Skip to main content
SpringerLink
Log in

Asymptotics of the solutions to theN-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space Γ(L 1(ℝn)) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL 1(ℝn) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L 1(ℝn)) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent characteristic equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. P. Maslov and S. É. Tariverdiev, “Asymptotics of the Kolmogorov-Feller equation for a system of a large number of particles,” in:Probability Theory, Mathematical Statistics, Theoretical Cybernetics [in Russian], Vol. 19, VINITI, Moscow (1982), pp. 85–126.

    Google Scholar 

  2. V. P. Maslov and A. M. Chebotarev, “Random fields corresponding to the Bogolyubov, Vlasov, and Boltzmann chains,”Teor. Mat. Fiz. [Theoret. and Math. Phys.],54, 78–88 (1983).

    MathSciNet  Google Scholar 

  3. V. P. Maslov and O. Yu. Shvedov, “An asymptotic formula for theN-particle density function asN→∞ and violation of the chaos hypothesis,”Russ. J. Math. Phys.,2, No. 2, 217–234 (1994).

    MathSciNet  Google Scholar 

  4. V. P. Maslov and O. Yu. Shvedov, “The spectrum of theN-particle Hamiltonian for largeN and superfluidity,”Dokl. Ross. Akad. Nauk [Russian Math. Dokl.],335, No. 1, 42–46 (1994).

    Google Scholar 

  5. V. P. Maslov and O. Yu. Shvedov, “Quantization in the vicinity of classical solutions in theN-particle problem and superfluidity,”Teor. Mat. Fiz. [Theoret. and Math. Phys.],98, No. 2, 266–288 (1994).

    MathSciNet  Google Scholar 

  6. V. P. Maslov and O. Yu. Shvedov, “Steady-state asymptotic solutions in the many-body problem and the derivation of integral equations with jumping nonlinearity,”Differentsial'nye Uravneniya [Differential Equations],31, No. 2, 312–325 (1995).

    MathSciNet  Google Scholar 

  7. V. P. Maslov and O. Yu. Shvedov, “The complex WKB method in Fock space,”Dokl. Ross. Akad. Nauk [Russian Math. Dokl.],340, No. 1, 42–47 (1995).

    MathSciNet  Google Scholar 

  8. J.-P. Serre,Lie Algebras and Lie Groups, Benjamin, New York-Amsterdam (1965).

    Google Scholar 

  9. A. A. Arsen'ev,Lectures on Kinetic Equations [in Russian], Nauka, Moscow (1992).

    Google Scholar 

  10. H. Tanaka, “Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,”Z. Wahrshr.,46, 67–105 (1978).

    MATH  Google Scholar 

  11. V. P. Maslov and A. M. Chebotarev, “Cluster expansions and secondary quantization,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],203, 168–185 (1994).

    Google Scholar 

  12. V. P. Maslov, “The equation of self-consistent field,” in:Current Problems of Mathematics, Itogi Nauki i Tekhniki [in Russian], Vol. 11, VINITI, Moscow (1978), pp. 153–234.

    Google Scholar 

  13. A. M. Chebotarev, “The logarithmic asymptotics of the solution to the Cauchy problem for the Boltzmann equation,”Mat. Model.,7, No. 10 (1995).

    Google Scholar 

  14. V. P. Maslov,Perturbation Theory and Asymptotic Methods [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  15. M. V. Fedoryuk,The Saddle-Point Method [in Russian], Nauka, Moscow (1977).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Electronics and Mathematics, USSR

    V. P. Maslov

Authors
  1. V. P. Maslov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995.

The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible.

This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maslov, V.P. Asymptotics of the solutions to theN-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations. Math Notes 58, 1166–1177 (1995). https://doi.org/10.1007/BF02305000

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02305000

Keywords

Search

Navigation