Convergence to equilibrium under a random Hamiltonian

TL;DR

This paper investigates how quickly subsystems reach equilibrium under a randomly chosen Hamiltonian, revealing that equilibration time relates inversely to average Bohr frequencies, with new methods for averaging over random bases.

Contribution

It introduces a novel approach to compute equilibration times by averaging over Haar-random Hamiltonians using group representation theory.

Findings

Equilibration time is inversely proportional to average Bohr frequencies.

Derived a method to compute averages over random bases using permutation group representations.

Provided explicit formulas for overlaps of operators in the context of random Hamiltonians.

Abstract

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.