Stable nonequilibrium probability densities and phase transitions for mean-field models in the thermodynamic limit
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 a 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 nonequilibrium 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 Gruener-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.
- Research Organization:
- Universidad de Sevilla, Spain
- OSTI ID:
- 6459414
- Journal Information:
- J. Stat. Phys.; (United States), Vol. 46:3/4
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
PARTICLE MODELS
FOKKER-PLANCK EQUATION
MEAN-FIELD THEORY
THERMODYNAMICS
BOUNDARY CONDITIONS
CHARGE DENSITY
CRYSTAL MODELS
NONLINEAR PROBLEMS
ORDER PARAMETERS
ORDER-DISORDER TRANSFORMATIONS
PERTURBATION THEORY
PROBABILITY
SPIN GLASS STATE
STATISTICAL MECHANICS
STOCHASTIC PROCESSES
SYNCHRONIZATION
TIME DEPENDENCE
WKB APPROXIMATION
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL MODELS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PHASE TRANSFORMATIONS
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics
645400 - High Energy Physics- Field Theory