Journal of Statistical Physics

, Volume 145, Issue 6, pp 1499–1512

Current Symmetries for Particle Systems with Several Conservation Laws



We consider stochastic interacting particle systems with more than one conservation law in a regime far from equilibrium. Using time reversal we derive symmetry relations for the stationary currents of the conserved quantities that are reminiscent of Onsager’s reciprocity relations. These relations are valid for a very large class of particles with only some mild assumption on the decay of stationary relations and imply that the coarse-grained macroscopic dynamics is governed by a system of hyperbolic conservation laws. An explicit expression for the conserved Lax entropy is obtained.


Interacting particle systems Hydrodynamic limit Time reversal Hyperbolic system of conservation laws 


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  1. 1.
    Evans, D.J., Searles, D.J.: The Fluctuation Theorem. Adv. Phys. 51, 1529–1585 (2002) CrossRefADSGoogle Scholar
  2. 2.
    Harris, R.J., Schütz, G.M.: Fluctuation theorems for stochastic dynamics. J. Stat. Mech. P07020 (2007) Google Scholar
  3. 3.
    Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995) CrossRefMATHADSMathSciNetGoogle Scholar
  4. 4.
    Lebowitz, J.L., Spohn, H.: A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333–365 (1999) CrossRefMATHADSMathSciNetGoogle Scholar
  5. 5.
    Tóth, B., Valkó, B.: Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws. J. Stat. Phys. 112, 497–521 (2003) CrossRefMATHGoogle Scholar
  6. 6.
    Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999) MATHGoogle Scholar
  7. 7.
    Schütz, G.M.: Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles. J. Phys. A 36, R339–R379 (2003) CrossRefMATHGoogle Scholar
  8. 8.
    Schütz, G.M., Tabatabaei, F.: Shocks in the asymmetric exclusion process with internal degree of freedom. Phys. Rev. E 74, 051108 (2006) CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Tabatabaei, F., Schütz, G.M.: Nonequilibrium field-induced phase separation in single-file diffusion. Diffus. Fundam. 4, 5.1–5.38 (2006) Google Scholar
  10. 10.
    Arndt, P.F., Heinzel, T., Rittenberg, V.: Spontaneous breaking of translational invariance in one-dimensional stationary states on a ring. J. Phys. A 31, L45 (1998) CrossRefMATHADSMathSciNetGoogle Scholar
  11. 11.
    Evans, M.R., Kafri, Y., Koduvely, H.M., Mukamel, D.: Phase separation and coarsening in one-dimensional driven diffusive systems: local dynamics leading to long-range Hamiltonians. Phys. Rev. E 58, 2764 (1998) CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Blythe, R.A., Evans, M.R.: Nonequilibrium steady states of matrix product form: a solver’s guide. J. Phys. A, Math. Theor. 40, R333–R441 (2007) CrossRefMATHADSMathSciNetGoogle Scholar
  13. 13.
    Rajewsky, N., Sasamoto, T., Speer, E.R.: Spatial condensation for an exclusion process on a ring. Physica A 279, 123 (2000) CrossRefADSGoogle Scholar
  14. 14.
    Clincy, M., Evans, M.R.: Phase transition in the ABC model. Phys. Rev. E 67, 066115 (2003) CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Popkov, V., Schütz, G.M.: Shocks and excitation dynamics in a driven diffusive two-channel system. J. Stat. Phys. 112, 523–540 (2003) CrossRefMATHGoogle Scholar
  16. 16.
    Popkov, V., Salerno, M.: Hierarchy of boundary-driven phase transitions in multispecies particle systems. Phys. Rev. E 83, 011130 (2011) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Serre, D.: Systems of Conservation Laws. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  18. 18.
    Yau, H.T.: Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22, 63–80 (1991) CrossRefMATHADSMathSciNetGoogle Scholar
  19. 19.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centro de Matemática, Computação e CogniçãoUniversidade Federal do ABCSanto AndréBrazil
  2. 2.Theoretical Soft Matter and Biophysics, Institute of Complex SystemsForschungszentrum JülichJülichGermany
  3. 3.Interdisziplinäres Zentrum für komplexe SystemeUniversität BonnBonnGermany

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