Time-dependent Fokker-Planck Research Articles

Time-dependent models for dark matter at the galactic center

The prospects for indirect detection of dark matter at the galactic center with $\ensuremath{\gamma}$-ray experiments like the space telescope GLAST, and air Cherenkov telescopes like HESS, CANGAROO, MAGIC and VERITAS depend sensitively on the mass profile within the inner parsec. We calculate the distribution of dark matter on subparsec scales by integrating the time-dependent Fokker-Planck equation, including the effects of self-annihilations, scattering of dark matter particles by stars, and capture in the supermassive black hole. We consider a variety of initial dark matter distributions, including models with very high densities (``spikes'') near the black hole, and models with ``adiabatic compression'' of the baryons. The annihilation signal after ${10}^{10}\text{ }\text{ }\mathrm{yr}$ is found to be substantially reduced from its initial value, but in dark matter models with an initial spike, order-of-magnitude enhancements can persist compared with the rate in spike-free models.

L p estimates on a time-inhomogeneous diffusion process

In this paper we consider the diffusion process X determined by the one-variable time-dependent Fokker-Planck equation, (∂∕∂t)P(y,t)=−g(t)(∂∕∂y)yP(y,t)+12f(t)2(∂2∕∂y2)P(y,t), where f,g:R+→R+ are two continuous functions, i.e., X satisfies Itô stochastic differential equation, dXt=f(t)dBt−g(t)Xtdt, where B is a standard Brownian motion starting at zero. We obtain Lp estimates on the process, and we show that ∥log(1+Jτ)∥p and ∥sup0⩽t⩽τ∣a(t)Xt∣∥p are equivalent for all stopping times τ of B, where Jt=∫0ta2(s)f2(s)ds and a:R+→R+ the solution to the equation (da∕dt)−g(t)a=−a3f2(t),a(0)=1.

Memoryless control of boundary concentrations of diffusing particles

Flux between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Diffusion theory has calculated that flux since the time of Fick (1855), and the flux has been known to arise from the stochastic behavior of Brownian trajectories since the time of Einstein (1905), yet the mathematical description of the behavior of trajectories corresponding to different types of boundaries is not complete. We consider the trajectories of noninteracting particles diffusing in a finite region connecting two baths of fixed concentrations. Inside the region, the trajectories of diffusing particles are governed by the Langevin equation. To maintain average concentrations at the boundaries of the region at their values in the baths, a control mechanism is needed to set the boundary dynamics of the trajectories. Different control mechanisms are used in Langevin and Brownian simulations of such systems. We analyze models of controllers and derive equations for the time evolution and spatial distribution of particles inside the domain. Our analysis shows a distinct difference between the time evolution and the steady state concentrations. While the time evolution of the density is governed by an integral operator, the spatial distribution is governed by the familiar Fokker-Planck operator. The boundary conditions for the time dependent density depend on the model of the controller; however, this dependence disappears in the steady state, if the controller is of a renewal type. Renewal-type controllers, however, produce spurious boundary layers that can be catastrophic in simulations of charged particles, because even a tiny net charge can have global effects. The design of a nonrenewal controller that maintains concentrations of noninteracting particles without creating spurious boundary layers at the interface requires the solution of the time-dependent Fokker-Planck equation with absorption of outgoing trajectories and a source of ingoing trajectories on the boundary (the so called albedo problem).

Effects of noise on the phase dynamics of nonlinear oscillators

Various properties of human rhythmic movements have been successfully modeled using nonlinear oscillators. However, despite some extensions towards stochastical differential equations, these models do not comprise different statistical features that can be explained by nondynamical statistics. For instance, one observes certain lag one serial correlation functions for consecutive periods during periodic motion. This work aims at an extension of dynamical descriptions in terms of stochastically forced nonlinear oscillators such as $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{\ensuremath{\xi}}+{\ensuremath{\omega}}_{0}^{2}\ensuremath{\xi}=n(\ensuremath{\xi},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\xi}})+q(\ensuremath{\xi},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\xi}})\ensuremath{\Psi}(t)$, where the nonlinear function $n(\ensuremath{\xi},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\xi}})$ generates a limit cycle and $\ensuremath{\Psi}(t)$ denotes colored noise that is multiplied via $q(\ensuremath{\xi},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\xi}})$. Nonlinear self-excited systems have been frequently investigated, particularly emphasizing stability properties and amplitude evolution. Thus, one can focus on the effects of noise on the frequency or phase dynamics that can be analyzed by use of time-dependent Fokker-Planck equations. It can be shown that noise multiplied via polynoms of arbitrary finite order cannot generate the desired period correlation but predominantly results in phase diffusion. The system is extended in terms of forced oscillators in order to find a minimal model producing the required error correction.

