We consider a system of linear oscillators, or quantum states, described by Random Matrix Theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Due to universal features of Random Matrix Theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.

21=5+16 pages, 5 figures + 16 SupMat figures, additional figures/data available at: https://www.quantware.ups-tlse.fr/QWLIB/nonlinrmt/index.html additional material in SupMat (new figure panels with new parameters and new SupMat sections)