Nonlocal Fokker-Planck Equation Research Articles

A mixed SOC-turbulence model for nonlocal transport and Lévy-fractional Fokker–Planck equation

The phenomena of nonlocal transport in magnetically confined plasma are theoretically analyzed. A hybrid model is proposed, which brings together the notion of inverse energy cascade, typical of drift-wave- and two-dimensional fluid turbulence, and the ideas of avalanching behavior, associable with self-organized critical (SOC) behavior. Using statistical arguments, it is shown that an amplification mechanism is needed to introduce nonlocality into dynamics. We obtain a consistent derivation of nonlocal Fokker-Planck equation with space-fractional derivatives from a stochastic Markov process with the transition probabilities defined in reciprocal space. The hybrid model observes the Sparre Andersen universality and defines a new universality class of SOC.

Elementary bifurcations for a simple dynamical system under non-Gaussian Lévy noises

Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies.A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian α-stable Lévy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by solving a nonlocal Fokker-Planck equation numerically. This allows numerically investigating phenomenological bifurcation or P-bifurcation, for stochastic differential equations with non-Gaussian Lévy noises.

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior.

Ensemble Modeling of Hydrologic and Hydraulic Processes at One Shot: Application to Kinematic Open-Channel Flow under Uncertain Channel Properties and Uncertain Lateral Flow Conditions by the Stochastic Method of Characteristics

A stochastic kinematic wave model for open channel flow is developed under uncertain channel properties and uncertain lateral flow conditions. Applying a known methodology, the Fokker-Planck equation (FPE) of the kinematic open-channel flow process under uncertain channel properties and uncertain lateral flow conditions is derived using the method of characteristics. Because every stochastic partial differential equation has a one-to-one relationship with a nonlocal Lagrangian-Eulerian Fokker-Planck equation (LEFPE), the LEFPE for the governing equation of any hydrologic or hydraulic process can be developed as the physically based stochastic model of the particular process. To quantify the ensemble behavior of a process, LEFPE provides a quantitative description of the time-space evolution of the probability density function of the state variables of the process at one shot. The nonlocal LEFPE reduces to the classical local FPE, which is more convenient to solve, under certain assumptions. The developed ...

Non-local quantum-kinetic equation for laser-excited semiconductors

A generalized non-local Fokker-Planck type kinetic equation obtained on the basis of quantum-mechanical formalism is used to study the conduction electron dynamics in model semiconductors excited by spatially inhomogeneous pulsed laser irradiation. In this approach the electrons are assumed to undergo a drift-diffusion motion in the energy-position manifold.

Inertial manifolds for a Smoluchowski equation on a circle

The existence of inertial manifolds for a Smoluchowski equation—a nonlinear and nonlocal Fokker–Planck equation which arises in the modelling of colloidal suspensions—is investigated. The difficulty due to first-order derivatives in the nonlinearity is circumvented by a transformation.

Numerical Analysis of Electronic Transport Characteristics in Dielectrics Irradiated by Ultrashort Pulsed Laser Using the Nonlocal Fokker-Planck Equation

ABSTRACT The ultimate goal of this article lies in investigating theoretically the electronic transport characteristics in fused silica (SiO2) irradiated by ultrashort pulsed lasers. The nonlocal type of multivariate Fokker-Planck equation is modeled on the basis of the Boltzmann transport formalism to describe the ultrashort pulsed laser-induced damage phenomena in the energy-position space, together with avalanche ionization, three-body recombination, and multiphoton ionization. From the results, it is observed that the recombination becomes prominent and contributes to reduce substantially the rate of increase in electron number density when the electron density exceeds a certain threshold. With very intense laser irradiation, a strong absorption of laser energy takes place and an initially transparent solid is converted to a metallic state, a phenomenon well known as laser-induced breakdown. It is also found that full ionization is provided at intensities above threshold, and all further laser energy is deposited within a thin skin depth. The absorption length is of the order of a wavelength at very high laser fluence, and it becomes thinner as laser fluence is larger. This is because the absorbed energy is no longer consumed for multiphoton ionization, but rather leads to a drastic increase in the absorption coefficient because of Joule heating.

Phase noise and timing jitter in oscillators with colored-noise sources

Phase noise or timing jitter in oscillators is of major concern in wireless and optical communications, being a major contributor to the bit-error rate of communication systems, and creating synchronization problems in other clocked and sampled-data systems. This paper presents the theory and practical characterization of phase noise in oscillators due to colored, as opposed to white, noise sources. Shot and thermal noise sources in oscillators can be modeled as white-noise sources for all practical purposes. The characterization of phase noise in oscillators due to shot and thermal noise sources is covered by a recently developed theory of phase noise due to white-noise sources. The extension of this theory and the practical characterization techniques to noise sources in oscillators, which have a colored spectral density, e.g., 1/f noise, is crucial for practical applications. In this paper, we first derive a stochastic characterization of phase noise in oscillators due to colored-noise sources. This stochastic analysis is based on a novel nonlinear perturbation analysis for autonomous systems, and a nonlocal Fokker-Planck equation we derive. Then, we calculate the resulting spectrum of the oscillator output with phase noise as characterized. We also extend our results to the case when both white and colored-noise sources are present. Our treatment of phase noise due to colored-noise sources is general, i.e., it is not specific to a particular type of colored-noise source.

Growth and accretion of mass in an astrophysical model, II

Radially symmetric solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles in a bounded container are studied. Conditions ensuring either global-in-time existence of solutions or their finite time blow up are g

Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker-Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the efficient construction of these schemes in higher dimensions while preserving their structural properties. Numerical experiments validate the schemes and show their ability to handle complicated phenomena typical in aggregation-diffusion equations, such as free boundaries, metastability, merging and phase transitions.