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On N. N. Bogolyubov's works in classical and quantum statistical mechanics

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Abstract

A review of N. N. Bogolyubov's works in classical and quantum statistical mechanics is presented.

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References

Section 1

  1. N. M. Krylov and N. N. Bogolyubov,Investigation of Longitudinal Stability of an Airplane [in Russian], Gosaviaavtotraktizdat, Moscow-Leningrad (1932).

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  2. N. M. Krylov and N. N. Bogolyubov,On Oscillations of Synchronous Machines. 2.On Stability of Parallel Work of n-Synchronous Machines [in Russian], Energovydav, Khar'kov (1932).

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  3. N. N. Bogolyubov,New Methods of Solving Some Mathematical Problems in Engineering [in Russian], Budvydav, Khar'kov (1933).

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  4. N. M. Krylov and N. N. Bogolyubov,New Methods of Nonlinear Mechanics and Their Application to the Study of Work of Electronic Generators [in Russian], Vol. 1, Gostekhteoretizdat, Moscow-Leningrad (1934).

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  5. N. M. Krylov and N. N. Bogolyubov,The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations [in Russian], Izd. VUAN, Kiev (1934).

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  6. N. M. Krylov and N. N. Bogolyubov,On Some Formal Expansions in Nonlinear Mechanics [in Ukrainian], Izd. VUAN, Kiev (1934).

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  7. N. M. Krylov and N. N. Bogolyubov,Introduction to Nonlinear Mechanics (Approximate and Asymptotic Methods of Nonlinear Mechanics) [in Russian], Izd. Ukrain. Akad. Nauk, Kiev (1937).

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  8. N. N. Bogolyubov, “One-frequency free oscillations in nonlinear systems with many degrees of freedom,“Sbornik Tr. Inst. Stroit. Mekhaniki Ukrain. Akad. Nauk, No. 10, 9–21 (1948).

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  9. N. N. Bogolyubov, “Synchronization of relaxation oscillations,“Nauch. Zap. Kiev. Univ.,9, issue 9, Mat Sb., No. 4, 5–28 (1950).

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  10. N. N. Bogolyubov, “Perturbation theory in nonlinear mechanics,“Sbornik Tr. Inst. Stroit. Mekhaniki Ukrain. Akad. Nauk, No. 14, 9–34 (1950).

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  11. N. N. Bogolyubov and Yu. A. Mitropol'skii,Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1955).

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  12. N. N. Bogolyubov and D. N. Zubarev, “Method of asymptotic approximation for systems with rotating phase and its application to the motion of charged particles in magnetic field,“Ukr. Mat. Zh.,1, No. 1, 5–17 (1955).

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  13. N. N. Bogolyubov and Yu. A. Mitropol'skii,Method of Integral Manifolds in Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrain. Acad. of Sciences, Kiev (1961).

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  14. N. N. Bogolyubov, “On quasiperiodic solutions in problems of nonlinear mechanics,“ in:Proc. of the First Summer Mathematical School (Kanev, 1963) [in Russian], Vol. 1, Naukova Dumka, Kiev (1964), pp. 11–101.

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  15. N. N. Bogolyubov, Yu. A. Mitropol'skii, and A. M. Samoilenko,Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).

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Section 2

  1. N. M. Krylov and N. N. Bogolyubov, “General theory of measure in nonlinear mechanics,“ in:Sbornik Tr. Nelin. Mekh. [in Russian], Izd. Ukrain. Akad. Nauk, Kiev (1937), pp. 55–112.

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  2. N. M. Krylov and N. N. Bogolyubov, “Influence of statistical change of parameters with respect to ergodic properties of dynamical nonconservative systems,“ in:Sbornik Tr. Nelin. Mekh. [in Russian], Izd. Ukrain. Akad. Nauk, Kiev (1937), pp. 154–171.

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  3. N. M. Krylov and N. N. Bogolyubov, “Influence of statistical change of parameters on the motion of dynamical conservative systems for sufficiently long periods of time,“ in:Sbornik Tr. Nelin. Mekh. [in Russian], Izd. Ukrain. Akad. Nauk, Kiev (1937), pp. 119–135.

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  4. N. M. Krylov and N. N. Bogolyubov, “La théorie generale de la mesure dans son application á l'étude des systámes dynamiques de la mécanique non-linéaire,“Ann. Math. Ser. 2,38, No. 1, 65–113 (1937).

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  5. N. M. Krylov and N. N. Bogolyubov, “On iterations with varying parameter,“ in:Sbornik Trudov po Nelin. Mekh., Zap. Kaf. Mat. Fiz. Inst. Stroit. Mekhaniki Ukrain. Akad. Nauk [in Ukrainian], Kiev (1937), pp. 191–200.

