Generalized Fokker-Planck Equation Research Articles

Generalized Fokker–Planck equation for superstatistical systems

Superstatistical systems are non-equilibrium systems in stationary states with large fluctuations of intensive quantities. Different effective statistical processes follow accordingly to the statistical properties of those fluctuations. We derive the generalized Fokker–Planck equation for such systems and the Erdélyi–Kober fractional operators emerge to be involved.

Controlling a generalized Fokker–Planck equation via inputs with nonlocal action

This paper concerns an optimal control problem (P) related to a generalized Fokker–Planck equation. Basic properties of the solutions to the generalized FP equation are derived via a semigroup approach in the space H−1(Rd). Problem (P) is proven to be deeply related to a stochastic optimal control problem (PS) for a McKean–Vlasov equation. The existence of an optimal control is obtained for the deterministic problem (P). The existence of an optimal control is established for an approximating optimal control problem (Ph) related to a backward Euler approximation of the generalized FP equation (with a constant discretization step h). One proves under additional hypotheses that “(Ph) converges to (P)” in a certain sense. First order necessary conditions for (Ph) are derived as well.

THE ROLE OF HIGHER MOMENTS ON THE DISTRIBUTION OF PARTICLES IN THE SPACE OF IMPULSES AT CYCLOTRON RESONANCES

The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It is shown that there are two qualitatively different regimes. The appearance of the first one is due to the overlap of nonlinear cyclotron resonances. The second mode is related to intermittency. The moments and spectra of each of these regimes are determined. It is shown that with an increase in the intensity of an external electromagnetic wave, the first regime appears at the beginning and only then the second regime appears. A characteristic feature of the second regime is intermittency. Steps appear on the time dynamics of pulses in the second mode. It is shown that the spectra in the second mode are narrower than in the first mode. A characteristic feature of the second regime (the regime with intermittency) is the fact that the higher moments turn out to be larger than the lower moments. In the first regime, the highest moments decrease rapidly. To find the particle momentum distribution function, the generalized Fokker-Planck equation was used. Solutions of this equation are written out for some important cases.

Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

Volterra-Prabhakar function of distributed order and some applications

The paper studies the exact solution of two kinds of generalized Fokker–Planck equations in which the integral kernels are given either by the distributed order function k1(t)=∫01t−μ/Γ(1−μ)dμ or the distributed order Prabhakar function k2(α,γ;λ;t)=∫01eα,1−μ−γ(λ;t)dμ, where the Prabhakar function is denoted as eα,1−μ−γ(λ;t). Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non-negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker–Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of k2(α,γ;λ;t) and generated by it the probability density function p2(x,t).

Kinetic theory of two-dimensional point vortices and fluctuation–dissipation theorem

We complete the kinetic theory of two-dimensional (2D) point vortices initiated in previous works. We use a simpler and more physical formalism. We consider a system of 2D point vortices submitted to a small external stochastic perturbation and determine the response of the system to the perturbation. We derive the diffusion coefficient and the drift by polarization of a test vortex. We introduce a general Fokker–Planck equation involving a diffusion term and a drift term. When the drift by polarization can be neglected, we obtain a secular dressed diffusion equation sourced by the external noise. When the external perturbation is created by a discrete collection of N point vortices, we obtain a Lenard–Balescu-like kinetic equation reducing to a Landau-like kinetic equation when collective effects are neglected. We consider a multi-species system of point vortices. We discuss the process of kinetic blocking in the single and multi-species cases. When the field vortices are at statistical equilibrium (thermal bath), we establish the proper expression of the fluctuation–dissipation theorem for 2D point vortices relating the power spectrum of the fluctuations to the response function of the system. In that case, the drift coefficient and the diffusion coefficient satisfy an Einstein-like relation and the Fokker–Planck equation reduces to a Smoluchowski-like equation. We mention the analogy between 2D point vortices and stellar systems. In particular, the drift of a point vortex in 2D hydrodynamics (Chavanis in Phys Rev E 58:R1199, 1998) is the counterpart of the Chandrasekhar dynamical friction in astrophysics. We also consider a gas of 2D Brownian point vortices described by N coupled stochastic Langevin equations and determine its mean and mesoscopic evolution. In the present paper, we treat the case of unidirectional flows, but our results can be straightforwardly generalized to axisymmetric flows.

Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay.

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.

Coupled Kansa and hybrid optimization methodological approach for Kolmogorov–Feller equations

Methodological approaches coupling a meshfree method with multiquadric radial basis function and a hybrid optimization method are elaborated to solve nonlinear stochastic differential equations with a Poisson white noise. The associated probability density function is governed by the Kolmogorov Feller equation named also Generalized Fokker–Planck equation. Actually, the shape parameter of RBFs affects the numerical solution strongly. Choosing arbitrary this parameter may lead to inaccurate solutions and thus parameter should be chosen. A hybrid optimization method is elaborated to predict an efficient shape parameter based on a well adapted residual error. The resulting coupled procedure is applied to the considered stationary and instationary generalized Fokker-Planck equations using an implicit time scheme with a regularized initial condition. The multiquadric radial basis function with optimized parameter is also coupled with the Monte Carlo method. The effectiveness and accuracy of the elaborated methodological approaches are demonstrated and the probability density functions are computed for several nonlinear stochastic differential equations under a Poisson white noise.

