Random-Matrix Model for Thermalization

An isolated quantum system is said to thermalize if {rm Tr} (A ρ(t)) → {rm Tr} \(A ρ_{rm eq}\) for time t → infty. Here ρ(t) is the time-dependent density matrix of the system, ρ_{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 ρ(t)) in the ensemble thermalize: For N → infty every such function tends to the value {rm Tr} \(A ρ_{rm eq}(infty\)) + {rm Tr} (A ρ(0)) g²(t). Here ρ_{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.