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Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System

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Abstract

We study spontaneous symmetry breaking in a one-dimensional driven two-species stochastic cellular automaton with parallel sublattice update and open boundaries. The dynamics are symmetric with respect to interchange of particles. Starting from an empty initial lattice, the system enters a symmetry broken state after some time T 1 through an amplification loop of initial fluctuations. It remains in the symmetry broken state for a time T 2 through a traffic jam effect. Applying a simple martingale argument, we obtain rigorous asymptotic estimates for the expected times 〈 T 1〉 ∝ Lln L and ln 〈 T 2〉 ∝ L, where L is the system size. The actual value of T 1 depends strongly on the initial fluctuation in the amplification loop. Numerical simulations suggest that T 2 is exponentially distributed with a mean that grows exponentially in system size. For the phase transition line we argue and confirm by simulations that the flipping time between sign changes of the difference of particle numbers approaches an algebraic distribution as the system size tends to infinity.

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References

  1. M.R. Evans, D.P. Foster, C. Godrèche and D. Mukamel, Spontaneous symmetry breaking in a one dimensional driven diffusive system, Phys. Rev. Lett. 74(2):208–211 (1995).

    Article  ADS  Google Scholar 

  2. M.R. Evans, D.P. Foster, C. Godrèche and D. Mukamel, Asymmetric exclusion model with two species: Spontaneous symmetry breaking, J. Stat. Phys. 80:69–102 (1995).

    Article  MATH  Google Scholar 

  3. C. Godrèche, J.M. Luck, M.R. Evans, D. Mukamel, S. Sandow and E.R. Speer, Spontaneous symmetry breaking: Exact results for a biased random walk model of an exclusion process, J. Phys. A 28(21):6039–6071 (1995).

    Article  MATH  ADS  Google Scholar 

  4. A. Rákos, M. Paessens and G.M. Schütz, Hysteresis in one-dimensional reaction-diffusion systems, Phys. Rev. Lett. 91:238302 (2003).

    Google Scholar 

  5. E. Levine and R.D. Willmann, Spontaneous symmetry breaking in a non-conserving two-species driven model, J. Phys. A 37:3333–3352 (2004).

    Article  MATH  ADS  Google Scholar 

  6. P.F. Arndt, T. Heinzel and V. Rittenberg, First-order phase transitions in one-dimensional steady states, J. Stat. Phys. 90:783–815 (1998).

    MATH  Google Scholar 

  7. M. Clincy, M.R. Evans and D. Mukamel, Symmetry breaking through a sequence of transitions in a driven diffusive system, J. Phys. A 34:9923–9937 (2001).

    Article  MATH  ADS  Google Scholar 

  8. D.W. Erickson, G. Pruessner, B. Schmittmann and R.K.P. Zia, Spurious phase in a model for traffic on a bridge, J. Phys. A 38(41): L659-L665 (2005).

    Google Scholar 

  9. E. Pronina and A.B. Kolomeisky, Spontaneous symmetry breaking in two-channel asymmetric exclusion processes with narrow entrances, J. Phys. A: Math. Theo. 40:2275–2286 (2007).

    Article  MATH  ADS  Google Scholar 

  10. V. Popkov and G.M. Schütz, Why spontaneous symmetry breaking disappears in a bridge system with PDE-friendly boundaries, J. Stat. Mech.: Theor. Exp., P12004 (2004).

  11. V. Popkov, Infinite reflections of shock fronts in driven diffusive systems with two species, J. Phys. A 37(5):1545–1557 (2004).

    Article  MATH  ADS  Google Scholar 

  12. V. Popkov and G.M. Schütz, Shocks and excitation dynamics in a driven diffusive two-channel system, J. Stat. Phys. 112:523–540 (2003).

    Article  MATH  Google Scholar 

  13. B. Tóth and B. Valkó, Onsager relations and Eulerian hydrodynamics for systems with several conservation laws J. Stat. Phys. 112(3-4): 497-521 (2003).

    Article  MATH  Google Scholar 

  14. M. Clincy, B. Derrida, M.R. Evans, Phase transition in the ABC model, Phys. Rev. E 67(6):066115 (2003).

    Google Scholar 

  15. V. Popkov, M. Salerno, Hydrodynamic limit of multichain driven diffusive models, Phys. Rev. E 69(4):046103 (2004).

    Google Scholar 

  16. J. Fritz and B. Tóth, Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas, Comm. Math. Phys. 249(1):1–27 (2004).

    Article  MATH  ADS  Google Scholar 

  17. B. Tóth, B. Valkó, Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit, Comm. Math. Phys. 256(1):111–157 (2005).

    Article  MATH  ADS  Google Scholar 

  18. G.M. Schütz, Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles J. Phys. A 36:R339-R379 (2003).

    Google Scholar 

  19. R.D. Willmann, G.M. Schütz, S. Groß kinsky, Dynamical origin of spontaneous symmetry breaking in a field-driven nonequilibrium system, Europhys. Lett. 71(4):542–548 (2005).

    Article  ADS  Google Scholar 

  20. Schütz G., Time-dependent correlation functions in a one-dimensional asymmetric exclusion process, Phys. Rev. E 47(6):4265–4277 (1993).

    Article  ADS  Google Scholar 

  21. A. Parmeggiani, T. Franosch and E. Frey, Phase coexistence in driven one-dimensional transport, Phys. Rev. Lett. 90(8):086601 (2003).

    Google Scholar 

  22. V. Popkov, A. Rákos, R.D. Willmann, A.B. Kolomeisky and G.M. Schütz, Localization of shocks in driven diffusive systems without particle number conservation, Phys. Rev. E 67(6):066117 (2003).

    Google Scholar 

  23. M.R. Evans, R. Juhasz and L. Santen, Shock formation in an exclusion process with creation and annihilation, Phys. Rev. E 68:026117 (2003).

    Google Scholar 

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Authors and Affiliations

  1. Statistical Laboratory, University of Cambridge, Cambridge, CB3 0WB, UK

    Stefan Großkinsky

  2. Institut für Festkörperforschung, Forschungszentrum Jülich, 52425, Jülich, Germany

    Gunter M. Schütz & Richard D. Willmann

Authors
  1. Stefan Großkinsky
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  2. Gunter M. Schütz
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  3. Richard D. Willmann
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Correspondence to Stefan Großkinsky.

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Großkinsky, S., Schütz, G.M. & Willmann, R.D. Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System. J Stat Phys 128, 587–606 (2007). https://doi.org/10.1007/s10955-007-9341-x

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  • DOI: https://doi.org/10.1007/s10955-007-9341-x

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