Dissipation effects in infinite-dimensional Hamiltonian systems
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).
KeywordsLagrange system system with infinite number of degrees of freedom final dynamics
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- 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
- 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.L. Shwartz, Cours d’analyse, Vol. 2, Hermann, Paris (1981).Google Scholar
- 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