Fokker-Planck Description Research Articles

The effect of quasi-accelerator modes on diffusion

We examine the diffusion in action of a generic Hamiltonian mapping over a range of actions for which the local standard mapping approximation exhibits stable accelerator modes. For the generic mapping the corresponding orbits are not completely stable, but the orbits may appear to be stable over many mappings periods. We examine the effect of these “quasi-accelerator mode” orbits on the diffusion. We find the most important physical effect in enhancing the diffusion to be the growth and decay of the locally stable mode (island) area, which leads to trapping and detrapping of phase space within the mode. An analytic theory is developed to incorporate this effect into a Fokker-Planck description of the diffusion. The solution for the distribution function, generated by a constant flux across the mode, is calculated and compared to numerical results. Provided a reasonable approximation to the locally stable island area is used in the theory, good agreement between theoretical and numerical results is obtained in a parameter range for which the effect of the mode is significant.

Mapping study of nonlinear electron motion in a coherent wave packet

Mapping equations are derived, with which nonlinear electron motion in a coherent wave packet for the arbitrary strength of the amplitude is investigated. In a small amplitude wave packet, electrons oscillate in velocity space, reflecting the coherent structure of the wave field. The motion is decorrelated in a moderately large amplitude wave packet, which exhibits a random walk. In a strong amplitude wave packet, electrons are trapped by the wave potential well, when a certain degree of correlation in the motion appears again. In that case, the mapping equations play an active role in the analysis of the particle dynamics because the Fokker-Planck description breaks down there. It is also shown, under a certain condition, that they can be reduced to the standard mapping, which is now well investigated.

A Fokker-Planck description of K-distributed noise

A Fokker-Planck description of the time evolution of a K-distributed noise process is presented, discussed and taken as the basis of an approximate analysis of the correlation properties of the noise. An equivalent set of stochastic differential equations is identified and exploited in a numerical simulation of the noise process.

Stochastic particle acceleration and statistical closures

In a recent paper, Maasjost and Elsässer (ME) concluded, from the results of numerical experiments and heuristic arguments, that the Bourret and the direct-interaction approximation (DIA) are “of no use in connection with the stochastic acceleration problem” because (1) their predictions were equivalent to that of the simpler Fokker-Planck (FP) theory, and (2) either all or none of the closures were in good agreement with the data. Here some analytically tractable cases are studied and used to test the accuracy of these closures. The cause of the discrepancy (2) is found to be the highly non-Gaussian nature of the force used by ME, a point not stressed by them. For the case where the force is a position-independent Ornstein-Uhlenbeck (i.e., Gaussian) process, an effective Kubo numberK can be defined. ForK≪1 an FP description is adequate, and conclusion (1) of ME follows; however, forK>1 the DIA behaves much better qualitatively than the other two closures. For the non-Gaussian stochastic force used by ME, all common approximations fail, in agreement with (2).

The rate of nucleation in a vapor–liquid transition determined from hydrodynamic fluctuation theory

A Fokker–Planck description, based upon hydrodynamic fluctuation theory, of homogeneous nucleation of liquid drops in a vapor phase is reformulated as a one-dimensional time-independent scattering problem. This result is then coupled with an elementary analysis of the interactions between distinct fluctuations to determine the concentration of nuclei produced in a nucleation experiment. An extended comparison between the assumptions of this formulation and those of the classical formulation is given.

Fokker-Planck and Langevin descriptions of fluctuations in uniform shear flow

Fokker-Planck and Langevin descriptions of fluctuations in uniform shear flow

Statistical properties of a laser driven by colored noise

We obtain an exact analytic solution of the equation of motion for a dye laser with a fluctuating pump parameter of finite correlation time. The exact solution is compared with recent experimental results on the intensity fluctuations of a dye laser. We also obtain a favorable comparison of the experiments to the results obtained from an approximate Fokker–Planck description of the laser.

