×

zbMATH — the first resource for mathematics

Infinite-dimensional equation Liouville with respect to measures. (English. Russian original) Zbl 1210.37046
Dokl. Math. 81, No. 3, 476-480 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 432, No. 1, 28-32 (2010).
An infinite system of equations with respect to time-dependent finite-dimensional probability distributions is obtained. This system is equivalent to the Liouville equation with respect to functions of a real argument taking values in the space of probability measures. As a consequence, the equations of this type as referred in this paper as Liouville equations with respect to measures. It is observed that in the absence of an interaction, all equations in this system turn out to be independent. Then, it is shown that the probability distribution on the configuration space tends to a product of independent uniform distributions. The system of equations under consideration substantially differs from the Bogoliubov infinite system of equations. It is proven that the solution of the Liouville infinite-dimensional equation with respect to measures is the pointwise weak limit of the measure-valued functions. Also, it is deduced that the Liouville equation with respect to functions is an equation for the first integrals of the Hamilton equation.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q70 PDEs in connection with mechanics of particles and systems of particles
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Poincaré, Selected Works (Nauka, Moscow, 1974), Vol. 3 [in Russian].
[2] N. N. Bogolyubov, Collection of Scientific Works (Nauka, Moscow, 2006), Vol. 5 [in Russian].
[3] N. Bourbaki, Éléments de Mathématique, Book 6: Intégration, Chapter 6: Intégration vectorielle, Chapter 7: Mesure de Haar, Chapter 8: Convolution et Représentations (Hermann, Paris, 1959, 1963; Nauka, Moscow, 1970).
[4] O. G. Smolyanov and S. V. Fomin, Usp. Mat. Nauk 31(4), 3–56 (1976).
[5] O. G. Smolyanov and H. von Weizsaecker, Acad. Sci. Paris Ser. I 321, 103–108 (1995).
[6] V. V. Kozlov, Thermal Equilibrium in the Sense of Gibbs and Poincaré (Izhevsk, 2002) [in Russian].
[7] V. V. Kozlov, Reg. Chaotic Dyn., No. 1, 23–34 (2004).
[8] V. V. Kozlov and O. G. Smolyanov, Teor. Veroyatn. Ee Primen. 51(2), 1–18 (2006).
[9] L. Accardi and O. G. Smolyanov, Dokl. Math. 79, 90–93 (2009) [Dokl. Akad. Nauk 424, 583–587 (2009)]. · Zbl 1261.46034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.