Advertisement

Ukrainian Mathematical Journal

, Volume 54, Issue 10, pp 1583–1601 | Cite as

Cumulant Representation of Solutions of the BBGKY Hierarchy of Equations

  • V. I. Herasymenko
  • T. V. Ryabukha
Article

Abstract

We construct a cumulant representation of solutions of the Cauchy problem for the BBGKY hierarchy of equations and for the dual hierarchy of equations. We define the notion of dual nonequilibrium cluster expansion. We investigate the convergence of the constructed cluster expansions in the corresponding functional spaces.

Keywords

Cauchy Problem Functional Space Cluster Expansion BBGKY Hierarchy Cumulant Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    D. Y. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems, Taylor and Francis, London (2002).Google Scholar
  2. 2.
    C. Cercignani, V. I. Gerasimenko, and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer, Dordrecht (1997).Google Scholar
  3. 3.
    D. Ya. Petrina and A. K. Vidybida, “Cauchy problem for the Bogolyubov chain of equations,” Tr. Mat. Inst. Akad. Nauk SSSR, 136, 370–378 (1975).Google Scholar
  4. 4.
    D. Ya. Petrina, “Mathematical description of the evolution of infinite systems of classical statistical physics. Locally perturbed one-dimensional systems,” Teor. Mat. Fiz., 38, No. 2, 230–250 (1979).Google Scholar
  5. 5.
    D. Ya. Petrina and V. I. Gerasimenko, “Mathematical description of the evolution of states of infinite systems of classical statistical mechanics,” Usp. Mat. Nauk, 38, Issue 3, 3–58 (1983).Google Scholar
  6. 6.
    V. P. Maslov and S. E. Tariverdiev, “Asymptotics of the KolmogorovJędrysekFeller equations for a system of a large number of particles,” in: VINITI Series in Probability Theory, Mathematical Statistics, and Theoretical Cybernetics [in Russian], Vol. 19, VINITI, Moscow (1982), pp. 85–126.Google Scholar
  7. 7.
    D. Y. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics, Naukova Dumka, Kiev (1985).Google Scholar
  8. 8.
    D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin, New York (1969).Google Scholar
  9. 9.
    E. G. D. Cohen, “The kinetic theory of dense gases,” in: E. G. D. Cohen (editor), Fundamental Problems in Statistical Mechanics, Vol. 2, North-Holland, Amsterdam, (1968), pp. 228–275.Google Scholar
  10. 10.
    M. S. Green, “Boltzmann equation from the statistical mechanical point of view,” J. Chem. Phys., 25, No. 5, 836–855 (1956).Google Scholar
  11. 11.
    M. V. Men'shikov, A. N. Kopylova, A. M. Revyakin, et al., Combinatorial Analysis. Problems and Exercises [in Russian], Nauka, Moscow (1982).Google Scholar
  12. 12.
    J. R. Dorfmann and E. G. D. Cohen, “Difficulties in the kinetic theory of the dense gases,” J. Math. Phys., 8, No. 2, 282–297 (1967).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • V. I. Herasymenko
    • 1
  • T. V. Ryabukha
    • 2
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev
  2. 2.Shevchenko Kiev UniversityKiev

Personalised recommendations