Entropy and Irreversibility in Classical and Quantum Mechanics
ABSTRACT
Review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and the important property of monotonicity of a probability is formulated. Using one thin inequality, the behavior of relative entropy in the classical case is considered. Further we pass to studying of the irreversibility phenomena in quantum problems. By new method is received the Lindblad’s equation and its physical essence is explained. Deep analogy between the classical Markov processes and development described by the Lindblad’s equation is conducted. Using method of comparison of the Lind-blad’s equation with the linear Langevin equation we receive a system of differential equations, which are more general, than the Caldeira-Leggett equation. Here we consider quantum systems without inverse influ-ence on a surrounding background with high temperature. Quantum diffusion of a single particle is consid-ered and possible ways of the permission of the Schrödinger’s cat paradox and the role of an external world for the phenomena with quantum irreversibility are discussed. In spite of previous opinion we conclude that in the equilibrium environment is not necessary to postulate the processes with collapses of wave functions. Besides, we draw attention to the fact that the Heisenberg’s uncertainty relation does not always mean the restriction is usually the product of the average values of commuting variables. At last, some prospects in the problem of quantum irreversibility are discussed.
Review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and the important property of monotonicity of a probability is formulated. Using one thin inequality, the behavior of relative entropy in the classical case is considered. Further we pass to studying of the irreversibility phenomena in quantum problems. By new method is received the Lindblad’s equation and its physical essence is explained. Deep analogy between the classical Markov processes and development described by the Lindblad’s equation is conducted. Using method of comparison of the Lind-blad’s equation with the linear Langevin equation we receive a system of differential equations, which are more general, than the Caldeira-Leggett equation. Here we consider quantum systems without inverse influ-ence on a surrounding background with high temperature. Quantum diffusion of a single particle is consid-ered and possible ways of the permission of the Schrödinger’s cat paradox and the role of an external world for the phenomena with quantum irreversibility are discussed. In spite of previous opinion we conclude that in the equilibrium environment is not necessary to postulate the processes with collapses of wave functions. Besides, we draw attention to the fact that the Heisenberg’s uncertainty relation does not always mean the restriction is usually the product of the average values of commuting variables. At last, some prospects in the problem of quantum irreversibility are discussed.
KEYWORDS
Markov Chains, Irreversibility in Classical and Quantum Mechanics, Lindblad Equation, Caldeira-Leggett Equation, Quantum Diffusion, Schrödinger’s Cat Paradox, Heisenberg’s Uncertainty Relation, Collapse of Wave Function, Effect of Sokolov
Markov Chains, Irreversibility in Classical and Quantum Mechanics, Lindblad Equation, Caldeira-Leggett Equation, Quantum Diffusion, Schrödinger’s Cat Paradox, Heisenberg’s Uncertainty Relation, Collapse of Wave Function, Effect of Sokolov
Cite this paper
nullV. Antonov and B. Kondratyev, "Entropy and Irreversibility in Classical and Quantum Mechanics," Journal of Modern Physics, Vol. 2 No. 6, 2011, pp. 519-532. doi: 10.4236/jmp.2011.26061.
nullV. Antonov and B. Kondratyev, "Entropy and Irreversibility in Classical and Quantum Mechanics," Journal of Modern Physics, Vol. 2 No. 6, 2011, pp. 519-532. doi: 10.4236/jmp.2011.26061.
References
