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Bulletin of the Crimean Astrophysical Observatory

Вероятностные модели в классической и квантовой механике

V.Yu. Terebizh, В.Ю. Теребиж

Published Jun 1, 2010• 0 citations
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Abstract

Stochastic models in classical and quantum mechanics, by V.Yu.Terebizh. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.

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Вероятностные модели в классической и квантовой механике: Исправления к опубликованной ранее статье

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Characteristic features of the stochastic models used in the classical mechanics, statistical physics and quantum theory are discussed. The arguments are given in favor of point of view that Nature, in itself, does not include probabilities. Researcher is forced to introduce the probabilistic concepts and corresponding models in conditions providing only partial predictability of a studied system behavior. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.

Bulletin of the Crimean Astrophysical Observatory•Jun 1, 2011•0 citations

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Foreword. The quantal fattening of fractals N.L. Balazs. How and when quantum phenomena become real Ph. Blanchard, A. Jadczyk. Chaotic dynamics in a periodically driven anharmonic oscillator Yu.L. Bolotin, V.Yu. Gonchar, M.Ya. Granovsky. Coherent and incoherent dynamics in a periodically driven bistable system T. Dittrich, F. Grossman, P. Hanggi, B. Oelschlagel, R. Utermann. The Ehrenfest theorem for Markov diffusions P. Garbaczewski. The quantum state diffusion model, asymptotic solutions, thermal equilibrium and Heisenberg picture N. Gisin. Stochastic representation of quantum dynamics Z. Haba. Level repulsion and exceptional points W.D. Heiss, W.-H. Steeb. Type-II intermittency in the presence of additive and multiplicative noise H. Herzel, F. Argoul, A. Arneodo. Aspects of Liouville integrability in quantum mechanics J. Hietarinta. KAM techniques for time dependent quantum systems H.R. Jauslin. Dissipation and noise in quantum mechanics N.G. van Kampen. Irregular scattering, number theory, and statistical mechanics A. Knauf. Band random matrices, kicked rotator and disordered systems L. Molinari. Robust scarred states D. Richards. The quasiclassical statistical description of quantum dynamical systems and quantum chaos Yu.P. Virchenko. Quantum measurement by quantum brain M. Jibu, K. Yasue. Time reversal and Gaussian measures in quantum physics J.C. Zambrini. Stochastic resonance in bistable systems with fluctuating barriers L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta, S. Santucci. Index.

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Stochastic models in classical and quantum mechanics

V. Yu. Terebizh

Characteristic features of the probabilistic models used in classical mechanics, statistical physics, and quantum theory are discussed. The viewpoint according to which there is no room for probabilities in Nature, as such, is consistently substantiated; the researcher is forced to introduce probabilistic concepts and the corresponding models in conditions that provide only partial predictability of the phenomena being studied. This approach allows one to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the theory of deterministic chaos, Boltzmann’s H -theorem, and paradoxes of quantum mechanics.

Bulletin of the Crimean Astrophysical Observatory•Jun 1, 2010•0 citations

Classical statistical physics and quantum mechanics

D. I. Blokhint︠s︡ev

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Given review is denoted to an important problem, connected with physical processes in the intermediate zone between microand macrocosmic. It is kept a row of new and original results. First of all, we come from the new view to the whole complex problem in the intermediate zone. The classical quantum mechanics describes the processes in microcosmic only and even is not tortured to touch complex evolutionary phenomena's in the intermediate world. As an effect, playing in her a central role the principle of Born on a probably interpretation of wave functions at all is not motivated and is received as some postulate. We come from а view, that is necessary consider real physical processes on scattering of wave packages in scales of the intermediate zone. Such physical approach shows degree of approximation and insufficiency of the Born principle. We present the method of description of such evolution, based on the nonlinear differential equation. Some Markov chain characteristics are make more precised and is formulated important for the given problems a characteristic of monotonic behavior of probability. Using one fine inequality, we are studied an evolution of relative entropy in the classical theory. Then we study the phenomena of irreversibility in quantum problems. Original derivation of the Lindblad equation for a wave density function is presented and, moreover, for the first time this equation is given physical motivation. We developed a deep analogy between classical Markov processes and process to evolutions, described by the Lindblad equation. Differential equation system, more general, than the known equation of evolution of Leggett and Caldeira, is derived. Quantum systems without inverse their in.uences upon surrounding background (that really for high temperatures) are considered. Quantum di.usion of single particles is studied also in detail. The famous paradox of Shredinger cat is discussed. A special role of the external world for phenomena with the quantum irreversibility is underlined. Against the opinion of Kadomtsev and Menskii, in equilibrium medium no need to postulate and consider processes with collapse of the wave functions. Is it pointed out, that the Heisenberg uncertainty relation not always gives usually understood restriction on making of an average of commuting values. Some prospects in the problem quantum irreversibility are planed.

