Quantum chaos and its kinetic stage of evolution

Usually reason of irreversibility in open quantum-mechanical system is interaction with a thermal bath, consisting form infinite number of degrees of freedom. Irreversibility in the system appears due to the averaging over all possible realizations of the environment states. But, in case of open quantum-mechanical system with few degrees of freedom situation is even more complicated. Should one still expect irreversibility, if external perturbation is just an adiabatic force without any random features? Problem is not clear yet. This is the main question we address in this review article. We prove that the key point in the formation of irreversibility in chaotic quantum-mechanical systems with few degrees of freedom is the complicated structure of energy spectrum. We shall consider quantum-mechanical system with parametrically dependent energy spectrum. In particular, we study energy spectrum of the Mathieu-Schrodinger equation. Structure of the spectrum is quite non-trivial, consists from the domains of non-degenerated and degenerated states, separated from each other by branch points. Due to the modulation of the parameter, system will perform transitions from one domain to other one. For determination of eigenstates for each domain and transition probabilities between them, we utilize methods of abstract algebra. We shall show that peculiarity of parametrical dependence of energy terms, leads to the formation of mixed state and to the irreversibility, even for small number of levels involved in the process. This last statement is important. Meaning is that we are going to investigate quantum chaos in essentially quantum domain. In the second part of the article, we will introduce the concept of random quantum phase approximation. Then, along with the methods of random matrix theory, we will use this assumption for the derivation of muster equation in the formal and mathematically strict way. The content of this article is based on our previous studies. However, in this review article some original material is also included. This part mainly concerns the discussion about possible experimental realization of theoretical concepts in the field of organic chemistry. (C) 2009 Elsevier B.V. Ail rights reserved.