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Hydrodynamics and fluctuations outside of local equilibrium: Driven diffusive systems

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Abstract

We derive hydrodynamic equations for systems not in local thermodynamic equilibrium, that is, where the local stationary measures are “non-Gibbsian” and do not satisfy detailed balance with respect to the microscopic dynamics. As a main example we consider thedriven diffusive systems (DDS), such as electrical conductors in an applied field with diffusion of charge carriers. In such systems, the hydrodynamic description is provided by a nonlinear drift-diffusion equation, which we derive by a microscopic method ofnonequilibrium distributions. The formal derivation yields a Green-Kubo formula for the bulk diffusion matrix and microscopic prescriptions for the drift velocity and “nonequilibrium entropy” as functions of charge density. Properties of the hydrodynamic equations are established, including an “H-theorem” on increase of the thermodynamic potential, or “entropy”, describing approach to the homogeneous steady state. The results are shown to be consistent with the derivation of the linearized hydrodynamics for DDS by the Kadanoff-Martin correlation-function method and with rigorous results for particular models. We discuss also the internal noise in such systems, which we show to be governed by a generalizedfluctuation-dissipation relation (FDR), whose validity is not restricted to thermal equilibrium or to time-reversible systems. In the case of DDS, the FDR yields a version of a relation proposed some time ago by Price between the covariance matrix of electrical current noise and the bulk diffusion matrix of charge density. Our derivation of the hydrodynamic laws is in a form—the so-called “Onsager force-flux form” which allows us to exploit the FDR to construct the Langevin description of the fluctuations. In particular, we show that the probability of large fluctuations in the hydrodynamic histories is governed by a version of the Onsager “principle of least dissipation,” which estimates the probability of fluctuations in terms of the Ohmic dissipation required to produce them and provides a variational characterization of the most probable behavior as that associated to least (excess) dissipation. Finally, we consider the relation of longrange spatial correlations in the steady state of the DDS and the validity of ordinary hydrodynamic laws. We also discuss briefly the application of the general methods of this paper to other cases, such as reaction-diffusion systems or magnetohydrodynamics of plasmas.

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References

  1. H. Mori,Phys. Rev. 111:694 (1958);112:1829 (1958).

    Google Scholar 

  2. D. N. Zubarev,Nonequilibrium statistical Thermodynamics (Consultants Bureau, New York, 1974).

    Google Scholar 

  3. J. A. McLennan,Introduction to Nonequilibrium Statistical Mechanics (Prentice-Hall, Englewood Cliffs, New Jersey, 1989).

    Google Scholar 

  4. Y. G. Sinai,Selecta Math. Sov. 7:279 (1988).

    Google Scholar 

  5. G. Eyink, Nonequilibrium statistical distributions, unpublished.

  6. G. L. Eyink, J. L. Lebowitz, and H. Spohn, Microscopic origin of hydrodynamic behavior: Entropy production and the steady-state, inChaos/Xaoc, Soviet-American Perspectives on Nonlinear Science, D. K. Campbell, ed. (American Institute of Physics, New York, 1990).

    Google Scholar 

  7. D. N. Zubarev and V. G. Morozov,Physica A 120:411 (1983).

    Google Scholar 

  8. R. Zwanzig,J. Chem. Phys. 33:1338 (1960).

    Google Scholar 

  9. A. Einstein,Ann. Phys. (Leipzig)22:180 (1907);33: 1275 (1910).

    Google Scholar 

  10. L. Onsager,Phys. Rev. 37:405 (1931);38:2265 (1931).

    Google Scholar 

  11. G. H. Wannier,Phys. Rev. 83:281 (1951);87:795 (1952).

    Google Scholar 

  12. G. H. Wannier,Bell. Syst. Techn. J. 32:170 (1953).

    Google Scholar 

  13. P. A. Markowich, C. A. Ringhofer, and C. Schmeisser,Semiconductor Equations (Springer, Vienna, 1990).

    Google Scholar 

  14. S. Katz, J. L. Lebowitz, and H. Spohn,J. Stat. Phys. 34:497 (1984).

    Google Scholar 

  15. B. Schmittmann and R. K. P. Zia, Statistical mechanics of driven diffusive systems, inPhase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1995).

    Google Scholar 

  16. R. Esposito, R. Marra, and H. T. Yau, Diffusive limit of asymmetric simple exclusion, inThe State of Matter, M. Aizenmann and H. Araki, eds (World Scientific, Singapore, 1994).

    Google Scholar 

  17. C. Landim, S. Olla, and H.-T. Yau, First-order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimensiond≥3, Preprint, Ecole Polytechnique, R.I. No. 307 (Novemeber 1994).

