Abstract
A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Grüner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.
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References
J. L. Lebowitz and P. G. Bergmann,Ann. Phys. (N.Y.)1:1 (1957).
R. C. Desai and R. Zwanzig,J. Stat. Phys. 19:1 (1978).
D. A. Dawson,J. Stat. Phys. 31:29 (1983).
L. L. Bonilla,Phys. Rev. B 35, to appear (1986); and in preparation.
L. L. Bonilla, J. M. Casado, and M. Morillo, in preparation.
W. Horsthemke and R. Lefever, inStochastic Nonlinear Systems, L. Arnold and R. Lefever, eds. (Springer, New York, 1981); also I. Prigogine and I. Stengers,Order out of Chaos (Bantam Books, Toronto, 1984).
H. Sompolinsky and A. Zippelius,Phys. Rev. B 25:6860 (1982).
D. S. Fisher,Phys. Rev. B 31:1396 (1985).
Y. Yamaguchi, K. Kometani, and H. Shimizu,J. Stat. Phys. 26:719 (1981).
H. Fukuyama and P. A. Lee,Phys. Rev. B 17:535 (1977).
R. Graham,Z. Phys. B 26:281 (1977); B. Jouvet and R. Phythian,Phys. Rev. A 19:1350 (1979).
H. Jauslin,J. Stat. Phys. 40:147 (1985).
B. J. Matkowsky and E. L. Reiss,SIAM J. Appl. Math. 33:230 (1977); J. G. Watson and E. L. Reiss,SIAM J. Appl. Math. 42:135 (1982).
J. L. Lebowitz, inLong Time Prediction in Dynamics, D. Horton, L. Reichl, and S. Szebehely, eds. (Wiley, New York, 1983), p. 3; J. L. Lebowitz and H. Spohn,Commun. Pure Appl. Math. 36:595 (1983); J. B. Keller and L. L. Bonilla,J. Stat. Phys. 42:1115 (1986).
Y. Kuramoto,Physica 106A:128 (1981).
L. Sneddon,Phys. Rev. B 29:719, 725 (1984).
G. Grüner and A. Zettl,Phys. Rep. 119:117 (1985).
M. O. Robbins, J. P. Stokes, and S. Bhattacharya,Phys. Rev. Lett. 55:2822 (1985).
J. J. Brey, J. M. Casado, and M. Morillo,Physica 128A:497 (1984).
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Bonilla, L.L. Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit. J Stat Phys 46, 659–678 (1987). https://doi.org/10.1007/BF01013379
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DOI: https://doi.org/10.1007/BF01013379