Abstract
We consider the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of a system of hard spheres. A new concept of averages over states of stochastic systems is introduced, in which the contribution of the hypersurfaces on which stochastic point particles interact is taken into account. We give a rigorous derivation of the infinitesimal operators of the semigroups of evolution operators.
Abstract
Визначена стохаатична динаміка, яка є границею Больцмана-Греда від гамільтонової динаміки системи пружних куль. Введено нову концепцію середніх від спостережуваних за станами стохастичних систем. В ньому враховуються вклади від гіперповерхонь, на яких взаємодіуть точкові стохастичні частки. Дано строге визначення інфінітезімальних операторов для півгрупи еволюційних операторів.
Similar content being viewed by others
Наслідок доведено
Petrina D. Ya., Petrina K. D. Stochastic dynamics and Boltzmann hierarchy.-Kiev, 1996.—51 p. —(Preprint/Inst. Math. Nat. Acad. Sci. Ukraine; 96.7).
Petrina D. Ya., Petrina K. D. Stochastic dynamics and Boltzmann hierarchy. I, II, III // Ukr. Math. J.—50, No 2.—P. 195–210;—50, No 3.—P. 372–387;—50, No 4.—P. 552–569.
Petrina D. Ya., Gerasimenko V. I., andMalyshev P. V. Mathematical foundation of classical statisticalmechanics. Continuous systems.—New York: Gordon and Breach, 1989.—386 p.
Cercignani C., Gerasimenko V. I., andPetrina D. Ya. Many-particle dynamics and kinetic equations.—Dordrecht: Kluwer, 1997.—244 p.
Cercignani C., Illner R., andPulvirenti M. The mathematical theory of dilute gases.—New York: Springer-Verlag, 1994.—254 p.
Author information
Authors and Affiliations
Additional information
This work was performed during the stay May–June 1997 of D. Ya. Petrina and during the stay September–December 1997 of K. D. Petrina as visiting professors at the Politecnico di Milano, supported by the Italian Consiglio Nazionale delle Ricerche. The authors would like to thank Prof. C. Cercignani for his comments and support.
Rights and permissions
About this article
Cite this article
Lampis, M., Petrina, D.Y. & Petrina, K.D. Stochastic dynamics as a limit of Hamiltonian dynamics of hard spheres. Ukr Math J 51, 614–635 (1999). https://doi.org/10.1007/BF02515237
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02515237