Numerical method for the nonlinear Fokker-Planck equation

Abstract

A practical method based on distributed approximating functionals ͑DAFs͒ is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo ͓Phys. Rev. E 54, 931 ͑1996͔͒ to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the ''flat'' region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schrödinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem.

Efficient path integral solution of Fokker–Planck equation: response, bifurcation and periodicity of nonlinear systems

“…Thus for the Gaussian response assumption to work one has to take extremely small time steps and space intervals in the PI procedure for improving the accuracy of the PDF. To overcome this difficulty, a new PI method based on an adjustable non-Gaussian transition PDF expressed as a product of a Gaussian PDF and a polynomial function, which is a truncated version of an infinite expansion [31] and the GaussLegendre integration scheme [11] is proposed in this article. This method makes use of knowledge of higher order moments or cumulants and allows for a coarse space grid size and longer time steps, thereby reducing the computational effort.…”

“…5 anyway, because of computational complexities associated with the higher order polynomials around the tails of the probability densities. In fact, attention is commonly limited to j = 3 and j = 4, which provide one anti-symmetric and one symmetric correction to the Gaussian PDF [31]. Also for N [ 4, the non-Gaussian transition PDF model based on Hermite polynomials are unrealizable since for certain combinations of c j there are ranges of x where probability is negative, which does not have physical meaning.…”

“…In some cases the knowledge of the actual or approximate values of higher order cumulants of the non-Gaussian process may be available in a particular problem, in which cases, an approximate non-Gaussian model can be found which is better than the Gaussian approximation. The best known approach for approximating a non-Gaussian PDF is to express it as a product of a Gaussian PDF and a polynomial function, which is a truncated version of an infinite expansion [31]. The procedure can also be extended for joint PDFs.…”

Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

“…A fundamental example of a strongly nonlinear Markov diffusion process is a generalized Ornstein-Uhlenbeck process involving a mean field force ℎ MF proportional to the mean ⟨ ( )⟩ of the process [9,15,16]. For = 1, the stochastic difference equation reads…”

Quantum wave packet dynamics with trajectories: Implementation with distributed approximating functionals

“…The method is based upon the use of distributed approximating functionals ͑DAFs͒, 23,24 which have been used by Hoffman, Kouri and co-workers since their introduction in 1991. DAFs have been successfully used in a number of studies for fitting, interpolation, and extrapolation of functions [25][26][27][28][29][30] ͑including potential energy surfaces͒, for solving differential and partial differential equations, [31][32][33][34] and for smoothing and filtering digital signals. 35,36 For the purposes of this study, it is important that DAFs can provide accurate values for derivatives both on and off the grid points.…”