•Jan 1, 2006•0 citations

Probability and quantum mechanics the conceptual foundations of stochastic mechanics

Francesco Guerra

Lecture notes in mathematics•Jan 1, 1984•4 citations
Stochastic models in classical and quantum mechanics, by V.Yu.Terebizh. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.
Characteristic features of the stochastic models used in the classical mechanics, statistical physics and quantum theory are discussed. The arguments are given in favor of point of view that Nature, in itself, does not include probabilities. Researcher is forced to introduce the probabilistic concepts and corresponding models in conditions providing only partial predictability of a studied system behavior. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.
Foreword. The quantal fattening of fractals N.L. Balazs. How and when quantum phenomena become real Ph. Blanchard, A. Jadczyk. Chaotic dynamics in a periodically driven anharmonic oscillator Yu.L. Bolotin, V.Yu. Gonchar, M.Ya. Granovsky. Coherent and incoherent dynamics in a periodically driven bistable system T. Dittrich, F. Grossman, P. Hanggi, B. Oelschlagel, R. Utermann. The Ehrenfest theorem for Markov diffusions P. Garbaczewski. The quantum state diffusion model, asymptotic solutions, thermal equilibrium and Heisenberg picture N. Gisin. Stochastic representation of quantum dynamics Z. Haba. Level repulsion and exceptional points W.D. Heiss, W.-H. Steeb. Type-II intermittency in the presence of additive and multiplicative noise H. Herzel, F. Argoul, A. Arneodo. Aspects of Liouville integrability in quantum mechanics J. Hietarinta. KAM techniques for time dependent quantum systems H.R. Jauslin. Dissipation and noise in quantum mechanics N.G. van Kampen. Irregular scattering, number theory, and statistical mechanics A. Knauf. Band random matrices, kicked rotator and disordered systems L. Molinari. Robust scarred states D. Richards. The quasiclassical statistical description of quantum dynamical systems and quantum chaos Yu.P. Virchenko. Quantum measurement by quantum brain M. Jibu, K. Yasue. Time reversal and Gaussian measures in quantum physics J.C. Zambrini. Stochastic resonance in bistable systems with fluctuating barriers L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta, S. Santucci. Index.
Characteristic features of the probabilistic models used in classical mechanics, statistical physics, and quantum theory are discussed. The viewpoint according to which there is no room for probabilities in Nature, as such, is consistently substantiated; the researcher is forced to introduce probabilistic concepts and the corresponding models in conditions that provide only partial predictability of the phenomena being studied. This approach allows one to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the theory of deterministic chaos, Boltzmann’s H -theorem, and paradoxes of quantum mechanics.
Given review is denoted to an important problem, connected with physical processes in the intermediate zone between microand macrocosmic. It is kept a row of new and original results. First of all, we come from the new view to the whole complex problem in the intermediate zone. The classical quantum mechanics describes the processes in microcosmic only and even is not tortured to touch complex evolutionary phenomena's in the intermediate world. As an effect, playing in her a central role the principle of Born on a probably interpretation of wave functions at all is not motivated and is received as some postulate. We come from а view, that is necessary consider real physical processes on scattering of wave packages in scales of the intermediate zone. Such physical approach shows degree of approximation and insufficiency of the Born principle. We present the method of description of such evolution, based on the nonlinear differential equation. Some Markov chain characteristics are make more precised and is formulated important for the given problems a characteristic of monotonic behavior of probability. Using one fine inequality, we are studied an evolution of relative entropy in the classical theory. Then we study the phenomena of irreversibility in quantum problems. Original derivation of the Lindblad equation for a wave density function is presented and, moreover, for the first time this equation is given physical motivation. We developed a deep analogy between classical Markov processes and process to evolutions, described by the Lindblad equation. Differential equation system, more general, than the known equation of evolution of Leggett and Caldeira, is derived. Quantum systems without inverse their in.uences upon surrounding background (that really for high temperatures) are considered. Quantum di.usion of single particles is studied also in detail. The famous paradox of Shredinger cat is discussed. A special role of the external world for phenomena with the quantum irreversibility is underlined. Against the opinion of Kadomtsev and Menskii, in equilibrium medium no need to postulate and consider processes with collapse of the wave functions. Is it pointed out, that the Heisenberg uncertainty relation not always gives usually understood restriction on making of an average of commuting values. Some prospects in the problem quantum irreversibility are planed.