Abstract
We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.
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References
N. N. Bogoliubov, Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhteorizdat, Moscow (1946).
N. N. Bogoliubov, “Problems of dynamical theory in statistical physics,” in: Studies in Statistical Mechanics (J. de Boer and G. E. Uhlenbeck, eds.), Vol. 1, North-Holland, Amsterdam (1962), pp. 1–118.
N. N. Bogoliubov, “On the stochastic processes in the dynamical systems,” Sov. J. Part. Nucl., 9, 205 (1978).
D. Ya. Petrina, Stochastic Dynamics and Boltzmann Hierarchy (De Gruyter Expos. Math., Vol. 48), Walter de Gruyter, Berlin (2009).
D. N. Zubarev, Nonequilibrium Statistical Thermodynamics [in Russian], Nauka, Moscow (1971); English transl., Consultants Bureau, New York, London (1974).
J. A. McLennan, Introduction to Nonequilibrium Statistical Mechanics, Prentice Hall, New York (1989).
B. C. Eu, Nonequilibrium Statistical Mechanics: Ensemble Method (Fund. Theor. Phys., Vol. 93), Kluwer, Dordrecht (1998).
R. Zwanzig, “Time-correlation functions and transport coefficients in statistical mechanics,” Ann. Rev. Phys. Chem., 16, 67–102 (1965).
R. Zwanzig, “The concept of irreversibility in statistical mechanics,” Pure and Appl. Chem., 22, 371–378 (1970).
R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford Univ. Press, Oxford, New York (2001).
G. Gallavotti, Nonequilibrium and Irreversibility, Springer, Cham (2014).
V. V. Kozlov, Gibbs Ensembles and Nonequilibrium Statistical Mechanics [in Russian], RKhD, Moscow (2008).
V. Vedenyapin, A. Sinitsyn, and E. Dulov, Kinetic Boltzmann, Vlasov, And Related Equations, Elsevier, Amsterdam (2011).
D. N. Zubarev, V. G. Morozov, and G. Röpke, Statistical Mechanics of Nonequilibrium Processes, Vol. 1, Basic Concepts, Kinetic Theory, Akademie, Berlin (1996); Vol. 2, Relaxation and Hydrodynamic Processes, Akademie, Berlin (1997).
A. L. Kuzemsky, Statistical Mechanics and the Physics of Many-Particle Model Systems, World Scientific, Singapore (2017).
A. L. Kuzemsky, “Theory of transport processes and the method of the nonequilibrium statistical operator,” Internat. J. Modern Phys. B, 21, 2821–2949 (2007).
A. L. Kuzemsky, “Generalized kinetic and evolution equations in the approach of the nonequilibrium statistical operator,” Internat. J. Modern Phys. B, 19, 1029–1059 (2005).
A. L. Kuzemsky, “Electronic transport in metallic systems and generalized kinetic equations,” Internat. J. Modern Phys. B, 25, 3071–3183 (2011).
M. Toda, R. Kubo, and N. Saitô, Statistical Physics I: Equilibrium Statistical Mechanics, Springer, Berlin (1992).
M. Toda, R. Kubo, and N. Saitô, Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer, Berlin (1991).
J. W. Gibbs, Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundations of Thermodynamics, Dover, New York (1960).
N. N. Bogolyubov, D. Ya. Petrina, and B. I. Khatset, “Mathematical description of the equilibrium state of classical systems on the basis of the canonical ensemble formalism,” Theor. Math. Phys., 1, 194–212 (1969).
A. L. Kuzemsky, “Thermodynamic limit in statistical physics,” Internat. J. Modern Phys. B, 28, 1430004 (2014).
R. A. Minlos, Introduction to Mathematical Statistical Physics (Univ. Lect. Ser., Vol. 19), Amer. Math. Soc., Providence, R. I. (2000).
R. Zwanzig, “Ensemble method in the theory of irreversibility,” J. Chem. Phys., 33, 1338–1341 (1960).
P. G. Bergmann and J. L. Lebowitz, “New approach to nonequilibrium processes,” Phys. Rev., 99, 578–587 (1955).
J. L. Lebowitz and P. G. Bergmann, “Irreversible Gibbsian ensembles,” Ann. Phys. (N. Y.), 1, 1–23 (1957).
J. L. Lebowitz, “Stationary nonequilibrium Gibbsian ensembles,” Phys. Rev., 114, 1192–1202 (1959).
J. L. Lebowitz and A. Shimony, “Statistical mechanics of open systems,” Phys. Rev., 128, 1945–1958 (1962).
N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations [in Russian], Nauka, Moscow (1974); English transl., Gordon and Breach, New York (1961).
Yu. A. Mitropolsky, Averaging Method in Nonlinear Mechanics, Naukova Dumka, Kiev (1971).