  6. N. N. Bogolyubov, “On some ergodic properties of continuous groups of transformations,“Nauch. Zap. Kiev. Univ. Fiz.-Mat. Sb.,4, Issue 5, 45–52 (1939).

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  7. N. M. Krylov and N. N. Bogolyubov, “On some problems of ergodic theory of stochastic systems,“Zap. Kaf. Mat. Fiz. Ukrain. Akad. Nauk,4, 243–287 (1939).

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Section 3

  1. N. M. Krylov and N. N. Bogolyubov, “On derivation of Fokker-Planck equations by perturbation-theory method based on spectral properties of the perturbation Hamiltonian,“Zap. Kaf. Mat. Fiz. Ukrain. Akad. Nauk,4, 5–80 (1939).

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  2. N. N. Bogolyubov, “On influence of random force on a harmonic vibrator,“Uch. Zap. Mosk. Univ., Issue 77, Fizika, Vol. 3, 51–73 (1945).

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  3. N. N. Bogolyubov, “On some limit distributions for sums depending on arbitrary phases,“Uch. Zap. Mosk. Univ., Issue 77, Fizika, Vol. 3, 43–50 (1945).

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  4. N. N. Bogolyubov,On Some Statistical Methods in Mathematical Physics [in Russian], Izd. Ukrain. Akad. Nauk, Kiev (1945).

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  5. N. N. Bogolyubov, “An elementary example of transition to statistical equilibrium in a system connected with a heat bath,“ in:On Some Statistical Methods in Mathematical Physics [in Russian], Izd. Ukrain. Akad. Nauk, Kiev (1945), pp. 115–137.

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Section 4

  1. N. N. Bogolyubov,Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow-Leningrad (1946).

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  2. N. N. Bogolyubov, “Kinetic equations,“Zh. éksp. Teor. Fiz.,16, No. 8, 691–702 (1946).

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  3. N. N. Bogolyubov, “Hydrodynamic equations in statistical mechanics,“Shornik Tr. Inst. Matem. Ukrain. Akad. Nauk, No. 10, 41–59 (1948).

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  4. N. N. Bogolyubov and B. I. Khatset, “On some mathematical problems in the theory of statistical equilibrium,“Dokl. Akad. Nauk SSSR,66, No. 3, 321–324 (1949).

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  5. N. N. Bogolyubov, D. Ya. Petrina, and B. I. Khatset, “Mathematical description of equilibrium state of classical systems on the basis of canonical formalism,“Teor. Mat. Fiz.,1, No. 2, 251–274 (1969).

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  6. D. Ruelle,Statistical Mechanics. Rigorous Results, Benjamin Inc., New York-Amsterdam (1969).

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  7. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev,Mathematical Foundations of Classical Statistical Mechanics [in Russian], Naukova Dumka, Kiev (1985).

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Section 5

  1. N. N. Bogolyubov,Lectures in Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems [in Ukrainian], Radyans'ka Shkola, Kiev (1949).

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  2. N. N. Bogolyubov and K. P. Gurov, “Kinetic equations in quantum mechanics,“Zh. éksp. i Teor. Fiz.,17, No. 7, 614–628 (1947).

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  3. J. Ginibre, “Reduced density matrices of quantum gases. I. Limit of infinite volume,“J. Math. Phys.,6, No. 2, 238–251 (1968); “Reduced density matrices of quantum gases. II. Cluster property,“J. Math. Phys.,6, No. 2, 252–262 (1968)

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  4. D. Ya. Petrina, “On solutions of kinetic Bogolyubov equations. Quantum statistics,“Teor. Mat. Fiz.,13, No. 3, 391–405 (1972).

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Section 6

  1. N. N. Bogolyubov, “On the theory of superfluidity,“Izv. Akad. Nauk SSSR. Ser. Fiz.,11, No. 1, 77–90 (1947).

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  2. N. N. Bogolyubov, “Energy levels of nonideal Bose-Einstein gas,“Vestnik Mosk. Univ., No. 7, 43–56 (1947).

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  3. N. N. Bogolyubov, “On the theory of superfluidity,“J. Phys.,11, No. 1, 23–32 (1947).

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  4. N. N. Bogolyubov, “On the theory of superfluidity,“Sbornik Tr. Inst. Matem. Ukrain. Akad. Nauk, No. 9, 89–103 (1948).

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Section 7

  1. N. N. Bogolyubov,On a New Method in the Theory of Superconductivity [in Russian], Preprint P-99, JINR, Dubna (1957); also in:Zh. éksp. Teor. Fiz.,34, issue 1, 73–79 (1958).

