Skip to main content
SpringerLink
Log in

Phase Transitions in Stochastic Models of Flow

  • Conference paper
Traffic and Granular Flow’05

Summary

In this talk I will review some very simple models of nonequilibrium systems known as the ‘Asymmetric Exclusion Process’ and the ‘Zero-Range Process’. These involve particles hopping stochastically on a lattice and thus form stochastic models of flow. Systems driven out of equilibrium can often exhibit behaviour not seen in systems in thermal equilibrium - for example phase transitions in one-dimensional systems. I shall show how examples of such transitions may be interpreted as jamming transitions in the context of traffic flow. More generally I shall discuss other instances of the condensation transition which is the phenomenon of a finite fraction of the driven conserved quantity condensing into a small spatial region. Criteria for the occurrence of condensation may be formulated and the detailed properties of the condensate such as its fluctuations have recently been elucidated.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Chowdhury, A. Schadschneider, and K. Nishinari, these proceedings

    Google Scholar 

  2. S. Katz, J.L. Lebowitz and H. Spohn, Phys. Rev. B 28, 1655 (1983); J. Stat. Phys 34, 497 (1984)

    Article  Google Scholar 

  3. B. Schmittmann and R.K.P. Zia, Statistical Mechanics of Driven Diffusive Systems vol. 17 of Domb and Lebowitz series, Academic Press, U.K. (1995)

    Google Scholar 

  4. S.A. Janowsky and J.L. Lebowitz, Physical Review A 45, 618 (1992)

    Article  Google Scholar 

  5. M.E.J. Newman and G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, U.K., (1999)

    MATH  Google Scholar 

  6. A.B. Bortz, M.H. Kalos and J.L. Lebowitz, J. Comp. Phys 17, 10 (1975)

    Article  Google Scholar 

  7. D.T. Gillespie, J. Phys. Chem. 81, 2340 (1977)

    Article  Google Scholar 

  8. F. Spitzer, Advances in Math. 5 246 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  9. M.R. Evans and T. Hanney, J. Phys. A: Math. Gen. 38, R195–R239 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. M.R. Evans, Pramana 64 859 (2005)

    Google Scholar 

  11. M.R. Evans and R.A. Blythe, Physica A 313 110 (2002); R. A. Blythe, PhD Thesis University of Edinburgh, U.K. (2001)

    Article  MATH  Google Scholar 

  12. J.M.J van Leeuwen and A. Kooiman, Physica A 184, 79 (1992)

    Google Scholar 

  13. M. Barma and R. Ramaswamy in Non-linearity and Breakdown in Soft Condensed Matter, edited by K.K. Bardhan, B. K. Chakrabarti and A. Hansen (Springer, Berlin 1993) p.309

    Google Scholar 

  14. D. Biron and E. Moses, Biophys. J. 86, 3284 (2004)

    Article  Google Scholar 

  15. Y. Kafri, E. Levine, D. Mukamel, G.M. Schütz and J. Török, Phys. Rev. Lett. 89, 035702 (2002)

    Article  Google Scholar 

  16. M.R. Evans, E. Levine, P.K. Mohanty and D. Mukamel, Euro. Phys. J. B 41, 223 (2004)

    Article  Google Scholar 

  17. O.J. O’Loan, M.R. Evans and M.E. Cates, Phys. Rev. E. 58, 1404 (1998)

    Article  Google Scholar 

  18. P. Bialas, Z. Burda and D. Johnston, Nucl. Phys. B 493, 505 (1997)

    Article  MATH  Google Scholar 

  19. S.N. Majumdar, S. Krishnamurthy, M. Barma, Phys. Rev. Lett. 81, 3691 (1998); Phys. Rev. E 61, 6337 (2000)

    Article  Google Scholar 

  20. M. R. Evans, Europhys. Lett. 36, 13 (1996)

    Article  Google Scholar 

  21. J. Krug and P.A. Ferrari, J. Phys. A: Math. Gen. 29, L465 (1996)

    Article  Google Scholar 

  22. J. Krug and J. Garcia, J. Stat. Phys. 99, 31 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. R. Rajesh and S. N. Majumdar, J. Stat. Phys. 99, 943 (2000); Phys. Rev. E 64, 036103 (2001).

    Article  MATH  Google Scholar 

  24. F. Zielen and A. Schadschneider, J. Stat. Phys 106, 173 (2002); J. Phys. A: Math. Gen 36, 3709 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. M.R. Evans, S.N. Majumdar and R.K.P. Zia, J. Phys. A: Math. Gen. 37, L275 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. R.K.P. Zia, M.R. Evans and S.N. Majumdar, J. Stat. Mech.: Theor. Exp. (2004) L10001

    Google Scholar 

  27. S.N. Majumdar, M.R. Evans and R.K.P. Zia, Phys. Rev. Lett 94, 180601 (2005)

    Article  Google Scholar 

  28. M.R. Evans, S.N. Majumdar and R.K.P. Zia, J. Stat. Phys. 123, 357 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. C. Godreche and J.M. Luck, J. Phys. A 38, 7215–7237 (2005)

    Article  MathSciNet  Google Scholar 

  30. M. R. Evans and T. Hanney, J. Phys. A: Math. Gen. 36, L441 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Grosskinsky and H. Spohn, Bull. Braz. Math. Soc. 34, 489–507 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. T. Hanney and M. R. Evans, Phys. Rev. E 69, 016107 (2004)

    Article  Google Scholar 

  33. K. Jain and M. Barma, Phys. Rev. Lett. 91, 135701 (2003)

    Article  Google Scholar 

  34. E.K.O. Hellén and J. Krug, Phys. Rev. E 66, 011304 (2002)

    Article  Google Scholar 

  35. S. Grosskinsky, G.M. Schütz, H. Spohn, J. Stat. Phys. 113, 389 (2003)

    Article  MATH  Google Scholar 

  36. C. Godrèche, J. Phys. A: Math. Gen. 36, 6313–6328 (2003)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SUPA, School of Physics, The University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, Scotland

    Martin R. Evans

Authors
  1. Martin R. Evans
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, 50937, Köln, Germany

    Andreas Schadschneider

  2. Campus Virchow Klinikum Centrum für Unfall- und Wiederherstellungschirurgie, Charité, Augustenburger Platz 1, 13353, Berlin, Germany

    Thorsten Pöschel

  3. Institut für Verkehrsforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Rutherfordstr. 2, 12489, Berlin, Germany

    Reinhart Kühne

  4. Physik von Transport und Verkehr, Universität Duisburg-Essen, Lotharstr. 1, 47048, Duisburg, Germany

    Michael Schreckenberg

  5. Theoretische Physik, Universität Duisburg-Essen, Lotharstr. 1, 47048, Duisburg, Germany

    Dietrich E. Wolf

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Evans, M.R. (2007). Phase Transitions in Stochastic Models of Flow. In: Schadschneider, A., Pöschel, T., Kühne, R., Schreckenberg, M., Wolf, D.E. (eds) Traffic and Granular Flow’05. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-47641-2_41

Download citation

Publish with us

Policies and ethics

Search

Navigation