A quadrature based method of moments for nonlinear Fokker–Planck equations

Abstract

Fokker-Planck equations which are nonlinear with respect to their probability densities and occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, fermions and bosons can be challenging to solve numerically. To address some underlying challenges, we propose the application of the direct quadrature based method of moments (DQMOM) for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations (NLFPEs). In DQMOM, probability density (or other distribution) functions are represented using a finite collection of Dirac delta functions, characterized by quadrature weights and locations (or abscissas) that are determined based on constraints due to evolution of generalized moments. Three particular examples of nonlinear Fokker-Planck equations considered in this paper include descriptions of: (i) the Shimizu-Yamada model, (ii) the Desai-Zwanzig model (both of which have been developed as models of muscular contraction) and (iii) fermions and bosons. Results based on DQMOM, for the transient and stationary solutions of the nonlinear Fokker-Planck equations, have been found to be in good agreement with other available analytical and numerical approaches. It is also shown that approximate reconstruction of the underlying probability density function from moments obtained from DQMOM can be satisfactorily achieved using a maximum entropy method.

Weiss mean-field approximation for multicomponent stochastic spatially extended systems

“…3 shows that the dependence l(t) obtained by the DAF-based method is slowly decreasing, which means the violation of the normalization condition. On the contrary, scheme (18) conserves asymptotically (at large times) the condition of normalization of probability density per unit.…”

“…Dependencies of the decimal algorithm of the relative error lg |ε(t)| (21) on time. The dashed line is the DAF-based method [11], the solid line is the finite-difference method (18). It can be seen from the plots presented in Fig.…”

“…[39,40]. (20), the dashed line is the solution obtained by method (18), and the line with a long dash is the DAF-based method [11].…”

“…[14] a finite-difference scheme is used in the differential part and the trapezoid rule in the integral part of NFPE. Finite element [15] and finite-difference methods [15,16], discrete singular convolution algorithm [17], direct quadrature based method of moments [18,19], pseudo-spectral method [20], path-integral [21,22] and eigenfunction expansion methods [23,24] and others [25,26] are also used to find numerical solutions of NFPEs.…”

Hybrid master equation for jump-diffusion approximation of biomolecular reaction networks

“…Proof. Using Leibniz integral rule and the boundary conditions gives Since F t (c) is a polynomial function and sufficiently many moments of p t (d, c) with respect to c exist, we can manipulate the integral as follows: Here, we want to draw the attention of the reader to the following mean which is used in our equations Using this equality and the properties of the FPE will give us the following equation [9,31]:…”