Abstract
We consider an open one dimensional lattice gas on sites i=1,..., N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N→∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x=i/N, has the asymptotic form exp[−N \(F\)({ρ})]; \(F\) is the large deviation functional. In contrast to equilibrium systems, for which \(F\) eq({ρ}) is just the integral of the appropriately normalized local free energy density, the \(F\) we find here for the nonequilibrium system is a nonlocal function of ρ. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of \(F\) in general SNS, where the long range correlations have been observed experimentally.
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REFERENCES
R. Schmitz, Fluctuations in nonequilibrium fluids, Phys. Rep. 171:1–58 (1988), and references therein.
J. R. Dorfman, T. R. Kirkpatrick, and J. V. Sengers, Generic long-range correlations in molecular fluids, Annu. Rev. Phys. Chem. 45:213–239 (1994).
H. Spohn, Long range correlations for stochastic lattice gases in a non-equilibrium steady state, J. Phys. A. 16:4275–4291 (1983).
W. B. Li, K. J. Zhang, J. V. Sengers, R. W. Gammon, and J. M. Ortiz de Zárate, Concentration fluctuations in a polymer solution under a temperature gradient, Phys. Rev. Lett. 81:5580–5583 (1998).
O. E. Lanford, Entropy and Equilibrium States in Classical Mechanics (Springer, Berlin, 1973).
S. Olla, Large deviations for Gibbs random fields, Probab. Theory Related Fields 77: 343–357 (1988).
R. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer, New York, 1985).
A. Martin-Löf, Statistical Mechanics and the Foundations of Thermodynamics (Springer, Berlin, 1979).
C. Cercignani, Ludwig Boltzmann: The Man Who Trusted Atoms (Oxford University Press, Oxford, 1998).
M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65:851–1112 (1993).
R. Graham, Onset of cooperative behavior in nonequilibrium steady states, in Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, G. Nicolis, G. Dewel, and J. W. Turner, eds. (Wiley, New York, 1981).
G. Eyink, Dissipation and large thermodynamic fluctuations, J. Statist. Phys. 61:533–572 (1990).
T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (Springer-Verlag, New York, 1999).
H. Spohn, Large Scale Dynamics of Interacting Particles (Springer-Verlag, Berlin, 1991).
G. Eyink, J. L. Lebowitz, and H. Spohn, Hydrodynamics of stationary nonequilibrium states for some lattice gas models, Comm. Math. Phys. 132:253–283 (1990).
G. Eyink, J. L. Lebowitz, and H. Spohn, Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state, Comm. Math. Phys. 140:119–131 (1991).
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems (Springer-Verlag, Berlin, 1999).
A. De Masi, P. Ferrari, N. Ianiro, and E. Presutti, Small deviations from local equilibrium for a process which exhibits hydrodynamical behavior II, J. Statist. Phys. 29:81–93 (1982).
B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26:1493–1517 (1993).
F. H. L. Essler and V. Rittenberg, Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, J. Phys. A 29:3375–3408 (1996).
N. Rajewsky and M. Schreckenberg, Exact results for one-dimensional cellular automata with different types of updates, Physica A 245:139–144 (1997).
T. Sasamoto, One dimensional partially asymmetric simple exclusion process with open boundaries: Orthogonal polynomials approach, J. Phys. A 32:7109–7131 (1999).
R. A. Blythe, M. R. Evans, F. Colaiori, and F. H. L. Essler, Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra, J. Phys. A 33:2313–2332 (2000).
L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91: 1505–1512, 1512–1515 (1953).
C. Kipnis, S. Olla, and S. R. S. Varadhan, Hydrodynamics and large deviations for simple exclusion processes, Comm. Pure Appl. Math. 42:115–137 (1989).
G. Eyink, J. L. Lebowitz, and H. Spohn, Hydrodynamics and fluctuations outside of local equilibrium, J. Statist. Phys. 83:385–472 (1996).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Fluctuations in stationary non equilibrium states of irreversible processes, Phys. Rev. Lett. 87:040601 (2001).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory for stationary non equilibrium states, cond-mat/0108040.
P. Garrido, J. L. Lebowitz, C. Maes, and H. Spohn, Long-range correlations for conservative dynamics, Phys. Rev. A 42:1954–1968 (1990).
G. Grinstein, D.-H. Lee, and S. Sachdev, Conservation laws, anisotropy, and ''selforganized criticality'' in noisy nonequilibrium systems, Phys. Rev. Lett. 64:1927–1930 (1990).
B. Schmittman and R. K. P. Zia, Statistical Mechanics of Driven Diffusive Systems (Academic Press, London, 1995).
B. Derrida, J. L. Lebowitz, and E. R. Speer, Free energy functional for nonequilibrium systems: An exactly solvable case, Phys. Rev. Lett. 87:150601 (2001).
J. Santos and G. M. Schutz, Exact time-dependent correlation functions for the symmetric exclusion process with open boundary, Phys. Rev. E 64:036107 (2001).
Philip Hartman, Ordinary differential equations, 2nd ed. (Boston, Birkhäuser, 1982).
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Derrida, B., Lebowitz, J.L. & Speer, E.R. Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process. Journal of Statistical Physics 107, 599–634 (2002). https://doi.org/10.1023/A:1014555927320
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DOI: https://doi.org/10.1023/A:1014555927320