Skip to main content
SpringerLink
Log in

Markov Jump Dynamics with Additive Intensities in Continuum: State Evolution and Mesoscopic Scaling

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We investigate stochastic (conservative) non-equilibrium jump dynamics of interacting particles in continuum. The corresponding evolutions of correlation functions are constructed. The mesoscopic scaling (Vlasov scaling) of the dynamics is studied and the corresponding kinetic equations for the particle densities are derived.

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. Albeverio, S., Kondratiev, Y., Röckner, M.: Analysis and geometry on configuration spaces. J. Func. Anal. 154, 444–500 (1998)

    Article  MATH  Google Scholar 

  2. Balescu, R.: Statistical Dynamics: Matter Out of Equilibrium. Imperial College Press, Singapore (1997)

    Book  MATH  Google Scholar 

  3. Belavkin, V., Maslov, V., Tariverdiev, S.: The asymptoyic dynamics of a system with a large number of particles described by Kolmogorov–Feller equations. Theor. Math. Phys. 49(3), 1043–1049 (1981)

    Article  MathSciNet  Google Scholar 

  4. Berns, C., Kondratiev, Yu., Kozitsky, Yu., Kutoviy, O.: Kawasaki dynamics of interacting particles in continuum: micro- and mesoscopic description. J. Dyn. Diff. Eqn. 25, 1027–1056 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berns, C., Kondratiev, Yu., Kutoviy, O.: Construction of a state evolution for Kawasaki dynamics in continuum. Anal. Math. Phys. 3(2), 97–117 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bogoliubov, N.N.: Problems If a Dynamical Theory in Statistical Physics Gostekhisdat, Moscow, 1946 (in Russian). English translation in de Boer, J., Uhlenbeck, G.E. (eds.) Studies in Statistical Mechanics, vol. 1, pp. 1–118. North-Holland, Amsterdam (1962)

  7. Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56(2), 101–113 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Dobrushin, R.L.: Vlasov equations. Funct. Anal. Appl. 13(2), 115–123 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  9. Finkelshtein, D., Kondratiev, Y., Oliveira, M.: Markov evolution and hierarchical equations in the continuum. I:one-component systems. J. Evol. Equ. 9, 197–233 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Finkelshtein, D., Kondratiev, Y., Kutoviy, O.: Vlasov scaling for stochastic dynamics of continuous systems. J. Stat. Phys. 141, 158–178 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Finkelshtein, D., Kondratiev, Y., Kutoviy, O.: Semigroup approach to non-equilibrium birth-and-death stochastic dynamics in continuum. J. Funct. Anal. 262(3), 1274–1308 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hansen, J.P., McDonald, I.R.: Theory of Simple Liquids. Academic Press, London (1990)

    MATH  Google Scholar 

  13. Ichimaru, S.: Basic Principles of Plasma Physics: A Statistical Approach. W.A. Benjamin, Reading, MA (1973)

    Google Scholar 

  14. Kondratiev, Y., Kuna, T.: Harmonic analysis on configuration space I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 201–233 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kondratiev, Y., Kutoviy, O.: On the metrical properties of the configuration space. Math. Nachr. 279(7), 774–783 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kondratiev, Y., Milnos, R., Zhizhina, E.: One-particle subspace of the generator of Glauber dynamics of continuous particle systems. Rev. Math. Phys. 16(9), 1–42 (2004)

    Article  MathSciNet  Google Scholar 

  17. Kondratiev, Y., Lytvynov, E., Röckner, M.: Equilibrium Kawasaki dynamics of continuous particle systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(2), 185–209 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kondratiev, Y., Lytvynov, E., Röckner, M.: Non-equilibrium stochastic dynamics in continuum: the free case. Condens. Matter Phys. 11(4(56)), 701–721 (2008)

  19. Kondratiev, Y., Kutoviy, O., Minlos, R.: On non-equilibrium stochastic dynamics for interacting particle systems in continuum. J. Func. Anal. 255, 200–227 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kondratiev, Y., Kuna, T., Oliveira, M.J., da Silva, J.L., Streit, L.: Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems. arXiv:0912.1312v1 (2009)

  21. Kozlov, V.V.: The generalized Vlasov kinetic equation. Rus. Math. Surv. 63(2), 691–726 (2008)

    Article  MATH  Google Scholar 

  22. Lebowitz, J.L., Montroll, E.W. (eds.): Nonequilibrium Phenomena I: The Boltzmann Equation. North-Holland Publishing Company, Amsterdam (1983)

    MATH  Google Scholar 

  23. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles, I. Arch. Ration. Mech. Anal. 59(3), 219–239 (1975)

    MathSciNet  Google Scholar 

  24. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles, II: characterization of correlation measures. Arch. Ration. Mech. Anal. 59(3), 241–256 (1975)

    MathSciNet  Google Scholar 

  25. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  26. Presutti, E.: Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin (2009). (Theoretical and Mathematical Physics)

    MATH  Google Scholar 

  27. Ruelle, D.: Statistical Mechanics. Rigorous Results. Benjamin, New York (1969)

    MATH  Google Scholar 

  28. Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52(3), 569–615 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  30. Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Berlin (1991)

    Book  Google Scholar 

Download references

Acknowledgments

This work was financially supported by the DFG through the SFB 701: “Spektrale Strukturen und Topologische Methoden in der Mathematik” and the IGK “Stochastics and Real World Models” which is cordially acknowledged by the authors.

Author information

Authors and Affiliations

  1. IGK “Stochastics and Real World Models”, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Christoph Berns

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Yuri Kondratiev & Oleksandr Kutoviy

  3. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Building E18-420, Cambridge, MA, 02139, USA

    Oleksandr Kutoviy

Authors
  1. Christoph Berns
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuri Kondratiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oleksandr Kutoviy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleksandr Kutoviy.

Ethics declarations

Conflict of Interest:

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Berns, C., Kondratiev, Y. & Kutoviy, O. Markov Jump Dynamics with Additive Intensities in Continuum: State Evolution and Mesoscopic Scaling. J Stat Phys 161, 876–901 (2015). https://doi.org/10.1007/s10955-015-1365-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-015-1365-z

Keywords

Search

Navigation