The correlated dichotomous noise as an exact M-Gaussian stochastic process

Gaussian statistics is a milestone and often a fundamental assumption in the treatment of complex phenomena. The relevance of Gaussian statistics is highlighted by the Central Limit Theorem (CLT), which states that the sum of a large number of independent random variables leads to a universal form of the probability density function (PDF). A proposal for a sound extension of the definition of “Gaussianicity” and, consequently, of the CLT (named M-CLT) was presented in a recent paper [M. Bianucci, Chaos, Sol. & Frac. 148 (2021) 110961]. This extension includes non commuting quantities (named q-numbers or operators) and can be done in different ways, by choosing different projection mappings (the M map) to be combined with the exponentials entering in the definitions of both the first and the second characteristic functions. In the same paper it was illustrated that there is a strict relationship between “M“-Gaussianicity and the generalized Fokker Planck Equation (FPE). Here we show that there exists a particular M-map for which the dichotomous noise is M-Gaussian and the vector field of the system of interest perturbed by such a noise is M-Gaussian as well. Thus, for such a system of interest, a generalized FPE, characterized by a memory kernel, can be obtained. This is an interesting result in itself, but it is also important for two practical reasons: first, according to the M-CLT, it follows that such time-non local FPE is the M-Gaussian limit expression to which converge, under some assumptions, the sum (or the integral) of stochastic first order differential operators. Second, it states that, being M-Gaussian, the dichotomous noise assumes a relevance, in the frame of statistical processes, comparable to that of the standard Gaussian ones.

Data driven adaptive Gaussian mixture model for solving Fokker-Planck equation.

The Fokker-Planck (FP) equation provides a powerful tool for describing the state transition probability density function of complex dynamical systems governed by stochastic differential equations (SDEs). Unfortunately, the analytical solution of the FP equation can be found in very few special cases. Therefore, it has become an interest to find a numerical approximation method of the FP equation suitable for a wider range of nonlinear systems. In this paper, a machine learning method based on an adaptive Gaussian mixture model (AGMM) is proposed to deal with the general FP equations. Compared with previous numerical discretization methods, the proposed method seamlessly integrates data and mathematical models. The prior knowledge generated by the assumed mathematical model can improve the performance of the learning algorithm. Also, it yields more interpretability for machine learning methods. Numerical examples for one-dimensional and two-dimensional SDEs with one and/or two noises are given. The simulation results show the effectiveness and robustness of the AGMM technique for solving the FP equation. In addition, the computational complexity and the optimization algorithm of the model are also discussed.

Interacting-fermion dynamics in Majorana phase space

The problem of fermion dynamics is studied using the $Q$ function for fermions. This is a probabilistic phase-space representation, which we express using Majorana operators and a phase-space variable of real antisymmetric matrices. We consider a general Hamiltonian with quadratic and quartic Majorana interactions and arbitrary couplings. Our model includes the Majorana-Hubbard and Fermi-Hubbard Hamiltonians, as well as general quantum field theories of interacting fermions. The relevant coefficients are calculated for a number of both local and nonlocal Hamiltonian models. We also give a detailed example of a four-Majorana interaction. Using the Majorana $Q$ function we derive a generalized Fokker-Planck equation, with results for the drift and diffusion terms. The diffusion term is shown to be traceless, which has a dynamical interpretation as a forward-backward stochastic process. We prove that an initial pure state has trajectories that remain on a surface of pure states. This approach leads to a model of quantum measurement in terms of an ontology with real vacuum fluctuations.

SOLUTION OF LOCAL FRACTIONAL GENERALIZED FOKKER–PLANCK EQUATION USING LOCAL FRACTIONAL MOHAND ADOMIAN DECOMPOSITION METHOD

In this paper, we solve the local fractional generalized Fokker–Planck equation. To solve the problem, local fractional Mohand transform with Adomian decomposition method is introduced due to its simple approach and less computational work. Furthermore, for the applicability of the technique, we illustrate some examples and their exact or approximate solutions with their graphical representations.

Convergence results for the time-changed fractional Ornstein–Uhlenbeck processes

In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein-Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein-Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorohod $J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H \to 1/2^+$ in the space of cadlag functions. Finally, we exploit some convergence properties of mild solutions of a generalized Fokker-Planck equation associated to the aforementioned processes, as $H \to 1/2^+$.

Stationary solution and H theorem for a generalized Fokker-Planck equation.

