Fokker-Planck Operator Research Articles

Low temperature extension of the generalized Zusman phase space equations for electron transfer.

In a previous paper [J. Chem. Phys. 119, 11864 (2003)], we derived a set of two coupled equations which describe electron transfer in the presence of dissipation at high temperature. Employing the low temperature extension of the Fokker-Planck operator, suggested by Haake and Reibold [Phys. Rev. A 32, 2462 (1985)] and Ankerhold [Europhys. Lett. 61, 301 (2003)], we show that one may extend the generalized Zusman equations in a similar manner to low temperature. Numerical simulation shows that addition of the temperature-dependent term which couples the coordinate and momentum causes an increase in the electron transfer rate as compared to the rate obtained from the previous high temperature equations. The increase in the rate comes from the increase in the equilibrium variances of the coordinate and momentum. The low temperature quantum theory allows for higher energy portions of phase space to contribute to the electron transfer rate where the rate is higher thus enhancing the overall rate.

Bimodal approximation for anomalous diffusion in a potential.

Exact and approximate solutions of the fractional diffusion equation for an assembly of fixed-axis dipoles are derived for anomalous noninertial rotational diffusion in a double-well potential. It is shown that knowledge of three time constants characterizing the normal diffusion, viz., the integral relaxation time, the effective relaxation time, and the inverse of the smallest eigenvalue of the Fokker-Planck operator, is sufficient to accurately predict the anomalous relaxation behavior for all time scales of interest.

CFTs of SLEs: the radial case

We present a relation between conformal field theories (CFT) and radial stochastic Schramm–Loewner evolutions (SLE) similar to that we previously developed for the chordal SLEs. We construct an important local martingale using degenerate representations of the Virasoro algebra. We sketch how to compute derivative exponants and the restriction martingales in this framework. In its CFT formulation, the SLE dual Fokker–Planck operator acts as the two-particle Calogero Hamiltonian on boundary primary fields and as the dilatation operator on bulk primary fields localized at the fixed point of the SLE map.

On the Harnack inequality for a class of hypoelliptic evolution equations

We give a direct proof of the Harnack inequality for a class of degenerate evolution operators which contains the linearized prototypes of the Kolmogorov and Fokker-Planck operators. We also improve the known results in that we find explicitly the optimal constant of the inequality.

Global L1 theory and regularity for the 3D nonlinear Wigner–Poisson–Fokker–Planck system

A global existence, uniqueness and regularity theorem is proved for the simplest Markovian Wigner–Poisson–Fokker–Planck model incorporating friction and dissipation mechanisms. The proof relies on Green function and energy estimates under mild formulation, making essential use of the Husimi function and the elliptic regularization of the Fokker–Planck operator.

Hydrodynamics of driven granular gases.

Hydrodynamic equations for granular gases driven by the Fokker-Planck operator are derived. Transport coefficients appeared in Navier-Stokes order change from the values of a free-cooling state to those of a steady state. The mismatch between the granular temperature and the constant temperature of the heat bath produces a nontrivial steady state.

The Fokker–Planck operator at a continuous phase transition

I consider a physical system described by a continuous field theory and enclosed in a large but finite cubical box with periodic boundary conditions. The system is assumed to undergo a continuous phase transition at some critical point. The \phi^4 theory that is a continuous version of the Ising model is such a system but there are many other examples corresponding to higher spin, higher symmetry etc. The eigenfunctions of the corresponding Fokker-Planck operator can be chosen, of course, to be eigenfunctions of the momentum operator. It is shown that the eigenvalues of the FP operator, corresponding to each eigenvalue q of the momentum operator, evaluated at a transition point of the finite system, accumulate at zero, when the size of the system tends to infinity. There are many reasonable ways of defining a critical temperature of a finite system, that tends to the critical temperature of the infinite system as the size of the system tends to infinity. The accumulation of eigenvalues is neither affected by the specific choice of critical temperature of the finite system nor by whether the system is below or above its upper critical dimension.

