Марковская и немарковская динамика квантовых и мезоскопических систем

Аннотация

Topicality and relevance of the theme of the dissertation. At present time, a theoretical study of the dynamic problems of nonequilibrium quantum systems is one of the fundamental problems of modem theoretical and experimental physics. The phenomena of fluctuations and dissipations of nonequilibrium quantum systems arc usually investigated in the framework of the standard technique of the theory of quantum Markov processes. It is known that the response of real physical, technical and biological systems to external random effects is a non-Markovian process, and the non-Markovian effect increases with the complexity of the system. Consequently, such systems can not be described by standard methods developed on the basis of Markov processes. In connection with this, the development of mathematical methods for describing non-Markov random processes becomes topical. In our Republic, great attention is paid to the development of theoretical physics and the conduct of fundamental research on these areas at the world level. In this respect, significant results were achieved, in particular: in solving theoretical problems of nonequilibrium quantum systems and their practical application, developing methods for the analytical determination and calculation of nonstationary transport coefficients, theoretical study of the nuclear reaction with heavy ions, the influence of external fields on open quantum systems and nonequilibrium processes. In accordance with the "Strategy of actions on further development of the Republic of Uzbekistan", the most important is to increase the efficiency of the physics sector of the nonequilibrium system and nanotechnology on the basis of theoretical and practical research in the field of nonequilibrium quantum systems and dissipative nonequilibrium quantum processes through the introduction of innovative technologies. At the present time, new experimental data have been obtained on the subbarrier fusion of nuclei in collisions of the low energies heavy ions, tunnel transitions with dissipation in crystals, in effects associated with strong, weak and superweak impact of magnetic fields of various nature on a variety of complex physical and biological objects, trends that can not be explained within existing traditional approaches and models. Theoretical explanation of a large number of such effects is an actual problem of modem theoretical physics. Taking into account non-Markovian effects in tunnel transitions is not completely solved by the problem of the nonequilibrium quantum systems, therefore, solving the problem of a quantum tunnel transitions with dissipation may prove to be important in the study of nuclear, atomic low-dimcnsional and low-tcmpcraturc mesoscopic systems. The development of a method to determining the time-dependent transport coefficients and the widths of the decay of mctastable states is necessary when describing and explaining experimental data on the nuclear reactions of complete fusion, quasi-fission, fission and capture of the nucleus by the nucleus. In addition, theoretical data on the tunnel transition with dissipation arc extremely necessary for the transport of electrons in quantum dots and wells, and also for solving one of the important problems of nuclear astrophysics-thc fusion of nuclei in superdense astronomical objects. This research work corresponds to the tasks approved in the state normative documents, in the Decrees of the President of the Republic of Uzbekistan No. PR-1443 “On the priorities of industry development of the Republic of Uzbekistan in 2011-2015” of 15 December 2010 and No.PR-2789 “On measures to further improve the activities of the Academy of Sciences, organization, management and funding research activities” of 17 February 2017, and the Decree of the President of the Republic of Uzbekistan No. PD-4958 “On the further improvement of the system of postgraduate education” of 16 February 2017. The aim of the research is to develop a formalism to describe the Markovian non-Markovian dynamics of quantum systems. The scientific novelty of the research is as follows: a system of nonlinear quantum stochastic equations arc obtained and solved analytically in the limit of the complete connection between a harmonic oscillator and a quantum thermostat in the rotating wave approximation; a technique for obtaining time-dependent transport coefficients has been developed, and analytical formulas have been obtained for calculating quantum fluctuations in nonequilibrium quantum systems; the Markovian quantum equation for the density matrix is derived from the generalized non-Markovian stochastic equation and quantum fluctuation-dissipative (FD) relations arc obtained; a system of nonlinear stochastic equations with allowance for an external magnetic field is obtained and analytically solved; the values of non-stationary friction and diffusion coefficients for a two-dimensional charged quantum harmonic oscillator in a homogeneous magnetic field arc calculated; an analytical expressions arc obtained for the asymptotics of transport coefficients and correlation functions for a two-dimensional charged oscillator; the power law of the decay of the correlation functions of an oscillator with a complete coupling in the low-temperature and long-time limit has been predicted; the role of quantum and non-Markovian effects in reactions with heavy ions is established.