A. M. Samoilenko, “N. N. Bogolyubov and non-linear mechanics,” Russ. Math. Surveys, 49, 109–154 (1994).
V. I. Arnol’d, “From averaging to statistical physics,” Proc. Steklov Inst. Math., 228, 184–190 (2000).
N. N. Bogoliubov, “On some problems connected with the foundations of statistical mechanics,” in: Proc. Intl. Symp. on Selected Topics in Statistical Mechanics (N. N. Bogolyubov Jr. et al., eds.), Joint Inst. Nucl. Res., Dubna (1982), pp. 9–18.
N. N. Bogoliubov and D. N. Zubarev, “The method of asymptotic approximation for systems with revolving phase and its application to the motion of charged particles in a magnetic field [in Russian],” Ukr. Matem. Zhur., 7, 5–17 (1955).
V. V. Kozlov and O. G. Smolyanov, “Information entropy in problems of classical and quantum statistical mechanics,” Dokl. Math., 74, 910–913 (2006).
E. T. Jaynes, Probability Theory: The Logic of Science, Cambridge Univ. Press, New York (2003).
L. M. Martyushev and V. D. Seleznev, “Maximum entropy production principle in physics, chemistry, and biology,” Phys. Rep., 426, 1–45 (2006).
A. L. Kuzemsky, “Probability, information, and statistical physics,” Internat. J. Theor. Phys., 55, 1378–1404 (2016).
N. N. Bogoliubov, “Quasiaverages in problems of statistical mechanics,” in: Statistical Physics and Quantum Field Theory (N. N. Bogoliubov, ed.) [in Russian], Nauka, Moscow (1973), pp. 7–80.
D. N. Zubarev, “Boundary conditions for statistical operators in the theory of nonequilibrium processes and quasiaverages,” Theor. Math. Phys., 3, 505–512 (1970).
A. L. Kuzemsky, “Bogoliubov’s vision: Quasiaverages and broken symmetry to quantum protectorate and emergence,” Internat. J. Modern Phys. B, 24, 835–935 (2010).
D. N. Zubarev and V. P. Kalashnikov, “Extremal properties of the nonequilibrium statistical operator,” Theor. Math. Phys., 1, 108–118 (1969).
D. N. Zubarev and V. P. Kalashnikov, “Construction of statistical operators for nonequilibrium processes,” Theor. Math. Phys., 3, 395–401 (1971).
D. N. Zubarev and V. P. Kalashnikov, “Derivation of the nonequilibrium statistical operator from the extremum of the information entropy,” Phys., 46, 550–554 (1970).
L. A. Pokrovskii, “Derivation of generalized kinetic equations with nonequlibrium statistical operator,” Sov. Phys. Dokl., 13, 1154 (1968).
K. Valasek and A. L. Kuzemsky, “Kinetic equations for a system weakly coupled to a thermal bath,” Theor. Math. Phys., 4, 826–832 (1970).
V. V. Kozlov, “Gibbs ensembles, equidistribution of the energy of sympathetic oscillators, and statistical models of thermostat,” Regul. Chaotic Dyn., 13, 141–154 (2008).
A. L. Kuzemsky, “Statistical theory of spin relaxation and diffusion in solids,” J. Low Temp. Phys., 143, 213–256 (2006).
K. Valasek, D. N. Zubarev, and A. L. Kuzemsky, “Schrödinger-type equation with damping for a dynamical system in a thermal bath,” Theor. Math. Phys., 5, 1150–1158 (1970).
A. L. Kuzemsky, “Works of D. I. Blokhintsev and the development of quantum physics,” PEPAN, 39, 137–172 (2008).
D. I. Blokhintsev, Collected Papers [in Russian], Vol. 1, Fizmatlit, Moscow (2009).
N. N. Bogolubov and N. N. Bogolubov Jr., Aspects of Polaron Theory [in Russian], Fizmatlit, Moscow (2004); English transl., World Scientific, Singapore (2008).
A. L. Kuzemsky and A. Pawlikowski, “Note on the diagonalization of a quadratic linear form defined on the set of second quantization fermion operators,” Rep. Math. Phys., 3, 201–207 (1972).
A. L. Kuzemsky and K. Walasek, “On the calculation of the natural width of spectral lines of atom by the methods of nonequilibrium statistical mechanics,” Lett. Nuovo Cimento, 2, 953–956 (1971).
S. Bloom and H. Margenau, “Quantum theory of spectral line broadening,” Phys. Rev., 90, 791–794 (1953).
D. I. Blokhintsev, “Calculation of the natural width of spectral lines by the stationary method [in Russian],” Zh. Eksp. Teor. Fiz., 16, 965–967 (1946).
A. L. Kuzemsky, “Generalized Van Hove formula for scattering of neutrons by the nonequilibrium statisticalmedium,” Internat. J. Modern Phys. B, 26, 1250092 (2012).
W. Marshall and S. W. Lovesey, Theory of Thermal Neutron Scattering, Oxford Univ. Press, Oxford (1971).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 194, No. 1, pp. 39–70, January, 2018.
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Kuzemsky, A.L. Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations. Theor Math Phys 194, 30–56 (2018). https://doi.org/10.1134/S004057791801004X
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DOI: https://doi.org/10.1134/S004057791801004X