  18. V. P. Kalashnikov,Phys. Lett. A 26:433 (1968).

    Google Scholar 

  19. N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Y. G. Sinai,Commun. Math. Phys. 154:569 (1993).

    Google Scholar 

  20. H. Spohn,Large Scale Dynamics of Interacting Particles (Springer-Verlag, New York, 1991).

    Google Scholar 

  21. P. J. Price, InFluctuation Phenomena in Solids, R. E. Burgess, ed. (Academic Press, New York, 1965), p. 355.

    Google Scholar 

  22. L. P. Kadanoff and P. C. Martin,Ann. Phys. (N.Y.) 24:419 (1963).

    Google Scholar 

  23. D. Forster,Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, Reading, Massachusetts, 1975).

    Google Scholar 

  24. R. Graham,Z. Phys. B 26:397 (1977).

    Google Scholar 

  25. R. Graham,Z. Phys. B 26:281 (1977).

    Google Scholar 

  26. H.-O. Georgii,Gibbs Measures and Phase Transitions (de Gruyter, Berlin, 1988).

    Google Scholar 

  27. L. D. Landau and E. M. Lifshitz,Electrodynamics of Continuous Media (Pergamon, London, 1960).

    Google Scholar 

  28. S. R. de Groot and P. Mazur,Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962).

    Google Scholar 

  29. H. B. Callen,Thermodynamics (Wiley, New York, 1960).

    Google Scholar 

  30. P. L. Garrido, J. L. Lebowitz, C. Maes, and H. Spohn,Phys. Rev. A 42: 1954 (1990).

    Google Scholar 

  31. J. L. Lebowitz and R. H. Schonmann,Prob. Theory Related Fields 77:49 (1988).

    Google Scholar 

  32. R. S. Ellis,Entropy, Large Deviations, and Statistical Mechanics (Springer, New York, 1985).

    Google Scholar 

  33. H. Künsch,Z. Wahrsch. Geb. 66:407 (1984).

    Google Scholar 

  34. G. L. Eyink, Entropy, statistical mechanics, and PDE's, unpublished.

  35. H. van Beijeren,J. Stat. Phys. 35:399 (1984).

    Google Scholar 

  36. L. Onsager and S. Machlup,Phys. Rev. 91:1505 (1953).

    Google Scholar 

  37. G. L. Eyink,J. Stat. Phys. 61:533 (1990).

    Google Scholar 

  38. H. Haken,Synergetics (Springer-Verlag, Berlin, 1978).

    Google Scholar 

  39. J. W. Dufty and J. M. Rubi,Phys. Rev. A 36:222 (1987).

    Google Scholar 

  40. J. A. Krommes and G. Hu,Phys. Fluids B 5:3908 (1993).

    Google Scholar 

  41. L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon Press, London, 1959).

    Google Scholar 

  42. R. Stratonovich,Nonlinear Nonequilibrium Thermodynamics I (Springer, Berlin, 1992).

    Google Scholar 

  43. A. Einstein,Ann. Phys. (Leipzig)17:549 (1905).

    Google Scholar 

  44. B. Callen and T. A. Welton,Phys. Rev.,83:34 (1951).

    Google Scholar 

  45. H. Nyquist,Phys. Rev. 32:110 (1928).

    Google Scholar 

  46. R. F. Fox and G. E. Uhlenbeck,Phys. Fluids 13:1893 (1970).

    Google Scholar 

  47. K. Tomita and H. Tomita,Prog. Theor. Phys. 51:1731 (1974).

    Google Scholar 

  48. S. V. Gantsevich, V. L. Gurevich, and R. Katillius,Nuovo Cimento 2:1 (1979).

    Google Scholar 

  49. A.-M. S. Tremblay, InRecent Developments in Nonequilibrium Thermodynamics (Springer, Berlin, 1984).

    Google Scholar 

  50. C. Landim, S. Olla, and H. T. Yau, Some properties of the diffusion coefficient for asymmetric simple exclusion process, Ecole Polytechnique, R.I. No. 327 (June 1995).

  51. M. Q. Zhang, J. S. Wang, J. L. Lebowitz, and J. L. Valles,J. Stat. Phys. 52: 1461 (1988).

    Google Scholar 

  52. R. K. P. Zia and B. Schmittmann, On singularities in the disordered phase of a driven diffusive system, preprint (1995).

  53. A. Aharony,Phys. Rev. B 8:3363 (1973).

    Google Scholar 

  54. J. Skalyo, B. C. Frazer, and Shirane,Phys. Rev. B 1:278 (1970).

    Google Scholar 

  55. E. R. Speer, The two species, totally asymmetric simple exclusion process, inOn Three Levels, M. Fannes, C. Maes, and A. Verbeure, eds. (Plenum Press, New York, 1994). p. 91.