Fokker-Planck and Langevin descriptions of fluctuations in uniform shear flow

The Boltzmann description of the preceding paper for tagged particle fluctuations in a nonequilibrium gas is further analyzed in the limit of small mass ratio between the gas and the tagged particles. For a large class of nonequilibrium states the Boltzmann-Lorentz collision operator for the tagged particle distribution is expanded to leading order in the mass ratio, resulting in a Fokker-Planck operator. The drift vector and diffusion tensor are calculated exactly for Maxwell molecules. The Fokker-Planck operator depends on the nonequilibrium state only through the hydrodynamic variables for the fluid. The diffusion tensor is a measure of the “noise” amplitude and is not simply determined from the nonequilibrium temperature; instead, it depends on the fluid stress tensor components as well. For the special case of uniform shear flow, the Fokker-Planck equation is of the linear type and may be solved exactly. The associated set of Langevin equations is also identified and used to describe spatial diffusion in the Lagrangian coordinates of the fluid. The effect of viscous heating on diffusion is discussed and the dependence of the diffusion coefficient on the shear rate is calculated.

Fokker-Planck description of multi-degree-of-freedom systems with correlated fluctuations

A Fokker-Planck equation is derived for a many-degree-of-freedom nonlinear Langevin equation driven by parametric gaussian fluctuations with finite correlation times. An oscillator with a fluctuating frequency is presented as an example.

One-dimensional harmonic liquid: A Fokker-Planck description of fluctuations from the nonequilibrium steady state

One-dimensional harmonic liquid: A Fokker-Planck description of fluctuations from the nonequilibrium steady state

One-dimensional harmonic liquid: A Fokker-Planck description of fluctuations from the nonequilibrium steady state

The one-dimensional harmonic liquid, subject to a temperature gradient, is studied using the Fokker-Planck (FP) equation. A formalism is set up for solution of the FP equation and calculation of (1) the phase-space distribution function for the nonequilibrium steady state (NESS), (2) the conditional probability for evolution through phase space in the NESS, (3) phase-function averages in the NESS, (4) the correlation of phase functions in the NESS, etc. For the harmonic liquid this formalism can be implemented without approximation. Some properties of the harmonic liquid in equilibrium are examined to illustrate use of the formalism; so are some phase-function averages in the NESS. The displacement-displacement correlation function $D(k,\ensuremath{\omega})$ in the NESS is calculated and found to have the well-established, interesting amplitude and frequency dependence. The completeness of the Fokker-Planck description makes it possible to identify the physical processes responsible for the behavior of $D(k,\ensuremath{\omega})$ and the dynamic structure factor $S(k,\ensuremath{\omega})$. The interesting features of light scattering from a liquid subject to a temperature gradient, seen in $S(k,\ensuremath{\omega})$ or $D(k,\ensuremath{\omega})$, are due to light scattering from width fluctuations induced by the gradient; the interesting features in light scattering from a liquid-supporting shear are due to an attenuation mechanism that arises because of the velocity field induced by the shear. The results in this paper constitute a partial demonstration of the usefulness of the method employed in handling the Fokker-Planck equation.

Stochastic particle acceleration

The propagation of test particles in a turbulent plasma is considered both analytically and numerically. We split the Green's function into a deterministic and a stochastic part and derive several exact relations between them; these relations show that partial summations of the perturbation series do not lead, in general, to a consistent closure for the hierarchy problem. The mean propagator in velocity space is calculated numerically for an acceleration field given by an Ornstein-Uhlenbeck process. If the relative coherent changee of the particle velocity remains smaller than 0.1% the Fokker-Planck description may be adequate for the time evolution of the propagator. Otherwise the picture of Brownian motion becomes worse, and incorporating memory effects or renormalization of the mass operator have also been found to disagree with the numerical experiment.