Numerical method for the nonlinear Fokker-Planck equation

A practical method based on distributed approximating functionals ~DAFs! is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo @Phys. Rev. E 54, 931 ~1996!# to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the ‘‘flat’’ region in the kinetic potential fails for the ‘

Solution of nonlinear Fokker-Planck equations.

A finite-difference method for solving a general class of linear and nonlinear time-dependent Fokker-Planck equations, which is based on a K-point Stirling interpolation formula, is suggested. It has a fifth-order convergence in time and a 2Kth-order convergence in space and allows one to achieve a given level of accuracy with a slow (or even without) increase in the number of grid points. The most appealing features of the method are perhaps that it is norm conserved, and equilibrium preserving in the sense that every equilibrium solution of the analytic equations is also an equilibrium solution of the discretized equations. The method is applied to a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1983)], which exhibits a phase transition. The results are compared with those obtained with other methods that rely on not too well controlled approximations. Our finite-difference scheme permits us to establish the region of validity and the limitations of those approximations. The nonlinearity of the system is found to be an obstacle for the application of Suzuki's scaling ideas, which are known to be suitable for linear problems. But what is most remarkable is that this nonlinearity allows for transient bimodality in a globally monostable case, even though there is no ``flat'' region in the potential. \textcopyright{} 1996 The American Physical Society.

D-T FUSION REACTIVITY VARIATION INDUCED BY NBI-PRODUCED NON-MAXWELLIAN DISTRIBUTION

By numerically solving the tow-dimensional time-dependent non-linear Fokker-Planck equations, the distribution functions are calculated and the D-T fusion reactivity are evaluated. For the tritium parallel injection case, the reactivity variation induced by the non-Maxwellian distribution is analyzed. It is pointed out that it is appropriate to define. The calculation resutlts show that η increases from 1 to maximum and then decreases during NBI heating. Either high input power or high beam energy can make η to decrease to a value less than 1. When η decreases to 1, Ｔ∥Ｔ is raised to about 10keV.

Fokker-Planck analysis of short-pulse laser heating in the hot-spot-model approximation.

The electron distribution function in high-power, short-pulse, laser-produced plasmas is predicted to be significantly non-Maxwellian during plasma heating. This directly affects radiation production and ionized-state populations and alters the plasma heating rates and transport coefficients. Here we numerically solve a time-dependent Fokker-Planck equation to calculate electron energy distributions for a near-critical-density selenium plasma irradiated by a 2 ps pulse width KrF laser with powers between 5\ifmmode\times\else\texttimes\fi{}${10}^{14}$ and 7\ifmmode\times\else\texttimes\fi{}${10}^{16}$ W ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$. Distributions are calculated locally, with plasma heating treated as in the hot-spot model [K. G. Whitney and J. Davis, J. App. Phys. 45, 5294 (1974)], which focuses on a small, stationary volume element heated by inverse bremsstrahlung and cooled by heat conduction and electron-ion energy exchange. Inelastic electron-ion collisions, previously seen to have little influence on the distribution, and bremsstrahlung cooling, which is negligible in magnitude, are not included. Intense, long-pulse inverse-bremsstrahlung heating produces non-Maxwellian distributions of a well-known, depleted-tail form [A. B. Langdon, Phys. Rev. Lett. 44, 575 (1980)]. We show that in the present case, because of short time scales and high plasma density, these do not accurately represent the electron distribution. We obtain actual distributions both during and several picoseconds after the laser pulse and determine the resulting effect on the laser absorption coefficient, heat conductivity, and electron-ion coupling rate as a function of the peak intensity of the laser pulse. It is found that the non-Maxwellian distribution results in greater heating; a reduction in the laser absorption coefficient is more than balanced by the reduction in thermal conductivity. Electron-ion coupling is generally unaffected (10%) by the non-Maxwellian distribution. These effects increase as a function of laser intensity.