    Google Scholar 

  56. B. Derrida, S. Janowsky, J. L. Lebowitz, and E. Speer,Europhys. Lett. 22: 651 (1993);J. Stat. Phys. 73:813 (1993).

    Google Scholar 

  57. B. M. Law, R. W. Gammon, and J. V. Sengers,Phys. Rev. Lett. 60:1554 (1988).

    Google Scholar 

  58. V. G. Morozov,Physica A 126:461 (1984).

    Google Scholar 

  59. R. Graham and H. Haken,Z. Phys. 243:289 (1971);245: 141 (1971).

    Google Scholar 

  60. D. Gabrielli, G. Jona-Lasinio, and C. Landim, Onsager reciprocity relations without microscopic reversibility, preprint [mp_arc@ftp.ma.utexas.edu, #95-366].

  61. C. Kipnis, S. Olla, and S. R. S. Varadhan,Commun. Pure Appl. Math. XLII:243 (1989).

    Google Scholar 

  62. R. Graham, InOrder and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, G. Nicolis, G. Dewel, and J. W. Turner, eds. (Wiley, New York, 1981).

    Google Scholar 

  63. R. Graham, InStochastic Processes in Nonequilibrium Systems, L. Garrido, P. Seglar, and P. J. Shephard, eds. (Springer, Berlin, 1978).

    Google Scholar 

  64. M. I. Freidlin and A. D. Wentzell,Random Perturbations of Dynamical Systems (Springer, New York, 1984).

    Google Scholar 

  65. D. A. Dawson and J. GärtnerStochastics 20:247 (1987).

    Google Scholar 

  66. Y. Oono,Prog. Theor. Phys. Suppl. 99:165 (1989).

    Google Scholar 

  67. E. B. Pitman and D. G. Schaeffer,Commun. Pure Appl. Math. 40:421 (1987).

    Google Scholar 

  68. A. De Masi, E. Presutti, and J. L. Lebowitz,J. Stat. Phys. 55:523 (1986).

    Google Scholar 

  69. G. Jona-Lasinio, C. Landim, and M. E. Vares,Prob. Theory Related Fields 97:339 (1993).

    Google Scholar 

  70. G. Jona-Lasinio,Ann. Inst. H. Poincaré 55(2):751 (1991).

    Google Scholar 

  71. R. Balescu,Transport Process in Plasmas (North-Holland, Amsterdam, 1988).

    Google Scholar 

  72. R. Balescu,Phys. Fluids B 3:564 (1991).

    Google Scholar 

  73. A. van Enter, R. Fernández, and A. Sokel,J. Stat. Phys. 72:879 (1993).

    Google Scholar 

  74. T. J. Liggett,Interacting Particle Systems (Springer, Berlin, 1985).

    Google Scholar 

  75. F. Rezakhanlou,Commun. Math. Phys. 140:417 (1991).

    Google Scholar 

  76. H. van Beijeren, R. Kutner, and H. Spohn,Phys. Rev. Lett. 54:2026 (1985).

    Google Scholar 

  77. J. Krug and H. Spohn, Kinetic roughening of growing surfaces inSolids Far From Equilibrium: Growth, Morphology and Defects, C. Godréche, ed. (Cambridge University Press, Cambridge, 1991).

    Google Scholar 

  78. Lin Xu, Diffusion limit for lattice gas with short-range interactions, Thesis, NYU (1993).

  79. H. O. Georgii,Canonical Gibbs Measures (Springer-Verlag, Berlin, 1979).

    Google Scholar 

  80. R. H. Kraichnan,Phys. Rev. 113:1181 (1959).

    Google Scholar 

  81. G. Gallavotti and E. Verboven,Nuovo Cimento 28:274 (1975).

    Google Scholar 

  82. M. Aizenman et al.,Commun. Math. Phys. 53:209 (1977).

    Google Scholar 

  83. M. Aizenman et al.,Commun. Math. Phys. 48:1 (1976).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Building No. 89, 85721, Tucson, Arizona

    Gregory L. Eyink

  2. Departments of Mathematics and Physics, Rutgers University, Hill Center-Busch Campus, 08903, New Brunswick, New Jersey

    Joel L. Lebowitz

  3. Theoretische Physik, Ludwig-Maximilians-Universität, München, Germany

    Herbert Spohn

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  1. Gregory L. Eyink
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  2. Joel L. Lebowitz
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  3. Herbert Spohn
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Eyink, G.L., Lebowitz, J.L. & Spohn, H. Hydrodynamics and fluctuations outside of local equilibrium: Driven diffusive systems. J Stat Phys 83, 385–472 (1996). https://doi.org/10.1007/BF02183738

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