Nonlinear Fokker-Planck Equation Research Articles

Applying the Adomian Method to Solve the Fokker–Planck Equation: A Case Study in Astrophysics

The objective of this study is to model astrophysical systems using the nonlinear Fokker–Planck equation, with the Adomian method chosen for its iterative and precise solutions in this context, applying boundary conditions relevant to data from the Rossi X-ray Timing Explorer (RXTE). The results include analysis of 156 X-ray intensity distributions from X-ray binaries (XRBs), exhibiting long-tail profiles consistent with Tsallis q-Gaussian distributions. The corresponding q-values align with the principles of Tsallis thermostatistics. Various diffusion hypotheses—classical, linear, nonlinear, and anomalous—are examined, with q-values further supporting Tsallis thermostatistics. Adjustments in the parameter α (related to the order of fractional temporal derivation) reveal the extent of the memory effect, strongly correlating with fractal properties in the diffusive process. Extending this research to other XRBs is both possible and recommended to generalize the characteristics of X-ray scattering and electromagnetic waves at different frequencies originating from similar astronomical objects.

Nonlinear Vlasov and Fokker-Planck Dynamics in Confined Systems with Drag: A Numerical Study

Abstract We explore numerically the behavior of a one-dimensional many-body system consisting of particles that interact through short range repulsive forces, and are also under the effects of drag forces, and of an external confining potential. The statistical dynamics of systems of this kind exhibits interesting links with the thermostatistical formalism based on the Sq non-additive entropies. In the regime of overdamped motion, these systems admit an effective description in terms of a non-linear Fokker-Planck equation. When the overdamped condition is relaxed, and inertial effects are explicitly taken into account, the system can be described by a Vlasov-like effective mean field dynamics. The Vlasov-like description of this type of systems has been recently investigated in the literature from an analytical point of view. In the present contribution we explore the behaviour of these system numerically, through direct molecular dynamics simulations. We consider examples of systems with four different short range repulsive forces, with a dependence on distance given by exponential, Heaviside, Lorentzian, and Bessel functions. The results of our numerical simulations are fully consistent with the predictions derived from the Vlasov-like mean field description. In particular, we verify that, in the asymptotic limit of large times, the system evolves towards a state exhibiting a spatial distribution of particles that coincides with the stationary solution of an appropriate nonlinear Fokker-Planck equation. This limit spatial distribution has the form of a q-Gaussian.

Controlling a Nonlinear Fokker–Planck Equation via Inputs with Nonlocal Action

Controlling a Nonlinear Fokker–Planck Equation via Inputs with Nonlocal Action

Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with Lévy noise

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Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

Hyper-Ballistic Superdiffusion of Competing Microswimmers.

Hyper-ballistic diffusion is shown to arise from a simple model of microswimmers moving through a porous media while competing for resources. By using a mean-field model where swimmers interact through the local concentration, we show that a non-linear Fokker-Planck equation arises. The solution exhibits hyper-ballistic superdiffusive motion, with a diffusion exponent of four. A microscopic simulation strategy is proposed, which shows excellent agreement with theoretical analysis.

Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs

Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Steady states of an Elo-type rating model for players of varying strength

In this paper we study the long-time behaviour of a kinetic formulation of an Elo-type rating model for a large number of interacting players with variable strength. The model results in a non-linear mean-field Fokker-Planck equation and we show the existence of steady states via a Schauder fixed point argument. Our proof relies on the study of a related linear equation using hypocoercivity techniques.

Entropy Production in a Fractal System with Diffusive Dynamics.

We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker-Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.

COMPOSITE LAGUERRE PSEUDOSPECTRAL METHOD FOR FOKKER-PLANCK EQUATIONS

A composite generalized Laguerre pseudospectral method for the nonlinear Fokker-Planck equations on the whole line is developed. Some composite generalized Laguerre interpolation approximation results are established. As an application, a composite Laguerre pseudospectral scheme is provided for the problems of the relaxation of fermion and boson gases. Convergence and stability of the scheme are proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis.

Nonlinear Fokker-Planck Equations, H-Theorem and Generalized Entropy of a Composed System.

We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker-Planck equations by considering the H-theorem. We use the H-theorem to obtain the conditions required to establish a suitable dependence for the system's interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system.

Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities.

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker-Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker-Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker-Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed.

Weakly interacting oscillators on dense random graphs

AbstractWe consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

Global strong solutions and large time behavior to a micro-macro model for compressible polymeric fluids near equilibrium

In this paper, we mainly study the global strong solutions to a micro-macro model for compressible polymeric fluids with small initial data and their long-time decay rates of all order spatial derivatives. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first present that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for d≥2. Moreover, for d≥3, we show a new critical Fourier estimation that allows us to derive the long-time decay rates of the L2 norm for all order spatial derivatives.

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

In this work, we obtain quantitative convergence of moderately interacting particle systems towards solutions of nonlinear Fokker-Planck equations with singular kernels. In addition, we prove the well-posedness for the McKean-Vlasov SDEs involving these singular kernels and the trajectorial propagation of chaos for the associated moderately interacting particle systems. Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. This seems to be the first time that such quantitative convergence results are obtained in Lebesgue and Sobolev norms for the aforementioned kernels. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the Sobolev regularity of infinite-dimensional stochastic convolution integrals, and we also exploit the regularity of the solutions of the limiting equation.

Existence of Optimal Control for Nonlinear Fokker–Planck Equations in \(\boldsymbol{L^1(\mathbb{R}^d)}\).

This work is concerned with the existence of optimal controllers for the Bolza optimal control problem governed by the nonlinear Fokker–Planck equation in with control input in the drift term. The solution to the control state system is a weak (mild) solution obtained from a vanishing viscosity approximation scheme. One obtains in particular the existence for the stochastic optimal control problem governed by McKean–Vlasov SDEs. For this problem, one proves the existence of a stochastic Markov optimal controller in feedback form.

Scaling limit of moderately interacting particle systems with singular interaction and environmental noise

We consider moderately interacting particle systems with singular interaction kernel and environmental noise. It is shown that the mollified empirical measures converge in strong norms to the unique (local) solutions of nonlinear Fokker-Planck equations. The approach works for the Biot-Savart and repulsive Poisson kernels.

Singularities in $L^1$-supercritical Fokker–Planck equations: A qualitative analysis

A class of nonlinear Fokker–Planck equations with superlinear drift is investigated in the L^1 -supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis with main focus on the isotropic case, where solutions will be shown to converge to the unique minimiser of the free energy as time tends to infinity. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the three-dimensional Kaniadakis–Quarati model for Bose–Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.

Canards in a bottleneck

In this paper, we investigate the stationary profiles of a nonlinear Fokker–Planck equation with small diffusion and nonlinear inflow and outflow boundary conditions. We consider corridors with a bottleneck whose width has a unique global nondegenerate minimum in the interior. In the small diffusion limit, the profiles are obtained constructively by using methods from geometric singular perturbation theory (GSPT). We identify three main types of profiles corresponding to: (i) high density in the domain and a boundary layer at the entrance, (ii) low density in the domain and a boundary layer at the exit, and (iii) transitions from high density to low density inside the bottleneck with boundary layers at the entrance and exit. Interestingly, solutions of the last type involve canard solutions generated at the narrowest point of the bottleneck. We obtain a detailed bifurcation diagram of these solutions in terms of the inflow and outflow rates. The analytic results based on GSPT are further corroborated by computational experiments investigating corridors with bottlenecks of variable width.