Stochastic Criteria for Ignition of Single Particles

Abstract A stochastic analysis of panicle ignition has been given using a Fokker-Planck equation. Under critical conditions the steady slate probability density of particle temperature is found to be bimodal. A numerical solution to the time-dependent Fokker-Planck equation yields an estimate of the time scale over which a transition (’stochastic ignition“) from the kinetic to the diffusion regime takes place.

Rapid temporal evolution of radiation from nonthermal electrons in solar flares

Solutions of the time dependent Fokker-Planck equation was found for accelerated electrons undergoing Coulomb collisions in a magnetized, fully ionized plasma. An exact solution was found for arbitrary pitch angle and energy distribution in a uniform background plasma. Then, for an inhomogeneous plasma, a solution was found for particles with small pitch angles. These solutions were used to calculate the temporal evolution of bremsstrahlung x-rays from short bursts of nonthermal electron beams, and these spectra were compared with observed high time resolution spectra of short timescale solar hard x-ray bursts. It is shown that the observed softening in time of the spectra rules out a homogeneous background and therefore the possibility of electrons being confined to the corona either because of converging magnetic field or high densities. The inhomogeneous solution was also applied to a model with constant coronal density and exponentially rising chromospheric density. The spectra are shown to be consistent with that produced by a collimated beam of electrons accelerated in the corona with certain given conditions. These conditions could be violated if large pitch angle electrons are present.

Simulation for electron acceleration by DC electric field in the presence of ion sound waves and associated hard X-ray emission

Time-dependent Fokker-Planck equation was numerically solved to demonstrate the dynamics of electrons in a uniform coronal loop with an applied axial DC electric field in the presence of ion-sound waves. This electric field is attributed to an anomalous resistivity due to the ion-sound turbulence caused by an initially given critical current density.

A theoretical determination of the diffusion-like ionisation time of Rydberg atoms

The non-stationary multistep (diffusion-like) ionisation of Rydberg atoms is considered theoretically. The investigation is based on the time-dependent Fokker-Planck equation and is related to collisional and microwave ionisation of Rydberg atoms. Analyti- cal expressions for the mean time and the higher moments of the distribution of the diffusive ionisation times are derived. The conditions for the occurrence of diffusive ionisation are obtained. The relation of the present study to stationary diffusive ionisation and the experimental investigation of the ionisation of Rydberg atoms is discussed.

Random superharmonic response of a Duffing oscillator

The third order superharmonic response of a Duffing oscillator to narrow band random excitation is analyzed. The analysis shows the effect of excitation bandwidth on the response and stability of the non-linear oscillator. In particular the analysis shows that multivalued superharmonic response can occur only when the excitation bandwidth is small. Stability of the response is examined by constructing a mean square phase plane from a time dependent Fokker-Planck equation and also by perturbing the stationary solutions.

Synchronization Failures in a Chain of PLL Synchronizers

In digital transmission systems employing PLL repeaters, timing jitter is produced mostly by intersymbol interference (ISI). Despite the fact that IS1 represents a narrow-band disturbance to the PLL for a long chain of N repeaters, it is responsible for cycle slips of the PLL repeater. It is demonstrated that the cycle slip rate is the key parameter determining the error performance of the system. It is shown that the accumulated jitter for a large chain becomes a narrow-band process, which can be replaced by an equivalent low-pass jitter-noise process, which lends itself to analytical treatment. Numerical results of the cycle slip rate are given, based on the solution of a time-dependent Fokker-Planck equation. The results are useful for a wide range of loop transfer functions and phase-detector characteristics. The theoretical findings are confirmed by experimental results.

Green's function approach to the solution of the time dependent Fokker-Planck equation with an absorbing boundary

The solution of the time dependent multivelocity Fokker-Planck equation in plane geometry with an absorbing boundary is formulated in terms of the infinite medium Green's function and the boundary distribution. The boundary distribution is to be determined by solving an integral equation. It is shown that the velocity moments such as density, current and energy density can be expressed in terms of three reduced distributions which depend on the longitudinal velocity component only. Numerical results for monovelocity pulsed and steady sources are presented.

