Journal of Statistical Physics

, Volume 38, Issue 3–4, pp 615–645 | Cite as

On the asymptotic behaviour of Spitzer's model for evolution of one-dimensional point systems

  • J. Fritz


A nearest-neighbor gradient dynamics of one-dimensional infinite particle systems is considered; the model admits a two-parameter family of stationary configurations. Some domains of attraction of stationary configurations are described, and the continuum (hydrodynamical) limit of the system is investigated. It is shown that the mean density of points satisfies a nonlinear diffusion equation in the hydrodynamical limit.

Key words

Gradient dynamics local stability lattice approximation hydrodynamical limit 


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  1. 1.
    F. Spitzer, Random processes defined through the interaction of an infinite particle system, inLecture Notes in Mathematics, Vol. 89 (Springer Verlag, Berlin, 1969), pp. 201–223.Google Scholar
  2. 2.
    A. Garcia and H. Kesten, Unpublished; see the footnote on p. 221 in Ref. 1.Google Scholar
  3. 3.
    R. L. Dobrushnin and J. Fritz, Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction,Commun. Math. Phys. 55:275–292 (1977).Google Scholar
  4. 4.
    R. Lang, On the asymptotic behaviour of infinite gradient systems,Commun. Math. Phys. 65:129–149 (1979).Google Scholar
  5. 5.
    K. H. Fichtner and W. Freudenberg, Asymptotic behaviour of time evolutions of infinite particle systems,Z. Wahrsch. Verw. Gebiete 54:141–159 (1980).Google Scholar
  6. 6.
    J. Fritz, Local stability and hydrodynamical limit of Spitzer's one-dimensional lattice model,Commun. Math. Phys. 86:363–373 (1982).Google Scholar
  7. 7.
    C. Boldrighini, R. L. Dobrushnin, and Yu. M. Suhov, One-dimensional hard-rod caricature of hydrodynamics,J. Stat. Phys. 31:577 (1983).Google Scholar
  8. 8.
    H. Rost, Hydrodynamik gekoppelter Diffusionen: Fluktuationen im Gleichgewicht, preprint, Heidelberg University, 1982.Google Scholar
  9. 9.
    C. Boldrighini, A. Pellegrinotti, and L. Triolo, Convergence to stationary states for infinite harmonic systems,J. Stat. Phys. 30:123–155 (1983).Google Scholar
  10. 10.
    J. L. Daleckii and M. G. Krein,Stability of solutions to differential equations in Banach spaces (Nauka, Moscow, 1970) (in Russian).Google Scholar
  11. 11.
    J. L. Lions, Quelques methodes de résolution des problèmes aux limites non linéaries (Dunod, Paris, 1969).Google Scholar
  12. 12.
    T. Kato,Perturbation theory for linear operators (Springer Verlag, Berlin, 1980).Google Scholar
  13. 13.
    I. S. Gradshtein and I. M. Ryzhik,Tables of Integrals, Series and Products (Academic Press, New York, 1980).Google Scholar
  14. 14.
    A. De Masi, N. Ianiro, A. Pellegrinotti, and E. Presutti, A survey of the hydrodynamical behaviour of many particle systems, inNonequilibrium Processes, Vol. 11, J. L. Lebowitz, E. W. Montroll, eds.,Studies in Statistical Mechanics (North-Holland, Amsterdam, 1984).Google Scholar
  15. 15.
    E. Scacciatelli, private communication.Google Scholar
  16. 16.
    E. Presutti and E. Scacciatelli, Time evolution of a one-dimensional point system: A note on Fritz's paper,J. Stat. Phys. 38:647–653.Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • J. Fritz
    • 1
  1. 1.Mathematical Institute, H.A.S.BudapestHungary

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