Random-Matrix Model for Thermalization

TL;DR

This paper demonstrates that quantum systems with Hamiltonians modeled by GOE and GUE random matrices exhibit thermalization behavior, with observable expectations approaching equilibrium values over time, characterized by specific oscillatory decay functions.

Contribution

It provides a rigorous analysis of thermalization in random-matrix quantum systems, including time symmetry and decay behavior, extending the eigenstate thermalization hypothesis.

Findings

All functions Tr(A ρ(t)) thermalize in GOE ensemble.

Decay of oscillations follows a 1/|t| law.

Time-reversal symmetry leads to symmetric thermalization.

Abstract

An isolated quantum system is said to thermalize if ${\rm Tr} (A \rho(t)) \to {\rm Tr} (A \rho_{\rm eq})$ for time $t \to \infty$. Here $\rho(t)$ is the time-dependent density matrix of the system, $\rho_{\rm eq}$ is the time-independent density matrix that describes statistical equilibrium, and $A$ is a Hermitean operator standing for an observable. We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension $N$), all functions ${\rm Tr} (A \rho(t))$ in the ensemble thermalize: For $N \to \infty$ every such function tends to the value ${\rm Tr} (A \rho_{\rm eq}(\infty)) + {\rm Tr} (A \rho(0)) g^2(t)$. Here $\rho_{\rm eq}(\infty)$ is the equilibrium density matrix at infinite temperature. The oscillatory function $g(t)$ is the Fourier transform of the average GOE level density and falls off as $1 / |t|$ for large $t$. With $g(t) = g(-t)$, thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble (GUE) of random matrices. Comparison with the ``eigenstate thermalization hypothesis'' of Ref.~\cite{Sre99} shows overall agreement but raises significant questions.