Random-Matrix Model for 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.