Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 053, 8 pages      cond-mat/0605364      https://doi.org/10.3842/SIGMA.2006.053

On Regularized Solution for BBGKY Hierarchy of One-Dimensional Infinite System

Tatiana V. Ryabukha
Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs'ka Str., Kyiv-4, 01601 Ukraine

Received October 31, 2005, in final form April 26, 2006; Published online May 14, 2006

Abstract
We construct a regularized cumulant (semi-invariant) representation of a solution of the initial value problem for the BBGKY hierarchy for a one-dimensional infinite system of hard spheres interacting via a short-range potential. An existence theorem is proved for the initial data from the space of sequences of bounded functions.

Key words: BBGKY hierarchy; cumulant; regularized solution.

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References

  1. Petrina D.Ya., Gerasimenko V.I., Malyshev P.V., Mathematical foundations of classical statistical mechanics. Continuous systems, 2nd ed., London - New York, Taylor & Francis Inc., 2002.
  2. Cercignani C., Gerasimenko V.I., Petrina D.Ya., Many-particle dynamics and kinetic equations, Kluwer Acad. Publ., 1997.
  3. Cercignani C., Illner R., Pulvirenti M., The mathematical theory of dilute gases, Applied Mathematical Sciences, Vol. 106, New York, Springer, 1994.
  4. Spohn H., Large scale dynamics of interacting particles, Springer, 1991.
  5. Petrina D.Ya., Mathematical description of the evolution of infinite systems of classical statistical physics. I. Locally perturbed one-dimensional systems, Teoret. Mat. Fiz., V.38, 1979, 230-262 (in Russian).
  6. Petrina D.Ya., Gerasimenko V.I., Mathematical description of the evolution of the state of infinite systems of classical statistical mechanics, Uspekhi Mat. Nauk, 1983, V.38, 3-58 (in Russian).
  7. Gerasimenko V.I., Ryabukha T.V., Dual nonequilibrium cluster expansions, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 2003, N 3, 16-22 (in Ukrainian).
  8. Gerasimenko V.I., Ryabukha T.V., Stashenko M.O., On the BBGKY hierarchy solutions for many-particle systems with different symmetry properties, in Proceedings of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv, 2004, V.50, Part 3, 1308-1313.
  9. Gerasimenko V.I., Ryabukha T.V., Cumulant representation of solutions of the Bogolyubov chains of equations, Ukrain. Mat. Zh., 2002, V.54, 1313-1328 (English transl.: Ukrainian Math. J., 2002, V.54, 1583-1601).
  10. Ruelle D., Statistical mechanics. Rigorous results, New York - Amsterdam, W.A. Benjiamin Inc., 1969.
  11. Cohen E.G.D., Cluster expansions and the hierarchy. I. Non-equilibrium distribution functions, Physica, 1962, V.28, 1045-1059.
  12. Green H.S., Piccirelli R.A., Basis of the functional assumption in the theory of the Boltzmann equation, Phys. Rev. (2), 1963, V.132, 1388-1410.
  13. Reed M., Simon B., Methods of modern mathematical physics. Vol. 1: Functional analysis, New York - London, Academic Press, 1972.
  14. Kaniadakis G., BBGKY hierarchy underlying many-particle quantum mechanics, Phys. Lett. A, 2003, V.310, 377-382, quant-ph/0303159.
  15. Tarasov V.E., Fractional systems and fractional Bogoliubov hierarchy equations, Phys. Rev. E, 2005, V.71, 011102, 12 pages, cond-mat/0505720.
  16. Illner R., Pulvirenti M., Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Comm. Math. Phys., 1986, V.105, 189-203.
  17. Illner R., Pulvirenti M., A derivation of the BBGKY-hierarchy for hard sphere particle systems, Transport Theory Statist. Phys., 1987, V.16, 997-1012.


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