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  2. N. N. Bogolyubov, “On a new method in the theory of superconductivity,“Nuovo Cim.,7, 794–805 (1958).

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  3. N. N. Bogolyubov, V. V. Tolmachev, and D. V. Shirkov,A New Method in the Theory of Superconductivity [in Russian], Izd. Akad. Nauk SSSR, Moscow (1958).

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  4. N. N. Bogolyubov, “Basic principles of the theories of superfluidity and superconductivity,“Vestnik Akad. Nauk SSSR, No. 8, 36–46 (1958).

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  5. N. N. Bogolyubov, “On superfluidity condition in the theory of nuclear matter,“Dokl. Akad. Nauk SSSR,119, No. 1, 52–55 (1958).

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  6. N. N. Bogolyubov,On the Compensation Principle and the Method of Self-Consistent Field [in Russian], Preprint P-267, JINR, Dubna (1958).

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  7. N. N. Bogolyubov,On the Variational Principle in the Problem of Many Bodies [in Russian], Preprint P-136, JINR, Dubna (1958).

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Section 8

  1. N. N. Bogolyubov, D. N. Zubarev, and Yu. A. Tserkovnikov, “Asymptotically exact solution for the model Hamiltonian of the theory of superconductivity,“Zh. éksp. Teor. Fiz.,39, No. 1, 120–129 (1960).

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  2. N. N. Bogolyubov,On the Problem of Model Hamiltonian in the Theory of Superconductivity [in Russian], Preprint P-511, JINR, Dubna (1960).

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  3. N. N. Bogolyubov, “On some problem of the theory of superconductivity,“Physica, Suppl. (Proc. Int. Congr. on Many-Particle Problems, Utrecht, 1960),26, 51–116 (1960).

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  4. R. Haag, “The mathematical structure of the Bardeen-Cooper-Schrieffer model,“Nuovo Cim.,25, 287–302 (1962).

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  5. N. N. Bogolyubov (Jr.), “Calculation of average energy for model systems,“Ukr. Mat. Zh.,17, No. 3, 3–15 (1965).

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  6. N. N. Bogolyubov (Jr.),The Method for Investigation of Model Hamiltonians [in Russian], Nauka, Moscow (1974).

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  7. N. N. Bogolyubov (Jr.) and B. I. Sadovnikov,Some Problems of Statistical Mechanics [in Russian], Vysshaya Shkola, Moscow (1975).

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  8. N. N. Bogolyubov and N. N. Bogolyubov (Jr.),Introduction to Quantum Statistical Mechanics [in Russian], Nauka, Moscow (1984).

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  9. N. N. Bogolyubov (Jr.), I. G. Brankov, V. A. Zagrebnov, A. M. Kurbatov, and N. S. Tonchev,The Method of Approximating Hamiltonian in Statistical Physics [in Russian], Izd. BAN, Sofia (1981).

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  10. N. N. Bogolyubov (Jr.) and D. Ya. Petrina, “On a certain class of model systems admitting reduction of Hamiltonian degree in the thermodynamic limit. I, II,“Teor. Mat. Fiz.,33, No. 2, 231–245 (1977);37, No. 2, 246–257 (1978).

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  11. D. Ya. Petrina, “On Hamiltonians of quantum statistical mechanics and model Hamiltonian of the theory of superconductivity,“Teor. Mat. Fiz.,4, No. 3, 394–410 (1970).

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  12. D. Ya. Petrina and V. P. Yatsishin, “On the model Hamiltonian of the theory of superconductivity,“Teor. Mat. Fiz.,10, No. 2, 283–300 (1972).

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Section 9

  1. N. N. Bogolyubov,Quasiaverages in Problems of Statistical Mechanics [in Russian], Preprint P-781, JINR, Dubna (1961).

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  2. N. N. Bogolyubov,Cluster Property and the Method of Quasiaverages [in Russian], Preprint P-549, JINR, Dubna (1961).

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  3. N. D. Mermin and H. Wagner, “Absence of ferromagnetism and antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,“Phys. Rev. Lett.,17, No. 22, 1133–1136 (1966).

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    Yu. A. Mitropolskii (Academician, Ukrainian Academy of Sciences) & D. Ya. Petrina (Corresponding Member, Ukrainian Academy of Sciences)

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  1. Yu. A. Mitropolskii
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  2. D. Ya. Petrina
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 155–201, February, 1993.

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Mitropolskii, Y.A., Petrina, D.Y. On N. N. Bogolyubov's works in classical and quantum statistical mechanics. Ukr Math J 45, 171–214 (1993). https://doi.org/10.1007/BF01060975

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