Nonlinear Fokker-Planck Equation Research Articles

Local well-posedness of a nonlinear Fokker–Planck model

Noise or fluctuations play an important role in the modeling and understanding of the behavior of various complex systems in nature. Fokker–Planck equations are powerful mathematical tools to study behavior of such systems subjected to fluctuations. In this paper we establish local well-posedness result of a new nonlinear Fokker–Planck equation. Such equations appear in the modeling of the grain boundary dynamics during microstructure evolution in the polycrystalline materials and obey special energy laws.

From the Boltzmann equation with non-local correlations to a standard non-linear Fokker-Planck equation

In this work, we study the formal connections between the non-linear Fokker-Planck Equation associated with the non-additive entropy and the Boltzmann Equation with the non-additive correlation functional. The collisional term following the q-algebra is adopted. In the derivation of the non-additive Fokker-Planck Equation, two constraints are imposed on the final result: i) that the entropic index q is a characteristic parameter of the non-additive systems with a value that does not change with time, and ii) that for q→1 a smooth transition for the standard Fokker-Planck Equation is obtained.

Nonlinear Fokker–Planck Equations with Time-Dependent Coefficients

An operator-based approach is used here to prove the existence and uniqueness of a strong solution to the time-varying nonlinear Fokker–Planck equation , in the Sobolev space , under appropriate conditions on and It is proved also that if is a density of a probability measure, so is for all . Moreover, we construct a weak solution to the McKean–Vlasov SDE associated with the Fokker–Planck equation such that is the density of its time marginal law.

The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation

One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, t\geq0, x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution $u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative initial data in $L^1$. These results are new in the literature on nonlinear Fokker-Planck equations arising in the mean field theory and are also relevant to the theory of stochastic differential equations. As a matter of fact, by the above convergence result, it follows that the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, has this equilibrium state as its unique invariant measure. Keywords: Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10.

Rigorous derivation of the Euler-alignment model with singular communication weights from a kinetic Fokker–Planck-alignment model

In this paper, we present a rigorous derivation of the isothermal Euler-alignment model with singular communication weights. We consider a hydrodynamic limit of a kinetic Fokker–Planck-alignment model, which is the nonlinear Fokker–Planck equation with the Cucker–Smale alignment force. Our analysis is based on the estimate of relative entropy between macroscopic quantities, together with careful analysis on the singular communication weights.

Global solutions to a nonlinear Fokker-Planck equation

<abstract><p>In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space $ H^2_{x}L^2_v $ by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, <italic>J. Stat. Phys.</italic>, <bold>173</bold> (2018), 222–241.), the regularity assumption on the initial data is much weaker.</p></abstract>

Shehu Variational Iteration Method For Solve Some Fractional Differential Equations

The idea proposed in the work is to extend the Shehu transform method to resolve the nonlinear fractional partial differential equations by combining them with the variational iteration method (VIM). We apply this technique to solve nonlinear fractional equations as nonlinear time fractional Fokker Planck equation.

Statistical dynamics of driven systems of confined interacting particles in the overdamped-motion regime

Systems consisting of confined, interacting particles doing overdamped motion admit an effective description in terms of nonlinear Fokker-Planck equations. The behavior of these systems is closely related to the S power-law entropies and can be interpreted in terms of the S-based thermostatistics. The connection between overdamped systems and the S measures provides valuable insights on diverse physical problems, such as the dynamics of interacting vortices in type-II superconductors. The S-thermostatistical approach to the study of many-body systems described by nonlinear Fokker-Planck equations has been intensively explored in recent years, but most of these efforts were restricted to systems affected by time-independent external potentials. Here, we extend this treatment to systems evolving under time-dependent external forces. We establish a lower bound on the work done by these forces when they drive the system during a transformation. The bound is expressed in terms of a free energy based on the S entropy and is satisfied even if the driving forces are not derivable from a potential function. It constitutes a generalization, for systems governed by nonlinear Fokker-Planck equations involving general time-dependent external forces, of the H-theorem satisfied by these systems when the external forces arise from a time-independent potential.