We investigate a family of generalized Fokker-Planck equations that contains Richardson and porous media equations as members. Considering a confining drift term that is related to an effective potential, we show that each equation of this family has a stationary solution that depends on this potential. This stationary solution encompasses several well-known probability distributions. Moreover, we verify an H theorem for the generalized Fokker-Planck equations using free-energy-like functionals. We show that the energy-like part of each functional is based on the effective potential and the entropy-like part is a generalized Tsallis entropic form, which has an unusual dependence on the position and can be related to a generalization of the Kullback-Leibler divergence. We also verify that the optimization of this entropic-like form subjected to convenient constraints recovers the stationary solution. The analysis presented here includes several studies about H theorems for other generalized Fokker-Planck equations as particular cases.

The Fokker–Planck equation for the time-changed fractional Ornstein–Uhlenbeck stochastic process

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

Reply to "Comment on 'Non-Markovian harmonic oscillator across a magnetic field and time-dependent force fields' ".

Recently in a paper by Hidalgo-Gonzalez and Jiménez-Aquino [Phys. Rev. E 100, 062102 (2019)PREHBM2470-004510.1103/PhysRevE.100.062102], the generalized Fokker-Planck equation (GFPE) for a Brownian harmonic oscillator in a constant magnetic field and under the action of time-dependent force fields, has been explicitly calculated using the characteristic function method. Although the problem is linear it is not easy to solve, however, the method of the characteristic function is effective and allows to obtain an exact and precise solution of the problem. Our theoretical result has been compared with the one reported by Das etal. in a recently submitted paper [arXiv:2011.09771] using another solution method. The proposed method consists in constructing the GFPE and then calculating each time-dependent coefficient associated with this equation. However, in a more complicated case, one cannot know a priori the exact number of terms that this equation must contain. The precise number is further provided by the characteristic function method.

Three transport models for charged particles in three-dimensional semiconductors driven by a fractional noise

In this work, three models of the motion of charged particles in three-dimensional semiconductors governed by a stochastic differential equation driven by a magnetic field and an intrinsic fractional Gaussian noise have been introduced. Based on the general expansion theorem for the Laplace transform, the expressions of the average position and the average velocity of a charged particle have been obtained. The expressions of the complex susceptibilities, the spectral amplification, the stationary form of current density, as well as power absorption also have been obtained. It is worthy to note that the cyclotron resonance, stochastic dynamics of a charged particle could be induced by fractional noise. Furthermore, the expressions of variances and the generalized Fokker–Planck equation (GFPE) for the non-Markovian dynamics also have been investigated.

Stationary distribution and probability density function of a stochastic SVIS epidemic model with standard incidence and vaccination strategies.

Considering the great effect of vaccination and the unpredictability of environmental variations in nature, a stochastic Susceptible-Vaccinated-Infected-Susceptible (SVIS) epidemic model with standard incidence and vaccination strategies is the focus of the present study. By constructing a series of appropriate Lyapunov functions, the sufficient criterion is obtained for the existence and uniqueness of the ergodic stationary distribution of the model. In epidemiology, the existence of a stationary distribution indicates that the disease will be persistent in a long term. By taking the stochasticity into account, a quasi-endemic equilibrium related to the endemic equilibrium of the deterministic system is defined. By means of the method developed in solving the general three-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In statistical significance, the explicit density function can reflect all dynamical properties of an epidemic system. Next, a simple result of disease extinction is obtained. In addition, several numerical simulations and parameter analyses are performed to illustrate the theoretical results. Finally, the corresponding results and conclusions are discussed at the end of the paper.

Generalized Fokker-Planck equations derived from nonextensive entropies asymptotically equivalent to Boltzmann-Gibbs.

We derive generalized Fokker-Planck equations (FPEs) based on two nonextensive entropy measures S_{±} that depend exclusively on the probability. These entropies have been originally obtained from the superstatistics framework, therefore they regard nonequilibrium systems outlined by a long-term stationary state in view of a spatiotemporally fluctuating intensive quantity. Moreover, entropies S_{±} as well as Boltzmann-Gibbs (BG) entropy S_{B} both pertain to the same asymptotical equivalence class, thus suggesting that S_{±} could depict a consistent thermodynamic generalization of BG. For these reasons, we assert that transport phenomena to be accounted for by our models shall coincide with the portrait given by the conventional FPEs for systems comprehending short-range interactions or a high number of accessible microstates, whereas, for systems composed of a small number of microstates, or those with long-range interactions, the governing equations of motion are to be the FPEs here derived, as long as the system fulfills the attributes mentioned above. We discuss the anomalous diffusion exhibited by the two generalized FPEs and also present some numerical applications. In particular, we find that there are models regarding biological sciences, for the study of congregation and aggregation behavior, the structure of which coincides with the one of our models.

Time-changed fractional Ornstein-Uhlenbeck process

Abstract We define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process.