Discretization methods for one-dimensional Fokker-Planck operators

Abstract Several new numerical methods for solving a general class of linear and nonlinear 1-dimensional time-dependent Fokker-Planck equations are suggested. These methods are positive, simple to implement, and equilibrium-preserving in the sense that every equilibrium solution of the analytic equations is also an equilibrium solution of the discretized equations. The new methods are all applied to the nonlinear problem of Compton and inverse Compton scattering, and numerical results are compared. Our numerical comparison includes results from a standard technique now in use (the Chang-Cooper method), and we discuss the relationship between this technique and the new methods. In addition, we point out certain shortcomings of the Chang-Cooper technique which these new methods remedy.

Fokker-Planck calculations for JET ICRF heating scenarios

Flux-averaged ICRF heating of the JET tokamak is considered, utilizing an anisotropic, time-dependent Fokker-Planck code. Alternative ICRF heating scenarios are considered, and fundamental minority deuterium and second-harmonic tritium and deuterium cases are selected for detailed analysis of wave accessibility, strong single-pass damping and enhanced ion-tail-produced fusions. It is found that enhanced fusions, resulting from heated minority deuterium ion tails, and second-harmonic deuterium heating, yielding fusion power gains near Q = 1, can be achieved near the plasma core with moderate absorbed power levels, pulse widths and energy confinement times. The corresponding case for second-harmonic tritium heating located on the major axis, with the associated fusion power gain, does not appear as attractive.

Monte Carlo Simulations of the 2+1 Dimensional Fokker-Planck Equation: Spherical Star Clusters Containing Massive, Central Black Holes

The dynamical behavior of a relaxed star cluster containing a massive, central black hole poses a challenging problem for the theorist and intriguing possibilities for the observer. The historical development of the subject is sketched and the salient features of the physical solution and its observational consequences are summarized.The full dynamical problem of a relaxed, self-gravitating, large N-body system containing a massive central black hole has all the necessary ingredients to excite the most dispassionate many-body, computational physicist: it is a time-dependent, multidimensional, nonlinear problem which must be solved over widely disparate length and time scales simultaneously. The problem has been tackled at various levels of approximation over the years. A new 2+1 dimensional Monte Carlo simulation code has been developed in appreciable generality to solve the time-dependent Fokker-Planck equation in E-J space for this problem. The code incorporates such features as (1) a particle “cloning and renormalization” scheme to provide a statistically reliable population of test particles in low density regions of phase space and (2) a time-step “adjustment” algorithm to ensure integration on local relaxation timescales without having to follow typical particles on orbital trajectories. However, critical regions in phase space (e.g. disruption “loss-cone” trajectories) can still be followed on orbital timescales. Numerical results obtained with this Monte Carlo scheme for the dynamical structure and evolution of globular star clusters and dense galactic nuclei containing massive black holes are reviewed.Recent dynamical integrations of the Einstein field equations for spherical, collisionless (Vlasov) systems in General Relativity suggest a possible origin for the supermassive black holes believed to power quasars and active galactic nuclei. This scenario is discussed briefly.

Solutions of the Fokker-Planck equation describing the thermalization of neutrons in a heavy gas moderator

The thermalization of neutrons is described by a transport equation with a second order differential operator with respect to the energy. First this equation is transformed to an one-variable Fokker-Planck equation. Next an eigenfunction expansion and a polynomial expansion are used to solve the time-dependent Fokker-Planck equation. The eigenvalues are obtained either by solving a Schrodinger equation or by calculating 2×2 matrix continued fractions. Explicit results for the approach to equilibrium of a pulse of neutrons as well as the stationary distribution for 1/v absorption are presented. It is shown that the theory of the photoelectromotive force in semiconductors also leads to the same problem.

Application of stochastic computer simulation to a solution of a nonlinear Fokker-Planck equation governing bacteria chemotaxis

A stochastic computer simulation technique has been used to solve a one-dimensional time-dependent Fokker-Planck (F-P) equation governing spacetime distribution of bacteria in a substrate (food) gradient. Because of consumption of the substrate by the bacteria the F-P equation becomes nonlinear due to coupling of the distribution with the self-generated substrate gradient. The simulation is efficient and numerically accurate for generating transient solutions. We are able not only to produce a transient solution displaying approach to the steady state solitary wave solution known from an analytical result of Keller and Segal but also to show an interesting dependence of the solitary wave propagation speed on the concentration dependence of the substrate consumption rate.