Quantum versus classical thermalization in many-body isolated systems*

In view of recent studies of the thermalization occurring in many-body isolated systems, we discuss the onset of relaxation towards a steady-state equilibrium, paying main attention to the systems with finite number of interacting particles. We have found that in the models of randomly interacting fermions and bosons, the effective number of components in the wave packets increases exponentially in time, provided the eigenstates are strongly chaotic in the Hilbert space defined by non-interacting particles. Our semi-analytical approach allows one to obtain the estimates for the rate of this increase and for the characteristic time of the saturation which can be considered as the time for a complete thermalization. Special interest has been payed to the correlations between occupation numbers, that increase in time during the scrambling of wave packets in Hilbert space. These correlations are responsible for the onset of the Bose-Einstein and Fermi-Dirac distributions. We discuss the relevance of the exponentially fast quantum dynamics to the Kolmogorov-Sinai entropy characterizing degree of chaos in the corresponding classical systems.