Journal of Statistical Physics

, Volume 152, Issue 1, pp 112–135 | Cite as

Cusp Singularities in Boundary-Driven Diffusive Systems

Article

Abstract

Boundary driven diffusive systems describe a broad range of transport phenomena. We study large deviations of the density profile in these systems, using numerical and analytical methods. We find that the large deviation may be non-differentiable, a phenomenon that is unique to non-equilibrium systems, and discuss the types of models which display such singularities. The structure of these singularities is found to generically be a cusp, which can be described by a Landau free energy or, equivalently, by catastrophe theory. Connections with analogous results in systems with finite-dimensional phase spaces are drawn.

Keywords

Boundary-driven diffusive systems Rare events Large deviations Phase-transitions Catastrophe theory 

References

  1. 1.
    Arnold, V.I.: Russ. Math. Surv. 30, 1–75 (1975) CrossRefGoogle Scholar
  2. 2.
    Berry, M.V., Upstill, C.: Prog. Opt. 18, 257–346 (1980) CrossRefGoogle Scholar
  3. 3.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Phys. Rev. Lett. 87, 040601 (2001) MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: J. Stat. Phys. 107 (2002) Google Scholar
  5. 5.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Phys. Rev. Lett. 94, 030601 (2005) ADSCrossRefGoogle Scholar
  6. 6.
    Bertini, L., Gabrielli, D., Lebowitz, J.: J. Stat. Phys. 121, 843 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: J. Stat. Mech. L11001 (2010) Google Scholar
  8. 8.
    Bodineau, T., Derrida, B.: Phys. Rev. E 72, 066110 (2005) MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Bunin, G., Kafri, Y., Podolsky, D.: EPL 99, 20002 (2012) ADSCrossRefGoogle Scholar
  10. 10.
    Bunin, G., Kafri, Y., Podolsky, D.: J. Stat. Mech. L10001 (2012) Google Scholar
  11. 11.
    Derrida, B.: J. Stat. Mech. P07023 (2007) Google Scholar
  12. 12.
    Derrida, B., Lebowitz, J.L., Speer, E.R.: J. Stat. Phys. 107 (2002) Google Scholar
  13. 13.
    Dieterich, W., Fulde, P., Peschel, I.: Adv. Phys. 29 (1980) Google Scholar
  14. 14.
    Dorfman, J.R., Kirkpatrick, T.R., Sengers, J.V.: Annu. Rev. Phys. Chem. 45, 213–239 (1994) ADSCrossRefGoogle Scholar
  15. 15.
    Dykman, M.I., Millonas, M.M., Smelyanskiy, V.N.: Phys. Lett. A 195, 53 (1994) MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Dykman, M.I., Luchinsky, D.G., McClintock, P.V.E., Smelyanskiy, V.N.: Phys. Rev. Lett. 77, 26 (1996) CrossRefGoogle Scholar
  17. 17.
    Freidlinand, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984) CrossRefGoogle Scholar
  18. 18.
    Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer, Berlin (1994) Google Scholar
  19. 19.
    Gilmore, R.: Catastrophe theory. In: Digital Encyclopedia of Applied Physics. Wiley, New York (2003) Google Scholar
  20. 20.
    Graham, R., Tél, T.: Phys. Rev. Lett. 52(2), 9–12 (1984) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Graham, R., Tél, T.: J. Stat. Phys. 35(5/6) (1984) Google Scholar
  22. 22.
    Graham, R., Tél, T.: Phys. Rev. A 31(2) (1985) Google Scholar
  23. 23.
    Graham, R., Tél, T.: Phys. Rev. A 33(2) (1986) Google Scholar
  24. 24.
    Hager, J.S., Krug, J., Popkov, V., Schütz, G.M.: Phys. Rev. E 63, 056110 (2001) ADSCrossRefGoogle Scholar
  25. 25.
    Hurtado, P.I., Garrido, P.L.: Phys. Rev. Lett. 107, 180601 (2011) ADSCrossRefGoogle Scholar
  26. 26.
    Jordan, A.N., Sukhorukov, E.V., Pilgram, S.: J. Math. Phys. 45, 4386–4417 (2004) MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Katz, S., Lebowitz, J.L., Spohn, H.: J. Stat. Phys. 34(3/4) (1984) Google Scholar
  28. 28.
    Kipnis, C., Marchioro, C., Presutti, E.: J. Stat. Phys. 27, 65 (1982) MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Le Doussal, P., Vinokur, V.M.: Physica C 254 (1995) Google Scholar
  30. 30.
    Luchinsky, D.G., McClintock, P.V.E., Dykman, M.I.: Rep. Prog. Phys. 61(8), 889–997 (1998) ADSCrossRefGoogle Scholar
  31. 31.
    Maier, R.S., Stein, D.L.: Phys. Rev. E 48(2), 931–938 (1993) ADSCrossRefGoogle Scholar
  32. 32.
    Maier, R.S., Stein, D.L.: Phys. Rev. Lett. 85, 1358 (2000) ADSCrossRefGoogle Scholar
  33. 33.
    Merhav, N., Kafri, Y.: J. Stat. Mech. P02011 (2010) Google Scholar
  34. 34.
    Moss, F., McClintock, P.V.E. (eds.): Noise in Nonlinear Dynamical Systems. Cambridge University Press, Cambridge (1989) Google Scholar
  35. 35.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes. The Art of Scientific Computing, 3nd edn. (2007) Google Scholar
  36. 36.
    Scheidl, S.: Z. Phys. B 97, 345–352 (1995) ADSCrossRefGoogle Scholar
  37. 37.
    Schulman, L.S., Revzen, M.: Collect. Phenom. 1, 43–49 (1972) MathSciNetGoogle Scholar
  38. 38.
    Spohn, H.: J. Phys. A: Math. Gen. 16, 4275 (1983) MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991) MATHCrossRefGoogle Scholar
  40. 40.
    Tailleur, J., Kurchan, J., Lecomte, V.: J. Phys. A: Math. Theor. 41, 505001 (2008) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Technion—Israel Institute of TechnologyHaifaIsrael

Personalised recommendations