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Theoretical and Mathematical Physics

, Volume 191, Issue 1, pp 537–557 | Cite as

Dissipation effects in infinite-dimensional Hamiltonian systems

  • S. M. Saulin
Article

Abstract

We show that the potential coupling of classical mechanical systems (an oscillator and a heat bath), one of which (the heat bath) is linear and infinite-dimensional, can provoke energy dissipation in a finitedimensional subsystem (the oscillator). Under natural assumptions, the final dynamics of an oscillator thus reduces to a tendency toward equilibrium. D. V. Treschev previously obtained results concerning the dynamics of an oscillator with one degree of freedom and a quadratic or (under some additional assumptions) polynomial potential. Later, A. V. Dymov considered the case of a linear oscillator with an arbitrary (finite) number of degrees of freedom. We generalize these results to the case of a heat bath (consisting of several components) and a multidimensional oscillator (either linear or nonlinear).

Keywords

Lagrange system system with infinite number of degrees of freedom final dynamics 

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References

  1. 1.
    D. Treschev, “Oscillator and thermostat,” Discrete Contin. Dyn. Syst., 28, 1693–1712 (2010).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. V. Dymov, “Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom,” Izv. Math., 76, 1116–1149 (2012).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    N. N. Bogoliubov, “An elementary example of establishing statistical equilibrium in a system connected to a heat bath [in Russian],” in: On Several Statistical Methods in Mathematical Physics [in Russian], Acad. Sci. UkrSSR, Kiev (1945), pp. 115–137.Google Scholar
  4. 4.
    A. I. Komech, “On the stabilization of interaction of a string with a nonlinear oscillator,” Mosc. Univ. Math. Bull., 46, 34–39 (1992).MathSciNetMATHGoogle Scholar
  5. 5.
    A. I. Komech, H. Spohn, and M. Kunze, “Long-time asymptotics for a classical particle interacting with a scalar wave field,” Commun. Partial Differ. Equations, 22, 307–335 (1997).MathSciNetMATHGoogle Scholar
  6. 6.
    A. I. Komech and H. Spohn, “Long-time asymptotics for a coupled Maxwell–Lorentz equations,” Commun. Partial Differ. Equations, 25, 559–584 (2000).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Ann. Phys., 149, 374–456 (1983).ADSCrossRefMATHGoogle Scholar
  8. 8.
    A. O. Caldeira and A. J. Leggett, “Influence of dissipation on quantum tunneling in macroscopic systems,” Phys. Rev. Lett., 46, 211–214 (1981).ADSCrossRefGoogle Scholar
  9. 9.
    A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Phys. A, 121, 587–616 (1983); Erratum, 130, 374 (1985).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    F. M. Ramazanoglu, “Approach to thermal equilibrium in the Caldeira–Leggett model,” J. Phys. A: Math. Theor., 42, 265303 (2009); arXiv:0812.2520v1 [quant-ph] (2008).ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. Cacheffo, M. H. Y. Moussa, and M. A. de Ponte, “The double Caldeira–Leggett model: Derivation and solutions of the master equations, reservoir-induced interactions, and decoherence,” Phys. A, 389, 2198–2217 (2010); arXiv:0903.2176v2 [quant-ph] (2009).MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. Ayyar and B. Müller, “Approach to equilibrium in the Caldeira–Leggett model,” Internat. J. Modern Phys. E, 22, 1350016 (2012); arXiv:1212.3538v1 [nucl-th] (2012).ADSCrossRefGoogle Scholar
  13. 13.
    J. F. R. Archilla, R. S. MacKay, and J. L. Marin, “Discrete breathers and Anderson modes: Two faces of the same phenomenon,” Phys. D, 134, 406–418 (1999).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    R. S. MacKay, “Discrete breathers: Classical and quantum,” Phys. A, 288, 174–198 (2000).MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. Koukouloyannis and R. S. MacKay, “Existence and stability of 3-site breathers in a triangular lattice,” J. Phys. A: Math. Gen., 38, 1021–1030 (2005).ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    R. Yamapi and R. S. MacKay, “Stability of synchronisation in a shift-invariant ring of mutually coupled oscillators,” Discrete Contin. Dyn. Syst. Ser. B, 10, 973–996 (2008).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    E. A. Gorin and D. V. Treschev, “Relative version of the Titchmarsh convolution theorem,” Funct. Anal. Appl., 46, 26–32 (2012).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    V. I. Bogachev, Fundamentals of Measure Theory [in Russian], Vol. 1, RKhD, Moscow (2006); English transl.: Measure Theory, Vol. 1, Springer, Berlin (2007).Google Scholar
  19. 19.
    L. Shwartz, Cours d’analyse, Vol. 2, Hermann, Paris (1981).Google Scholar
  20. 20.
    V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981); English transl. prev. ed. (Pure Appl. Math., Vol. 3), Marcel Dekke, New York (1971).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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