Large-aspect-ratio limit of neoclassical transport theory

This paper presents a comprehensive description of neoclassical transport theory in the banana regime for large-aspect-ratio flux surfaces of arbitrary shapes. The method of matched-asymptotic expansions is used to obtain analytical solutions for plasma distribution functions and to compute transport coefficients. The method provides justification for retaining only the part of the Fokker-Planck operator that involves the second derivative with respect to the cosine of the pitch angle for the trapped and barely circulating particles. It leads to a simple equation for the freely circulating particles with boundary conditions that embody a discontinuity separating particles moving in opposite directions. Corrections to the transport coefficients are obtained by generalizing an existing boundary layer analysis. The system of moment and field equations is consistently taken in the cylinder limit, which facilitates the discussion of the treatment of dynamical constraints. It is shown that the nonlocal nature of Ohm's law in neoclassical theory renders the mathematical problem of plasma transport with changing flux surfaces nonstandard.

Analytic calculation of the longitudinal dynamic susceptibility of uniaxial superparamagnetic particles in a strong uniform DC magnetic field

A simple analytic equation which allows one to obtain a qualitative understanding of the physical processes underlying the behavior of the longitudinal complex susceptibility χ ||( ω) of an assembly of noninteracting uniaxial superparamagnetic particles subjected to a strong external uniform DC magnetic field is derived in the context of Brown's model of magnetic relaxation. It is shown that a knowledge of 3 time constants characterizing the magnetization relaxation, viz. the integral relaxation time τ, the effective relaxation time τ ef, and the inverse of the smallest eigenvalue λ 1 of the Fokker–Planck operator are sufficient to accurately predict the spectrum of χ ||( ω) in all frequency ranges of interest as well as the behavior of the equilibrium correlation function of the longitudinal component of the magnetization C ||( t) in the time domain at all values of the barrier and external field parameters.

The Milne problem for the linear Fokker–Planck operator with a force term

AbstractThis paper deals with the mathematical analysis of the linear stationary Fokker–Planck equation in a half‐space (also called ‘Milne’ problem), in presence of an external electrostatic force field. We prove existence, uniqueness and asymptotic properties of the solution. Copyright © 2003 John Wiley & Sons, Ltd.

Hypoelliptic regularity in kinetic equations

We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced by Hörmander for hypoelliptic operators. In contrast with averaging, the number of derivatives does not depend on the L p space considered. Three type of proofs are provided: one relies on the Fourier transform, another one uses Hörmander's commutators, and the last uses a characteristics commutator. Regularity of averages in velocity are deduced. We apply our method to the linear Fokker–Planck operator and recover the known optimal regularity, by direct estimates using Hörmander's commutator.

Modeling of high energy ion implantation based on splitting of the Boltzmann transport equation

For the straight-ahead form of the Boltzmann transport equation, describing high energy heavy ion transport in matter, the splitting in the Boltzmann–Fokker–Planck equation is considered. The splitting consists of decomposition of differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. An approximation of the scattering integral with singular differential cross-section by the Fokker–Planck operator of second order, including the energy straggling term, is considered. For this form of the Fokker–Planck equation boundary conditions, preserving the number of particles in flux, and the energy dissipated in target are suggested. An effective algorithm for numerical integration of the Boltzmann–Fokker–Planck equation, which preserves the number of particles in flux, and the energy deposited in target is developed. The convergence of the implantation profile, nuclear and electronic energy deposition, calculated from the Boltzmann–Fokker–Planck equation, to the respective exact distributions, calculated from the Monte-Carlo method, was investigated for various values of the splitting parameter for the different ion-target mass ratios and for a large energy interval. For the universal potential, it is shown the existence of an optimal value of splitting parameter, at which solutions of the Boltzmann–Fokker–Planck equation accurately approximate solutions of the exact equation with minimal computational efforts.

Solutions of a class of non-Markovian Fokker-Planck equations.

We show that a formal solution of a rather general non-Markovian Fokker-Planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the Markovian equation with the same Fokker-Planck operator. This allows us to classify memory kernels into safe ones, for which the solution is always a probability density, and dangerous ones, when this is not guaranteed. The first situation describes random processes subordinated to a Wiener process, while the second one typically corresponds to random processes showing a strong ballistic component. In this case the non-Markovian Fokker-Planck equation is only valid in a restricted range of parameters, initial and boundary conditions.

