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Conservation of Local Equilibrium for Attractive Systems

  • Claude Kipnis
  • Claudio Landim
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 320)

Abstract

In Chapter 1 we introduced the concept of local equilibrium and proved the conservation of local equilibrium for a superposition of independent random walks. Then, from Chapter 4 to Chapter 8, we proved a weaker version of local equilibrium for a large class of interacting particle systems: we showed that the empirical measure π t N converges in probability to an absolutely continuous measure whose density is the solution of some partial differential equation. The purpose of this chapter is to to show that in the case of attractive processes, the conservation of local equilibrium may be deduced from a law of large numbers for local fields, i.e., from the convergence in probability of the averages
$${N^{ - d}}\sum\limits_x {H(x/N){\tau _x}} \Psi ({\eta _t})to\int_{{T^d}} {H(u)\tilde \Psi } (\rho (t,u))du$$
for every t ≥ 0, every continuous function H and every bounded cylinder function ψ. Here ρ(t, u) is the solution of the hydrodynamic equation. This statement is slightly stronger than the convergence of the empirical measures since it involves all local fields.

Keywords

Invariant Measure Local Equilibrium Product Measure Entropy Solution Class Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Comments and References

  1. Rezakhanlou, F. (1991): Hydrodynamic limit for attractive particle systems on Zd. Commun. Math. Phys. 140, 417–448MathSciNetMATHCrossRefGoogle Scholar
  2. Andjel, E.D. (1982): Invariant measures for the zero-range process. Ann. Probab. 10, 525–547MathSciNetMATHCrossRefGoogle Scholar
  3. Liggett, T.M. (1985): Interacting Particle Systems, Springer-Verlag, New YorkMATHCrossRefGoogle Scholar
  4. Landim, C. (1991b): Hydrodynamical limit for asymmetric attractive particle systems on Z d. Ann. Inst. H. Poincaré, Probabilités 27, 559–581MathSciNetMATHGoogle Scholar
  5. Landim, C. (1993): Conservation of local equilibrium for attractive particle systems on Z d. Ann. Probab. 21, 1782–1808MathSciNetMATHCrossRefGoogle Scholar
  6. Ferrari, P.A. (1992): Shock fluctuations in asymmetric simple exclusion. Probab. Th. Rel. Fields 91, 81–101MATHCrossRefGoogle Scholar
  7. Ferrari, P.A. (1986): The simple exclusion process as seen from a tagged particle. Ann. Probab. 14, 1277–1290MathSciNetMATHCrossRefGoogle Scholar
  8. De Masi, A., Ferrari, P. A., Goldstein, S., Wick, W. D. (1985): Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. Contemp. Math. 41, 71–85CrossRefGoogle Scholar
  9. De Masi, A., Kipnis, C., Presutti, E., Saada, E. (1989): Microscopic structure at the shock in the asymmetric simple exclusion. Stochastics 27, 151–165MATHGoogle Scholar
  10. Benassi, A., Fouque, J.P., Saada, E., Vares, M.E. (1991): Asymmetric attractive particle systems on Z: hydrodynamical limit for monotone initial profiles. J. Stat. Phys. 63, 719–735MathSciNetCrossRefGoogle Scholar
  11. Alexander, F.J., Cheng, Z., Janowsky, S.A., Lebowitz, J.L. (1992): Shock fluctuations in the two—dimensional asymmetric simple exclusion process. J. Stat. Phys. 68, 761–785MathSciNetMATHCrossRefGoogle Scholar
  12. Ferrari, P.A., Fontes, L.R.G. (1994b): Shock fluctuations in the asymmetric simple exclusion process. Probab. Th. Rel. Fields 99, 305–319MathSciNetMATHCrossRefGoogle Scholar
  13. De Masi, A., Kipnis, C., Presutti, E., Saada, E. (1989): Microscopic structure at the shock in the asymmetric simple exclusion. Stochastics 27, 151–165MATHGoogle Scholar
  14. Caprino, S., De Masi, A., Presutti, E., Pulvirenti, M. (1990): A stochastic particle system modeling the Carleman equation: Addendum. J. Stat. Phys. 59, 535–537CrossRefGoogle Scholar
