Skip to main content
SpringerLink
Log in

Scaling Limits of a Tagged Particle in the Exclusion Process with Variable Diffusion Coefficient

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in ℤ with variable diffusion coefficient. The scaling limits are obtained from a similar result for the current through −1/2 for a zero-range process with bond disorder. For the CLT, we prove convergence to a fractional Brownian motion of Hurst exponent 1/4.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambjörsson, T., Lizana, L., Silbey, R.: Dynamics of two hardcore interacting particles with different diffusion constants in one dimension. Preprint (2008). Available online at http://arxiv.org/pdf/0803.2485

  2. Arratia, R.: The motion of a tagged particle in the simple symmetric exclusion system on Z. Ann. Probab. 11(2), 362–373 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aslangul, C.: Single-file diffusion with random diffusion constants. J. Phys. A 33(5), 851–862 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Benois, O., Kipnis, C., Landim, C.: Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stoch. Process. Their Appl. 55(1), 65–89 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brzank, A., Schütz, G.M.: Phase transition in the two-component symmetric exclusion process with open boundaries. J. Stat. Mech. Theory Exp. 2007(08), P08028 (2007)

    Article  Google Scholar 

  6. De Masi, A., Ferrari, P.A.: Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process. J. Stat. Phys. 107(3–4), 677–683 (2002)

    Article  MATH  Google Scholar 

  7. Faggionato, A., Martinelli, F.: Hydrodynamic limit of a disordered lattice gas. Probab. Theory Relat. Fields 127(4), 535–608 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fritz, J.: Hydrodynamics in a symmetric random medium. Commun. Math. Phys. 125(1), 13–25 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Gonçalves, P., Jara, M.: Scaling limit of gradient systems in random environment. J. Stat. Phys. 131(4), 691–716 (2008). Available online at http://www.springerlink.com/content/y2l1v5jnl5654371/fulltext.pdf

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Harris, T.E.: Diffusion with “collisions” between particles. J. Appl. Probab. 2, 323–338 (1965)

    Article  Google Scholar 

  11. Jankowski, H.: Nonequilibrium density fluctuations for the zero range process with colour. Preprint (2006). Available online at http://arxiv.org/abs/math/0607505

  12. Jara, M.D., Landim, C.: Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 42(5), 567–577 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jara, M.D., Landim, C.: Quenched nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 44(2), 341–361 (2008). Available online at http://fr.arxiv.org/abs/math/0603653

    Article  Google Scholar 

  14. Jara, M.D., Landim, C., Sethuraman, S.: Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes. Probab. Theory Relat. Fields (2008, to appear). Available online at http://fr.arxiv.org/abs/math/0703226

  15. Kipnis, C.: Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14(2), 397–408 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104(1), 1–19 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Landim, C., Olla, S., Volchan, S.B.: Driven tracer particle in one-dimensional symmetric simple exclusion. Commun. Math. Phys. 192(2), 287–307 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Landim, C., Volchan, S.: Equilibrium fluctuations for a driven tracer particle dynamics. Stoch. Process. Their Appl. 85(1), 139–158 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Quastel, J.: Bulk diffusion in a system with site disorder. Ann. Probab. 34(5), 1990–2036 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rost, H., Vares, M.E.: Hydrodynamics of one dimensional nearest neighbor model. Contemp. Math. 41, 329–342 (1985)

    MathSciNet  Google Scholar 

  21. Valle, G.: Evolution of the interfaces in a two dimensional Potts model. Electron. J. Probab. 12, 354–386 (2007)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Matemática, Campus de Gualtar, 4710-057, Braga, Portugal

    Patrícia Gonçalves

  2. FYMA, Université Catholique de Louvain, 2, Chemin du Cyclotron, 1348, Louvain-la-Neuve, Belgium

    Milton Jara

Authors
  1. Patrícia Gonçalves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milton Jara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Milton Jara.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gonçalves, P., Jara, M. Scaling Limits of a Tagged Particle in the Exclusion Process with Variable Diffusion Coefficient. J Stat Phys 132, 1135–1143 (2008). https://doi.org/10.1007/s10955-008-9595-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9595-y

Keywords

Search

Navigation