Quantum mappings and the problem of stochasticity in quantum systems

Discrete mappings for a dynamical quantum system possessing in the classical limit (ħ = 0) the property of stochasticity are analysed. The quasiclassical approximation is shown to be valid for the limited time and the asymptotic behaviour of the system for large times is to be determined by the quantum effects essentially. Contrary to the classical case when the correlation function decreases exponentially, the correlations decay with a powerlaw in the quantum case. This results in zero entropy of the quantum dynamical system while the entropy is nonzero in the classical limit case. It is shown that there is a quasiclassical region in which the quantum system dynamics is close to the stochastic dynamics of the corresponding classical system for a finite time. The parameter separating this quasiclassical region from the region of essentially quantum dynamics is obtained.