Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.     

Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker–Planck and Fractional Equations

The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function \(f_j(r,t)\) of a particle moving on a substrate with time delays \(\tau _j\). Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability \(P_j\) is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, \(P_j \propto f_j^{\nu - 1}\), the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, \(P_j \propto \tau _j^{-1-\alpha }\) (with \(0< \alpha < 2\)), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.     

Fokker–Planck equation analysis of randomly excited nonlinear energy harvester

The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker–Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results.     

A review of Vlasov–Fokker–Planck numerical modeling of inertial confinement fusion plasma

The interaction of intense lasers with solid matter generates a hot plasma state that is well described by the Vlasov–Fokker–Planck equation. Accurate and efficient modeling of the physics in these scenarios is highly pertinent, because it relates to experimental campaigns to produce energy by inertial confinement fusion on facilities such as the National Ignition Facility. Calculations involving the Vlasov–Fokker–Planck equation are computationally intensive, but are crucial to proper understanding of a wide variety of physical effects and instabilities in inertial fusion plasmas. In this topical review, we will introduce the background physics related to Vlasov–Fokker–Planck simulation, and then proceed to describe results from numerical simulation of inertial fusion plasma in a pedagogical manner by discussing some key numerical algorithm developments that enabled the research to take place. A qualitative comparison of the techniques is also given.     

Similarity solutions of the Fokker–Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker–Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker–Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker–Planck equations are presented, and their properties analyzed.     

On Dynamics of Brans–Dicke Theory of Gravitation

We study longstanding problem of cosmological clock in the context of Brans–Dicke theory of gravitation. We present the Hamiltonian formulation of the theory for a class of spatially homogeneous cosmological models. Then, we show that formulation of the Brans–Dicke theory in the Einstein frame allows how an identification of an appropriate cosmological time variable, as a function of the scalar field in the theory, can be emerged in quantum cosmology. The classical and quantum results are applied to the Friedmann–Robertson–Walker cosmological models.     

Numerical approximation of the Vlasov–Maxwell–Fokker–Planck system in two dimensions

A numerical method is developed for solving the Vlasov–Maxwell–Fokker–Planck system in two spatial dimensions. This system of equations is a model for a collisional plasma in the presence of a self consistent electromagnetic field. The numerical procedure is a type of deterministic particle method and is an extension to include the full electromagnetic field of the approximation method of Wollman and Ozizmir [S. Wollman, E. Ozizmir, Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in two dimensions, J. Comput. Phys. 228 (2009) 6629–6669]. In addition, the long time asymptotic behavior of solutions is studied. It is determined that the solution to the Vlasov–Maxwell–Fokker–Planck system converges to the same steady state solution as that for the Vlasov–Poisson–Fokker–Planck system.     

Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in two dimensions

A numerical method is developed for approximating the solution to the Vlasov–Poisson–Fokker–Planck system in two spatial dimensions. The method generalizes the approximation for the system in one dimension given in [S. Wollman, E. Ozizmir, Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. The numerical procedure is based on a change of variables that puts the convection–diffusion equation into a form so that finite difference methods for parabolic type partial differential equations can be applied. The computational cycle combines a type of deterministic particle method with a periodic interpolation of the solution along particle trajectories onto a fixed grid. computational work is done to demonstrate the accuracy and effectiveness of the approximation method. Parts of the numerical procedure are adapted to run on a parallel computer.     

The Fokker–Planck Equation, and Stationary Densities

The most general local Markovian stochastic model is investigated, for which it is known that the evolution equation is the Fokker–Planck equation. Special cases are investigated where uncorrelated initial states remain uncorrelated. Finally, stochastic one-dimensional fields with local interactions are studied that have kink-solutions.     

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.     

Stability analysis of implicit time discretizations for the Compton-scattering Fokker–Planck equation

The Fokker–Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker–Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.     

A Vlasov–Fokker–Planck code for high energy density physics

OSHUN is a parallel relativistic 2D3P Vlasov–Fokker–Planck code, developed primarily to study electron transport and instabilities pertaining to laser-produced—including laser-fusion—plasmas. It incorporates a spherical harmonic expansion of the electron distribution function, where the number of terms is an input parameter that determines the angular resolution in momentum-space. The algorithm employs the full 3D electromagnetic fields and a rigorous linearized Fokker–Planck collision operator. The numerical scheme conserves energy and number density. This enables simulations for plasmas with temperatures from MeV down to a few eV and densities from less than critical to more than solid. Kinetic phenomena as well as electron transport physics can be recovered accurately and efficiently.