Communications in Mathematical Physics

, Volume 86, Issue 3, pp 363–373 | Cite as

Local stability and hydrodynamical limit of Spitzer's one dimensional lattice model

  • J. Fritz


An infinite system of ordinary differential equations is considered, the right hand side is just the negative gradient of potential energy of a one-dimensional system of unbounded spins interacting by a symmetric and convex pair potential. Constant configurations are stationary points and the mean spin is conserved. It is shown that each of these stationary points has its own domain of attraction, the initial distribution need not be translation invariant. As a consequence we obtain that the mean spin satisfies the heat equation in the hydrodynamical limit.


Neural Network Potential Energy Complex System Ordinary Differential Equation Nonlinear Dynamics 
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  1. 1.
    Spitzer, F.: Random processes defined through the interaction of an infinite particle system. In: Lecture Notes in Mathematics, Vol. 89, 201–223. Berlin, Heidelberg, New York: Springer 1969Google Scholar
  2. 2.
    Garcia, A., Kesten, H.: Unpublished, see footnote on p. 221 in [1]Google Scholar
  3. 3.
    Dobrushin, R.L., Fritz, J.: 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.
    Lang, R.: On the asymptotic behaviour of infinite gradient systems. Commun. Math. Phys.65, 129–149 (1979)Google Scholar
  5. 5.
    Fichtner, K.H., Freudenberg, W.: Asymptotic behaviour of time evolutions of infinite particle systems. Z. Wahrsch. Verw. Gebiete54, 141–159 (1980)Google Scholar
  6. 6.
    Boldrighini, C., Dobrushin, R.L., Souhov, Ju.M.: The asymptotics of some degenerate models for the evolution of infinite particle systems. Itogi Nauki Tekh.14, 148–255 Moscow VINITI (1979) (in Russian)Google Scholar
  7. 7.
    Boldrighini, C., Dobrushin, R.L., Souhov, Ju.M.: Hydrodynamical limit for a degenerate model of classical statistical mechanics. (in Russian) (to appear)Google Scholar
  8. 8.
    Fritz, J.: Infinite lattice systems of interacting diffusion processes, Existence and regularity properties. Z. Wahrsch. Verw. Gebiete59, 291–309 (1980)Google Scholar
  9. 9.
    Elbert, A.: Private communicationGoogle Scholar
  10. 10.
    Gradshtein, I.S., Ryzhik, I.M.: Table of integrals, series and products. New York, London, Toronto, Sydney, San Francisco: Academic Press 1980Google Scholar
  11. 11.
    Daleckii, J.L., Krein, M.G.: Stability of solutions to differential equations in Banach spaces. Moscow: Nauka 1970 (in Russian)Google Scholar
  12. 12.
    Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  13. 13.
    Feller, W.: An introduction to probability theory and its applications. Vol. II. New York, London, Sydney: Wiley 1966Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • J. Fritz
    • 1
  1. 1.Mathematical InstituteBudapestHungary

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