Non-Markovian Model for the Approach to Equilibrium

Abstract

This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse-graining and time-smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.

Long-time asymptotics of the long-range Emch-Radin model

“…Performing the trace in the σ eigenbasis of , the diagonal form of ρ(0) implies that only the diagonal elements of the operator e −iH N Ae iH N are required, and it is this crucial ingredient which allows us to obtain, similar to the calculation in [14], the exact result…”

“…In a recent Letter [13], the author has reported results on the dynamics of a long-range interacting version of the Emch-Radin model [14,15]. This quantum mechanical spin model is particularly amenable to analytic calculations and is known to show relaxation to equilibrium in a non-Markovian way [14].…”

“…This quantum mechanical spin model is particularly amenable to analytic calculations and is known to show relaxation to equilibrium in a non-Markovian way [14]. Unlike in the papers of Emch and Radin, long-range interaction are permitted, i.e.…”

Approach to Equilibrium for a Class of Random Quantum Models of Infinite Range

“…Mean-field models, e.g., quantum analogues of the Sherrington-Kirkpatrick (SK) spin-glass have also been studied, and the thermodynamic limit proved for the SK model in a transverse external field [2]. In this paper we consider the dynamics of a special class of random quantum models of infinite range, which may be viewed as a version with random couplings of the model introduced by Emch [3] and generalized by Radin [4].…”

Many-Body Dynamics of Dipolar Molecules in an Optical Lattice

“…We show results for size g ¼ 10 clusters, where the method is converged for the system shown here. The convergence of the method is analyzed in the Supplemental Material [18], where we vary the cluster size for the present problem as well as by applying the MACE to the Ising limit (J z ≠ 0, J ⊥ ¼ 0), where an analytic solution exists [12,[30][31][32].…”