Spectral Properties of Hypoelliptic Operators

We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = sum_{i=1}^m X_i^T X_i + X_0 + f, where the X_j denote first order differential operators, f is a function with at most polynomial growth, and X_i^T denotes the formal adjoint of X_i in L^2. For any e > 0 we show that an inequality of the form |u|_{delta,delta} <= C(|u|_{0,eps} + |(K+iy)u|_{0,0}) holds for suitable delta and C which are independent of y in R, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Herau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x >= |y|^tau-c, tau in (0,1], c in R}.

The Fokker-Planck operator as an asymptotic limit in anisotropic media

We derive the Fokker-Planck operator describing a highly forward-peaked scattering process in the linear transport equation, in anisotropic media, as a formal asymptotic limit of the exact integral operator. The resulting operator, being both convective and diffusive in angle and energy variables, reduces the degenerate nature of the Fokker-Planck system compared to the isotropic media case where the corresponding limit is only convective in energy and diffusive in angle.

Nonlinear fluid closure: Three-mode slab ion temperature gradient problem with diffusion

The three-mode slab ion temperature gradient problem was considered. Starting from the drift kinetic equation with nonlinear term and diffusion, the hierarchy of fluid equations up to fourth moment was developed. As a closure, the nonlinear fluid closure by N. Mattor and S. Parker [Phys. Rev. Lett. 79, 3419 (1997)] was applied. Numerical solutions of the system of fluid equations have been obtained and analyzed. The time evolution of electrostatic potential shows that nonlinear fluid closure is able to capture particle trapping, which is important for fusion plasmas. Great attention was paid to studies of the role of diffusion. Diffusion here represents effects of background turbulence and can be described by a Fokker–Planck operator [A. Zagorodny and J. Weiland, Phys. Plasmas 6, 2359 (1999)]. The three wave system can be considered as a system of test waves in a turbulent background. This system can be used to study situations of varying partial coherence.

NUMERICAL RESOLUTION OF AN ION–ELECTRON COLLISION OPERATOR IN AXISYMMETRICAL GEOMETRY

ABSTRACT In this paper, we are interested in the kinetic model which describes the kinetic phenomena in the field of the Inertial Confinement Fusion (ICF) at the center of a microball of deuterium–tritium when the hot point is created. Especially, we recall the notion of entropic average designed to solve a non linear Fokker–Planck operator describing the ion–electron interactions in axisymmetrical geometry. Finally, we show numerical results.

Trace formula for noisy flows

A trace formula is derived for the Fokker–Planck equation associated with Ito stochastic differential equations describing noisy time-continuous nonlinear dynamical systems. In the weak-noise limit, the trace formula provides estimations of the eigenvalues of the Fokker–Planck operator on the basis of the Pollicott–Ruelle resonances of the noiseless deterministic system, which is assumed to be non-bifurcating. At first order in the noise amplitude, the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo-energy of the Onsager–Machlup–Freidlin–Wentzell scheme.

Thermodynamics and fractional Fokker-Planck equations

The relaxation to equilibrium in many systems that show strange kinetics is described by fractional Fokker-Planck equations (FFPEs). These can be considered as phenomenological equations of linear nonequilibrium theory. We show that the FFPEs describe a system whose noise in equilibrium fulfills the Nyquist theorem. Moreover, we show that for subdiffusive dynamics, the solutions of the corresponding FFPEs are probability densities for all cases in which the solutions of the normal Fokker-Planck equation (with the same Fokker-Planck operator and with the same initial and boundary conditions) exist. The solutions of the FFPEs for superdiffusive dynamics are not always probability densities. This fact means only that the corresponding kinetic coefficients are incompatible with each other and with the initial conditions.

Derivation of the basic system of kinetic equations governing superparamagnetic relaxation by the use of the adjoint Fokker-Planck operator

The basic equations governing the kinetics of superparamagnets (and dielectric lattices in the Debye approximation) are derived by taking advantage of the properties of the adjoint Fokker-Planck operator. The equations obtained, which duly reduce to those derived by direct averaging of the corresponding stochastic Landau-Lifshitz-type equations, are expressed in terms of the coefficients determining the action of the angular momentum operator and the spin variables on the spherical harmonics. The structure of the equations so obtained directly reflects that of the adjoint Fokker-Planck operator, which permits to trace back easily the origin of the different contributions (precession and relaxation). This makes the method specially suitable for generalization to more elaborate descriptions of rotational relaxation.