the approach from Refs. [35,36,41,42]. Rather, the two approaches are fundamentally different: One requires a generic eigenbasis constellation of H and A but admits any ρð0Þ, and the other requires a generic eigenbasis constellation of ρð0Þ and A but admits any H.…”

“…Furthermore, our findings (14) and (15) significantly generalize previously known typicality results for arbitrary (even time-dependent) Hamiltonians. As a by-product, we thus obtained a unifying framework for several key aspects of thermalization, such as the validation of the eigenstate thermalization hypothesis [8,21,23,25] by means of the random matrix theory [22], recent explorations of "equilibration" [9][10][11][12][13][14][15] and "canonical typicality" [26,33,[38][39][40], the long-lasting misjudgment of von Neumann's work [35,36,41,42], and its rehabilitation in Ref. [3].…”

Quantum ergodicity for a class of non-generic systems

“…The validity of this theorem, however, was heavily debated later in the literature [13][14][15]. It was just recently that this theorem was revisited carefully in Ref.…”