Approach to equilibrium for a class of random quantum models of infinite range

TL;DR

This paper studies how random quantum models with infinite-range interactions approach equilibrium, showing that Gaussian randomness accelerates convergence and that exponential decay occurs under certain conditions.

Contribution

It extends the analysis of equilibrium approach in quantum models from $l_1$ to $l_2$ potentials and compares the effects of Gaussian and Bernoulli distributions on decay rates.

Findings

Gaussian distribution leads to faster approach to equilibrium.

Exponential decay occurs for Gaussian distributions in both short and long-range potentials.

Exponential decay is observed for Bernoulli potentials in the $l_2$ class.

Abstract

We consider random generalizations of a quantum model of infinite range introduced by Emch and Radin. The generalization allows a neat extension from the class $l_1$ of absolutely summable lattice potentials to the optimal class $l_2$ of square summable potentials first considered by Khanin and Sinai and generalised by van Enter and van Hemmen. The approach to equilibrium in the case of a Gaussian distribution is proved to be faster than for a Bernoulli distribution for both short-range and long-range lattice potentials. While exponential decay to equilibrium is excluded in the nonrandom $l_1$ case, it is proved to occur for both short and long range potentials for Gaussian distributions, and for potentials of class $l_2$ in the Bernoulli case. Open problems are discussed.