Semigroup decay for the linearized kinetic ellipsoidal Fokker-Planck equation

In this paper, we deal with the semigroup decay for the linearized kinetic ellipsoidal Fokker-Planck equation in the torus. This model is an extension of the nonlinear kinetic Fokker-Planck equation in order to give a correct Prandtl number in the Navier-Stokes limit. Due to the diffusion coefficient is replaced by a non diagonal temperature tensor, this makes the linearized operator for the nonlinear kinetic ellipsoidal Fokker-Planck equation with more complicated form. By taking advantage of the H1 type hypocoercivity techniques, we prove that the solutions converge exponential to the equilibrium with explicit rates.

Physically guided deep learning solver for time-dependent Fokker–Planck equation

The Fokker–Planck (FP) equation is of significant relevance in stochastic dynamics. To solve the FP equation, this paper proposes a physically guided deep learning-based method. The proposed method uses physical modeling as the constraint for guiding the deep neural networks to learn the solutions of the FP equation from unlabeled data. Four numerical examples were used to verify the accuracy and effectiveness of the proposed method for solving linear FP equations and nonlinear FP equations. Moreover, we examined the robustness of the trained model. Finally, extra observation data was used to replace the boundary conditions and initial conditions in complicated forms. Meanwhile, the effect of missing boundary conditions or initial conditions to the proposed method was further analyzed. The results demonstrated that observation data can be used as promising substitution for the initial and boundary conditions while the proposed method can still obtain accurate numerical solutions in the absence of some conditions.

Lie Symmetries of the Nonlinear Fokker-Planck Equation Based on Weighted Kaniadakis Entropy

The paper studies the Lie symmetries of the nonlinear Fokker-Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted Kaniadakis entropy, are found. The MaxEnt problem associated to the weighted Kaniadakis entropy is given a complete solution, together with the thermodynamic relations which extend the known ones from the non-weighted case. Several different, but related, arguments point out a subtle dichotomous behavior of the Kaniadakis constant k, distinguishing between the cases k∈(−1,1) and k=±1. By comparison, the Lie symmetries of the NFPEs based on Tsallis q-entropies point out six “exceptional” cases, for: q=12, q=32, q=43, q=73, q=2 and q=3.

"Lie symmetries of the nonlinear Fokker-Planck equation based on weighted Tsallis entropy"

"We determine the nonlinear Fokker-Planck equation in one dimension, based on a weighted Tsallis entropy and we derive its associated Lie symmetries. The corresponding Lyapunov functions and Breg- man divergences are also found, together with a formula linking the drift function, the diffusion function and a diffusion constant. We solve the MaxEnt problem associated to the weighted Tsallis entropy."

H-theorems for systems of coupled nonlinear Fokker-Planck equations

Nonlinear diffusion and Fokker-Planck equations constitute valuable tools in the study of diverse phenomena in complex systems. Processes described by these equations are closely related to thermostatistical formalisms based on generalized entropic functionals. Inspired by these relations, we explore the behavior of systems of coupled, nonlinear Fokker-Planck equations. In particular, we establish an H-theorem for a wide family of this type of systems. This H-theorem is formulated in terms of an appropriate free-energy–like functional. The nonlinear evolution equations discussed here include, as particular instances, those governing the dynamics of interacting multi-species, many-body systems in the overdamped-motion regime.

JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L p and L $\infty$ norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher order results, and in particulat to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called five gradients inequality allows to recover some inequalities that can be deduced from the Bakry-Emery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller-Segel chemiotaxis model. 1 Short introduction The goal of this paper is to present some estimates on evolution PDEs in the space of probability densities which share two important features: they include a linear diffusion term, and they are gradient flows in the Wasserstein space W2. These PDEs will be of the form $\partial$t$\rho$ -- $\Delta$$\rho$ -- $\nabla$ $\times$ ($\rho$$\nabla$u[$\rho$]) = 0, complemented with no-flux boundary conditions and an intial condition on $\rho$0. We will in particular concentrate on the Fokker-Plack case, where u[$\rho$] = V and V is a fixed function (with possible regularity assumptions) independent of $\rho$, on the case where u[$\rho$] = W * $\rho$ is obtained by convolution and models interaction between particles, and on the parabolic-elliptic Keller-Segel case where u[$\rho$] is related to $\rho$ via an elliptic equation. This last case models the evolution of a biological population $\rho$ subject to diffusion but attracted by the concentration of a chemo-attractant, a nutrient which is produced by the population itself, so that its distribution is ruled by a PDE where the density $\rho$ appears as a source term. Under the assumption that the production rate of this nutrient is much faster than the motion of the cells, we can assume that its distribution is ruled by a statical PDE with no explicit time-dependence, and gives rise to a system which is a gradient flow in the variable $\rho$ (the parabolic-parabolic case, where the time scale for the cells and for the nutrient are comparable, is also a gradient flow, in the product space W2 x L 2 , but we will not consider this case). Since we mainly concentrate on the case of bounded domains, in the Keller-Segel case the term u[$\rho$] cannot be expressed as a convoluton and requires ad-hoc computations. In all the paper, the estimates will be studied on a time-discretized version of these PDEs, consisting in the so-called JKO (Jordan-Kinderleherer-Otto) scheme, based on iterated optimization problems involving the Wasserstein distance W2. We will first present 0-order estimates, on the L p and L $\infty$ norms of the solution. This is just a translation into the JKO language of well-known properties of these equations. The main goal of this part is hence to

