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Boundary Driven Zero-Range Processes in Random Media

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Abstract

The stationary states of boundary driven zero-range processes in random media with quenched disorder are examined, and the motion of a tagged particle is analyzed. For symmetric transition rates, also known as the random barrier model, the stationary state is found to be trivial in absence of boundary drive. Out of equilibrium, two further cases are distinguished according to the tail of the disorder distribution. For strong disorder, the fugacity profiles are found to be governed by the paths of normalized α-stable subordinators. The expectations of integrated functions of the tagged particle position are calculated for three types of routes.

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References

  1. Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)

    Article  MATH  Google Scholar 

  2. Andjel, E.D.: Invariant measures for the zero-range process. Ann. Prob. 10, 525–547 (1982)

    MATH  Google Scholar 

  3. Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Springer, Berlin (1999)

    MATH  Google Scholar 

  4. Bertini, L., de Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for non-equilibrium stationary states. J. Stat. Phys. 107, 635–675 (2002)

    Article  MATH  Google Scholar 

  5. Evans, M.R., Hanney, T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A: Math. Gen. 38, R195–R240 (2005)

    Article  MATH  ADS  Google Scholar 

  6. Levine, E., Mukamel, D., Schütz, G.M.: Zero-Range process with open boundaries. J. Stat. Phys. 120, 759–778 (2005)

    Article  MATH  Google Scholar 

  7. Evans, M.R.: Bose–Einstein condensation in disordered exclusion models and relation to traffic flow. Europhys. Lett. 36, 13–18 (1996)

    Article  ADS  Google Scholar 

  8. Krug, J., Ferrari, P.A.: Phase transitions in driven diffusive systems with random rates. J. Phys. A: Math. Gen. 29, L465–L471 (1996)

    Article  ADS  Google Scholar 

  9. Havlin, S., Ben-Avraham, D.: Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987)

    Article  ADS  Google Scholar 

  10. Hughes, B.D.: Random Walks and Random Environments, vol. 2: Random Environments. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  11. Zeitouni, O.: Random walks in random environments. J. Phys. A: Math. Gen. 39, R433–R464 (2006)

    Article  MATH  ADS  Google Scholar 

  12. Isichenko, M.B.: Percolation, statistical topography, and transport in random media. Rev. Mod. Phys. 64, 961–1043 (1992)

    Article  ADS  Google Scholar 

  13. Kawazu, K., Kesten, H.: On birth and death processes in symmetric random environment. J. Stat. Phys. 37, 561–576 (1984)

    Article  MATH  Google Scholar 

  14. Ferrari, P.: Limit theorems for tagged particles. Markov Process. Relat. Fields 2, 17–40 (1996)

    MATH  Google Scholar 

  15. Saada, E.: Processus de zéro-rangé avec particule marquée. Ann. Inst. H. Poincaré Probab. Stat. 26, 5–17 (1990)

    MATH  Google Scholar 

  16. Siri, P.: Asymptotic behaviour of a tagged particle in an inhomogeneous zero-range process. Stoch. Process. Appl. 77, 139–154 (1998)

    Article  MATH  Google Scholar 

  17. Sethuraman, S.: On diffusivity of a tagged particle in asymmetric zero-range dynamics. Ann. Inst. H. Poincaré Probab. Stat. 43, 215–232 (2007)

    Article  MATH  Google Scholar 

  18. Derrida, B., Lebowitz, J.L., Speer, E.R.: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys. 107, 599–634 (2002)

    Article  MATH  Google Scholar 

  19. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  20. Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Wolters–Noordhoff, Groningen (1971)

    MATH  Google Scholar 

  21. Billingsley, P.: Convergence of probability measures. Wiley, New York (1968)

    MATH  Google Scholar 

  22. Gihman, I.I., Skorohod, A.V.: The theory of stochastic processes, I. Springer, Berlin (1974)

    MATH  Google Scholar 

  23. Skorohod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl. 1, 261–290 (1956)

    Article  Google Scholar 

  24. Skorohod, A.V.: Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2, 138–171 (1957)

    Article  Google Scholar 

  25. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  26. Pitman, J.: Combinatorial Stochastic Processes: Ecole d’Eté de Probabilités de Saint-Flour XXXII, 2002. Springer, Berlin (2006)

    MATH  Google Scholar 

  27. Regazzini, E., Lijoi, A., Prünster, I.: Distributional results for means of normalized random measures with independent increments. Ann. Stat. 31, 560–585 (2003)

    Article  MATH  Google Scholar 

  28. Gurland, J.: Inversion formulae for the distribution of ratios. Ann. Math. Stat. 19, 228–237 (1948)

    Google Scholar 

  29. Perman, M., Pitman, J., Yor, M.: Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92, 21–39 (1992)

    Article  MATH  Google Scholar 

  30. Pitman, J.: Some developments of the Blackwell–MacQueen urn scheme. In: Ferguson, T.S., Shapley, L.S., MacQueen, J.B. (eds.) Statistics, Probability and Game Theory; Papers in Honor of David Blackwell. Institute of Mathematical Statistics, Hayward (1996)

    Google Scholar 

  31. Pitman, J., Yor, M.: The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900 (1997)

    Article  MATH  Google Scholar 

  32. Pitman, J.: Poisson–Kingman partitions. Technical Report No. 625, Department of Statistics, University of California, available via http://www.stat.berkeley.edu/tech-reports/

  33. Talagrand, M.: Spin Glasses: a Challenge for Mathematicians. Springer, Berlin (2003)

    Google Scholar 

  34. Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B 24, 2613–2626 (1981)

    Article  ADS  Google Scholar 

  35. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 2: Itô Calculus, 2nd edn. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  36. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

  37. Kook, K., Serfozo, R.: Travel and sojourn times in stochastic networks. Ann. Appl. Probab. 3, 228–252 (1993)

    MATH  Google Scholar 

  38. Baccelli, F., Brémaud, P.: Elements of Queuing Theory, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  39. Whit, W.: A review of L=λ W and extensions. Queuing Syst. 9, 235–268 (1990)

    Article  Google Scholar 

  40. Serfozo, R.: Introduction to Stochastic Networks. Springer, New York (1999)

    MATH  Google Scholar 

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  1. Department of Physics, University of Jyväskylä, P.O. Box 35, 40014, Jyväskylä, Finland

    Otto Pulkkinen

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  1. Otto Pulkkinen
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Correspondence to Otto Pulkkinen.

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Pulkkinen, O. Boundary Driven Zero-Range Processes in Random Media. J Stat Phys 128, 1289–1305 (2007). https://doi.org/10.1007/s10955-007-9361-6

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  • DOI: https://doi.org/10.1007/s10955-007-9361-6

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