Entropy production and wave packet dynamics in the Fock space of closed chaotic many-body systems

Highly excited many-particle states in quantum systems such as nuclei, atoms, quantum dots, spin systems, quantum computers etc., can be considered as ``chaotic'' superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is due to a very high level density of many-body states that are easily mixed by a residual interaction between particles (quasi-particles). For such systems, we have derived simple analytical expressions for the time dependence of energy width of wave packets, as well as for the entropy, number of principal basis components and inverse participation ratio, and tested them in numerical experiments. It is shown that the energy width $\Delta (t)$ increases linearly and very quickly saturates. The entropy of a system increases quadratically, $S(t) \sim t^2$ at small times, and after, can grow linearly, $S(t) \sim t$, before the saturation. Correspondingly, the number of principal components determined by the entropy, $N_{pc} \sim exp{(S(t))}$, or by the inverse participation ratio, increases exponentially fast before the saturation. These results are explained in terms of a cascade model which describes the flow of excitation in the Fock space of basis components. Finally, a striking phenomenon of damped oscillations in the Fock space at the transition to an equilibrium is discussed.