  15. Ferrari, P.A. (1992): Shock fluctuations in asymmetric simple exclusion. Probab. Th. Rel. Fields 91, 81–101MATHCrossRefGoogle Scholar
  16. Ferrari, P.A., Fontes, L.R.G. (1994b): Shock fluctuations in the asymmetric simple exclusion process. Probab. Th. Rel. Fields 99, 305–319MathSciNetMATHCrossRefGoogle Scholar
  17. Ferrari, P.A., Fontes, L.R.G. (1996): Poissonian approximation for the tagged particle in asymmetric simple exclusion. J. Appl. Prob. 33, 411–419MathSciNetMATHCrossRefGoogle Scholar
  18. Rezakhanlou, F. (1995): Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincaré, Analyse non Linéaire 12, 119–153MathSciNetMATHGoogle Scholar
  19. Kipnis, C., Landim, C., Olla, S. (1995): Macroscopic properties of a stationary non-equilibrium distribution for a non-gradient interacting particle system. Ann. Inst. H. Poincaré, Probabilités 31, 191–221MathSciNetMATHGoogle Scholar
  20. Kipnis, C., Léonard, C. (1995): Grandes Déviations pour un système hydrodynamique asymétrique de particules indépendantes. Ann. Inst. H. Poincaré, Probabilités 31, 223248Google Scholar
  21. Liggett, T.M. (1975): Ergodic theorems for the asymmetric simple exclusion process. Trans. Amer. Math. Soc. 213, 237–260MathSciNetMATHCrossRefGoogle Scholar
  22. Andjel, E.D. (1986): Convergence to a non extremal equilibrium measure in the exclusion process. Probab. Th. Rel. Fields 73, 127–134MathSciNetMATHCrossRefGoogle Scholar
  23. Andjel, E.D., Bramson, M., Liggett, T.M. (1988): Shocks in the asymmetric exclusion process. Probab. Th. Rel. Fields 78, 231–247MathSciNetMATHCrossRefGoogle Scholar
  24. Derrida, B., Domany, E., Mukamel, D. (1992): An exact solution of a one—dimensional asymmetric exclusion model with open boundaries. J. Stat. Phys. 69, 667–687MathSciNetMATHCrossRefGoogle Scholar
  25. Schütz, G., Domany, E. (1993): Phase transition in an exactly soluble one—dimensional exclusion model. J. Stat. Phys. 72, 277–296MATHCrossRefGoogle Scholar
  26. Derrida, B., Evans, M.T., Mallick, K. (1995): Exact diffusion constant for a one—dimensional asymmetric exclusion model with open boundaries. J. Stat. Phys. 79, 833–874MathSciNetMATHCrossRefGoogle Scholar
  27. Derrida, B., Janowsky, S.A., Lebowitz, J.L., Speer, E.R. (1993): Exact solutions of the totally asymmetric simple exclusion process: shock profiles. J. Stat. Phys. 73, 813–842MathSciNetMATHCrossRefGoogle Scholar
  28. Speer, E. (1994): The two species totally asymmetric simple exclusion process. In M. Fannes, C. Maes and A. Verbeure, editors, On Three Levels: Micro-, Meso-and Macro-Approaches in Physics. Volume 324 of Nato ASI series B, pages 91–102.Google Scholar
  29. Ferrari, P.A., Galves, A., Landim, C. (1994): Exponential waiting times for a big gap in a one-dimensional zero range process. Ann. Probab. 22, 284–288MathSciNetMATHCrossRefGoogle Scholar
  30. Foster, D.P., Godrèche, C. (1994): Finite—size effects for phase segregation in a two-dimensional asymmetric exclusion model with two species. J. Stat. Phys. 76, 1129–1151MATHCrossRefGoogle Scholar
  31. Fritz, J., Funaki, T., Lebowitz, J.L. (1994): Stationary states of random Hamiltonian systems. Probab. Th. Rel. Fields 99, 211–236MathSciNetMATHCrossRefGoogle Scholar
  32. Schütz, G. (1993): Generalized Bethe ansatz solution of a one—dimensional asymmetric exclusion process on a ring with blockage. J. Stat. Phys. 71, 471–505MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Claude Kipnis
    • 1
    • 2
  • Claudio Landim
    • 1
    • 2
  1. 1.Instituto de Matematica Pura e AplicadaRio de JaneiroBrasil
  2. 2.CNRS UPRES-A 6085Université de RouenMont Saint Aignan CedexFrance

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