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Tracer diffusion in lattice gases

Abstract

It has been proved that a tracer particle in a reversible lattice gas converges to Brownian motion. However, only in a few particular cases has a strictly positive self-diffusion coefficientD been established. Here we supply the missing piece and show thatD>0 in general. The exceptions are one-dimensional lattice gases with nearest neighbor jumps only, for whichD=0. The proof establishes a variational formula forD which could be used to obtain realistic bounds.

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References

  1. 1.

    J. Perrin,Atoms (Van Nostrand, New York, 1916).

    Google Scholar 

  2. 2.

    K. W. Kehr and K. Binder, Simulation of diffusion in lattice gases and related kinetic phenomena, inApplications of the Monte Carlo Method in Statistical Physics, K. Binder, ed. (Springer, Berlin, 1984), pp. 181–221.

    Chapter  Google Scholar 

  3. 3.

    C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion,Commun. Math. Phys. 106:1–19 (1986).

    MathSciNet  ADS  Article  Google Scholar 

  4. 4.

    A. DeMasi, P. A. Ferrari, S. Goldstein, and W. D. Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments,J. Stat. Phys. 55:787–855 (1989).

    MathSciNet  ADS  Article  Google Scholar 

  5. 5.

    R. Arratia, The motion of a tagged particle in the simple symmetric exclusion process on ℤ,Ann. Prob. 11:362–373 (1983).

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    T. M. Liggett,Interacting Particle Systems (Springer, Berlin, 1985).

    MATH  Book  Google Scholar 

  7. 7.

    H. O. Georgii,Canonical Gibbs Measures (Springer, Berlin, 1979).

    MATH  Google Scholar 

  8. 8.

    H. van Beijeren, Mode-coupling theory for purely diffusive systems,J. Stat. Phys. 35:399–112 (1984).

    ADS  MATH  Article  Google Scholar 

  9. 9.

    H. van Beijeren, Transport properties of stochastic Lorentz models,Rev. Mod. Phys. 54:195–234 (1982).

    ADS  Article  Google Scholar 

  10. 10.

    H. van Beijeren and H. Spohn, Transport properties of the one-dimensional stochastic Lorentz model. I velocity autocorrelation,J. Stat. Phys. 31:231–254 (1983).

    ADS  Article  Google Scholar 

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Spohn, H. Tracer diffusion in lattice gases. J Stat Phys 59, 1227–1239 (1990). https://doi.org/10.1007/BF01334748

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Key words

  • Stochastic lattice gases
  • self-diffusion
  • variational formula