Autocorrelation functions of nonlinear Fokker-Planck equations

The European Physical Journal B - We compute autocorrelation functions from nonlinear Fokker-Planck equations that describe nonlinear families of Markov diffusion processes and illustrate this...     

Exactly soluble multidimensional Fokker-Planck equations with nonlinear drift

The time-dependent analytic solutions of three classes of multidimensional Fokker-Planck equations with nonlinear drift are presented together with eigenvalues which are complex and depend essentially on the correlation functions of the fluctuations.     

Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations

Consequences of the connection between nonlinear Fokker-Planck equations and entropic forms are investigated. A particular emphasis is given to the feature that different nonlinear Fokker-Planck equations can be arranged into classes associated with the same entropic form and its corresponding stationary state. Through numerical integration, the time evolution of the solution of nonlinear Fokker-Planck equations related to the Boltzmann-Gibbs and Tsallis entropies are analyzed. The time behavior in both stages, in a time much smaller than the one required for reaching the stationary state, as well as towards the relaxation to the stationary state, are of particular interest. In the former case, by using the concept of classes of nonlinear Fokker-Planck equations, a rich variety of physical behavior may be found, with some curious situations, like an anomalous diffusion within the class related to the Boltzmann-Gibbs entropy, as well as a normal diffusion within the class of equations related to Tsallis’ entropy. In addition to that, the relaxation towards the stationary state may present a behavior different from most of the systems studied in the literature.     

Hurst exponents for interacting random walkers obeying nonlinear Fokker-Planck equations

Anomalous diffusion of random walks has been extensively studied for the case of non-interacting particles. Here we study the evolution of nonlinear partial differential equations by interpreting them as Fokker-Planck equations arising from interactions among random walkers. We extend the formalism of generalized Hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples. They include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable: the usual Fisher equation which contains a quadratic nonlinearity, a generalization of the Fisher equation with density-dependent diffusion constant, and the Nagumo equation which incorporates a cubic rather than a quadratic nonlinearity. We estimate the generalized Hurst exponents.     

Parametric solution method for self-consistency equations and order parameter equations derived from nonlinear Fokker-Planck equations

We propose a parametric approach to solve self-consistency equations that naturally arise in many-body systems described by nonlinear Fokker-Planck equations in general and nonlinear Vlasov-Fokker-Planck equations of Haissinski type in particular. We demonstrate for the Hess-Doi-Edwards model and the McMillan model of nematic and smectic liquid crystals that the parametric approach can be used to compute bifurcation diagrams and critical order parameters for systems exhibiting one or more than one order parameters. In addition, we show that in the context of the parametric approach solutions of the Haissinski model can be studied from the perspective of a pseudo order parameter.     

Quasi-exactly solvable Fokker-Planck equations

We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schrödinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the sl(2)-based systems in Turbiner’s classification are listed. We also present one sl(2)-based example which is not listed in Turbiner’s scheme.     

Asymptotic solutions of Fokker-Planck equations for nonlinear irreversible processes and generalized Onsager-Machlup theory

Fokker-Planck equations for nonlinear processes are solved asymptotically in the limit k→0 where k is the Boltzmann constant. It is shown that the leading asymptotic solutions for conditional (two-gate) distribution functions simply correspond to generalizations of the Onsager-Machlup theory to nonlinear processes. The asumptotic solution method used in the paper is similar to the well-known W.K.B. method in quantum mechanics. A stability criterion of nonlinear irreversible processes is also considered and compared with the Glansdorff-Prigogine stability criterion.     

Exact Fokker-Planck equations from stochastic master equations

A method for deriving exact Fokker-Planck equations from stochastic master equations by expanding the probability distribution in terms of Poisson distribution is given. It is applied to two non-linear chemical processes to obtain the steady state distributions.     

Fokker-Planck equations for eignstate distributions

A simple model of relaxation phenomena is defined with a variable strength of interaction, where the interaction term is given by a Gaussian unitary ensemble. A set of Fokker-Planck equations are derived which describe the gradual delocalization of the eigenstates with respect to the unperturbed energy with increasing strength of interaction. The effect of localization on the time evolution in the model is a nonergodic property: the system has a memory of the initial state.     

The Fokker-Planck equations in lattice gauge theories

The stochastic quantization in lattice gauge theories (LGT) is discussed by using Langevin equations and Fokker-Planck equations. It is shown that the evolution equations in the stochastic process reduce to the Schwinger-Dyson equation when the lattice system reaches equilibrium.     

Fokker-Planck approximation for N-dimensional nonmarkovian langevin equations

The Fokker-Planck approximation for n-dimensional nonmarkovian Langevin equations is discussed through an expansion in powers of the correlation time of the noise. Exact cases are considered and an application to brownian motion is presented.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

On the solutions in elliptic functions of integrable nonlinear equations

A new solution in elliptic functions for the KdV equation is constructed using the method proposed by Belokolos and the author.     

On the application of truncated generalized Fokker-Planck equations

The Pawula theorem states that the generalized Fokker-Planck equation with finite derivatives greater than two leads to a contradiction to the positivity of the distribution function. Though negative values are inconsistent from a logical point of view, we show that such distribution functions with negative values can be very useful from a practical point of view. For a Poisson-process, where the exact solution is known, we compare the solution of the second order Fokker-Planck equation to the solutions of Fokker-Planck equations of finite order. It turns out that for certain parameters the approximations of the distribution function and the moments are much better for some higher order and that the magnitude of negative values may be very small in the relevant region of variables.     

Canonically invariant formulation of Langevin and Fokker-Planck equations

We present a canonically invariant form for the generalized Langevin and Fokker-Planck equations. We discuss the role of constants of motion and the construction of conservative stochastic processes. Received : 24 July 1997 / Revised : 30 October 1997 / Accepted : 26 January 1998     

Limit cycles and detailed balance in Fokker-Planck equations

A class of non-linear Fokker Planck equations with exactly known steady state solution is investigated and its relation with the models considered earlier in the literature is discussed. A characterisation of the absence of detailed balance and possible existence of limit cycles is given. The implications of detailed balance on the existence and the character of limit cycle behaviour are studied. It is shown that detailed balance does not determine the existence or non-existence of limit cycles but rather their character.