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[1] M.B. Mensky, “Dissipation and decoherence of quantum systems”, UFN, vol. 173, P.1199, 2003. [2] S. Karlin. “Mathematical Methods and Theory in Games, Programming, and Economics”, Dover Publications, 1992. [3] Chzhun KayLay, “Uniform chains of the Markov”, Мoscow: Mir, 1964 (in Russian). [4] M. S. Bartlett, “Introduction to theory of the probability processes”,Мoscow: Inostrannaya literatura, 1958 (in Russian). [5] M. Klein, “Entropy and the Ehrenfest urn model”, Physica, vol. 22, no. 569, 1956. [6] M. Loeve, “Probability theory”, Springer; 4th edition, 1977. [7] A. Marshal, I. Olkin, “Inequality”, Academic Press, New York, 1979. [8] S. Chandrasekhar, “Principles of stellar dynamics”, Dover, New York, 1942. [9] B.B. Kadomcev, “Dynamics and information”, editing of the journal YFN, Мoscow, 1997; 1999 (in Russian). [10] G.B. Lesovik, “Reservoir as source of quantum mechanical probability”, Letter in JETP, vol. 74, P. 528, 2001. [11] G. Lindblad, “On the generators of quantum dynamical semi groups”, Commun. Math. Phys., vol. 48, P.119, 1976. [12] R. Balesku, “Equilibrium and nonequilibrium statistical mechanics”, ohn Wiley & Sons, New York, 1975. [13] E. M. Lifshic, L. P. Pitaevskiy, “Fizicheskaya kinetika” (Physical kinetics), Мoscow: Nauka, 1979 (in Russian). [14] J.V. Pulé “The Bloch equations”, Commun. Math. Phys., vol. 38, № 3, P. 241-256, 1974. [15] H. Bateman, A. Erdelyi, “Higher transcendental functions”, V. 3, MC GRAW-HILL OK COMPANY, NEW YORK, 1953. [16] P. K. Suetin, “Classical orthogonal polynomials”, Мoscow: Nauka, 1979 (in Russian). [17] A. O. Caldeira, A. J. Leggett, “Path integral approach to quantum Brownian motion” Physica A, vol. 121, P. 587, 1983. [18] A. O. Caldeira, A. J. Leggett, “Influence of damping on quantum interference: an ex aptly soluble model”, Phys. Rev. A, vol. 31, P.1059, 1985. [19] “Philosophical questions of modern physics”, M.: ADVO AN USSR, 1952 (in Russian). [20] I. A. Akchurin, “Theory of the elementary particles and theory to information – Philosophical problems of physics of elementary particles”, Мoscow: Nauka, 1964 (in Russian). [21] A.A. Grib, “The Bell’s Inequality and experimental check of quantum correlation on macroscopic distances” UFN, Vol. 142, P. 619, 1984. [22] M.A. Popov, “In protection of the quantum idealism”, UFN, vol.173, P. 1382, 2003. [23] I. Bialynicki-Birula, J. Mycielski, “Nonlinear wave mechanics”, Ann. Phys., New York, vol. 100, P. 62, 1976. [24] C. G. Shull, D. K. Atwood, J. Arthur, M. Horne, “Search for a Nonlinear Variant of the Schr?dinger Equation by Neutron Interferometry”, Phys. Rev., Lett., vol. 44, P. 765, 1980. [25] R.G?hler, A.Y., Klei, A. Zeilinger, “Neutron optical tests of nonlinear wave mechanics”, Phys. Rev., Ser. A, vol. 23, P.1611, 1981. [26] B. P. Kondratyev, V. A. Antonov, “The solution of the Schr?dinger’s cat paradox. Attempt of development of nonlinear quantum mechanics”, Izhevsk, Udmurt State University, 1994 (in Russian). [27] Y.A. Panovko, I.I. Gubanova, “Stability and oscillations of the springy systems”, Мoscow: Nauka, 1979 (in Russian). [28] “Bases of the autocontrol”, under ed. V. S. Pugachev, Moscow: Fizmatgiz, 1963 (in Russian). [29] S. E. Shnoll, V. A. Kolombet, E. V. Pozharskiy, T. A. Zinchenko, I. M. Zvereva, A. A. Konradov, “About realization of discrete conditions during fluctuations in macroscopical processes”, UFN, vol. 10, P. 1129, 1998. [30] S.E. Shnoll, “Macroscopical fluctuations of the form discrete distributions cosmophysical reasons”, Biophysics, vol. 46, P. 775, 2001. [31] B.B. Kadomcev, “Irreversible in quantum mechanics”, UFN, vol. 173, P. 1221, 2003. [32] A.Y. Hinchin, “Mathematical basis statistical mechanics”, Мoscow: Gostehizdat, 947 (in Russian). [33] L.P. Kadanoff, Baym G. “Quantum statistical mechanics”, Мoscow: Mir, 1964 (in Russian). [34] I. D. Abella, N. A. Kurnit, S. R. Hartmann, ?Photon ehoes?, Phys. Rev., vol.141, P. 391, 1966. [35] V.N. Pavlenko, “The eho phenomena in plasma”, UFN, № 3, vol.141, P. 393, 1983. [36] M. Zaslavsky, “Statistics of the energy spectrum”, UFN, vol.129, P. 211, 1979. [37] M Schroeder, “Fractals, chaos, power laws: minutes from an infinite”, paradise. New York: W.H. Freeman; 1991. [38] K. N. Drabovich, “Captivated atomic particles in action”, UFN, №3, vol.158, P. 499 1989. [39] Yu. L. Sokolov “Interference method of the measurement parameters atomic conditions”, UFN, № 5, vol. 169, P. 559, 1999. [40] A.M. Mishin, “Problems of research of the Universe”, Iss. 23. Publishing house of St. Petersburg State University, 2001 (in Russian). [41] I. M. Beterov, P.B. Lerner, “Spontaneous and forced radiation of Rudberg’s atom in resonator”, UFN, №4, vol. 159, P. 665, 1989. [42] W. H. Itano, D. J. Heinzen, J. J. Bottlinger, D. J. Wineland, “Quantum Zeno effect”, Phys. Rev. Ser A., 41, P. 2295, 1990. [43] B. Zhdanov, A.P. Chubenko, Materials of the seminar devoted to the 80 anniversary from the date of birth of M.I. Podgoretsky, Dubna, P. 32, 2000 (in Russian). [44] L. M. Gindilis, “Model of the contact, rather then proof of the probe”, Earth and Universe, № 2, P. 78, 1976 (in Russian).