The Non-Linear Fokker–Planck Equation in Low-Regularity Space

We construct the global existence and exponential time decay rates of mild solutions to the non-linear Fokker–Planck equation near a global Maxwellians with small-amplitude initial data in the low regularity function space Lk1LT∞Lv2 where the regularity assumption on the initial data is weaker.

Linearization of nonlinear Fokker-Planck equations and applications

Let P be the space of probability measures on Rd. We associate a coupled nonlinear Fokker-Planck equation on Rd, i.e. with solution paths in P, to a linear Fokker-Planck equation for probability measures on the product space Rd×P, i.e. with solution paths in P(Rd×P). We explicitly determine the corresponding linear Kolmogorov operator L˜t using the natural tangent bundle over P with corresponding gradient operator ∇P. Then it is proved that the diffusion process generated by L˜t on Rd×P is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by L˜t in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurability conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of “Nemytskii-type”. As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schrödinger type PDEs on Rd×P2, through the Feynman-Kac formula for the corresponding diffusion processes.

Nonlinear Fokker-Planck equation with reflecting boundary conditions

We study the existence and uniqueness of a mild solution to a nonlinear Fokker-Planck equation with reflecting boundary conditions, by using a monotonicity approach. Then we prove that the mild solution is also a distributional one.

Existence, Stability and Regularity of Periodic Solutions for Nonlinear Fokker–Planck Equations

We consider a class of nonlinear Fokker–Planck equations describing the dynamics of an infinite population of units with mean-field interaction. Relying on a slow–fast viewpoint and on the theory of approximately invariant manifolds we obtain the existence of a stable periodic solution for the PDE, consisting of probability measures. Moreover we establish the existence of a smooth isochron map in the neighborhood of this periodic solution.

Stochastic consensus dynamics for nonconvex optimization on the Stiefel manifold: Mean-field limit and convergence

We study a consensus-based method for minimizing a nonconvex function over the Stiefel manifold. The consensus dynamics consists of stochastic differential equations for interacting particle system, whose trajectory is guaranteed to stay on the Stiefel manifold. For the proposed model, we prove the mean-field limit of the stochastic system toward a nonlinear Fokker–Planck equation on the Stiefel manifold. Moreover, we provide a sufficient condition on the parameter and the initial data, so that the solution to the Fokker–Planck equation is asymptotically concentrated on the point near a global optimizer. To implement our consensus-based optimization (CBO) algorithm, we provide two algorithms; one is improved from the algorithm suggested in our previous work, and the other is based on an entirely different approach, namely the Cayley transformation. We validate the CBO algorithms on the various test problems on the Stiefel manifold.

Interplay between nonlinear Fokker–Planck equation and stochastic differential equation

We derive the stochastic differential equation (SDE) in Itô’s calculus corresponding to nonlinear Fokker–Planck equation where the nonlinearity appearing in this evolution equation can be interpreted as providing an effective description of a system of particles interacting. We deduce the Itô’s stochastic differential equation correspondent to nonlinear Fokker–Planck equation obtained in the framework of the non-additive statistical mechanics. Even though is used the Stratonovich stochastic differential equation to make the connection with non-linear Fokker–Planck equation, the connection made here using the Itô’s calculus, is therefore, more consistent.