Fokker-Planck equation for a nonlinear oscillator

Starting from a Hamiltonian model introduced by Zwanzig, a nonlinear Fokker-Planck equation for the distribution function of a particle interacting with a bath of harmonic oscillators is derived. The simplicity of the model allows explicit calculations and also a careful discussion of the conditions under which a Fokker-Planck description of the problem is valid. Connections with previous and more formal derivations are analysed.

Analytical and numerical results for the steady state in cooperative resonance fluorescence

The cooperative resonance fluorescence steady state is discussed within the context of an operator master equation which conserves total pseudospin. Emphasis throughout is on quantum fluctuations and their significance in relation to a background of factorised dynamics. Atom-atom correlations are shown to play a fundamental role for systems driven beyond the linear regime. Use of the atomic coherent state representation yields a Fokker-Planck description closely allied to the dynamics for a classical angular momentum oscillator. For intense incident fields the quantum-mechanical steady state is understood in terms of diffusion both around and between classical trajectories on the Bloch sphere. In the limit of infinite systems simple closed-form expressions for steady-state features are derived. Coherent and incoherent fluorescent intensities are obtained together with the second-order correlation function for fluorescent light. Specific features are illustrated by numerical results for systems of from two to fifty atoms.

Covariant lagrangian of Onsager and Machlup without discretization

A well defined notion of generalized Onsager-Machlup Lagrangian in the covariant formalism of general diffusion processes is obtained without any discretization procedure. Main ideas are to clarify some basic links between the conventional Fokker-Planck description and a Schrödingerlike description, and to exploit afterwards the well established path integral formalism of quantum mechanics. The first step is realized with the help of a modified version of Nelson's stochastic mechanics, that is, “thermal mechanics”, which seems well adapted to nonequilibrium statistical thermodynamics. A natural notion of deterministic approximation for the general diffusion process is also obtained in the present framework.

A Fokker-Planck equation for the fluctuations of the heat flux

The problem of heat fluctuations around the steady state is considered in the framework of extended irreversible thermodynamics. A new version using a Fokker-Planck description is carried out. Averages for the fluctuations around the steady state and correlation functions for these fluctuations are calculated.

Coupling of translational and reactive dynamics for a Fokker–Planck model

The coupling of translational and reactive dynamics is investigated for a Fokker–Planck description of relative particle motion in a solvent. A division into inner and outer relative coordinate spatial regions separated by a boundary layer is made. The inner region is characterized by a deep potential well associated with bound, or reacted, particles and a potential barrier. The outer zone is characterized by less rapidly varying forces which include solvent structural effects. With this division, a Fokker–Planck source equation for unreacted particles is derived. The source in this equation incorporates the complete inner region dynamics involving both reactive and nonreactive trajectories. This equation is then reduced via projection operator techniques to a spatial Smoluchowski source equation for unreacted particles in the outer region for the case of slow reaction. Validity conditions involving velocity relaxation are given. The sources here include both inner region short range repulsive effects and a reversible chemical reaction. The outer Smoluchowski description includes caging effects. The equivalent boundary condition formulation is given; thus, the radiative boundary condition relating diffusive and reactive fluxes is derived. New expressions for the reaction rate constants are found and examined in terms of more fundamental rate kernels and Fokker–Planck barrier crossing dynamics. The rate constants are evaluated for a Kramers-type treatment of the inner region. These rate constants include effects of velocity nonequilibrium. They reduce to transition state results under intermediate inner region friction conditions.

Fokker-Planck description of classical systems with application to critical dynamics

Classical systems with mixed canonical and dissipative dynamics are studied in terms of generalized Langevin equations for fluctuating variables. This class of systems embraces all models studied in recent papers on critical dynamics. The probability distributions satisfying the associated Fokker-Planck equation are used to build up a Green's function formalism which together with the diagram technique established earlier is applied to derive recursion relations of critical dynamics. A generalized fluctuation-dissipation theorem is proven and used to demonstrate the consistency between static and dynamic recursion relations. Special attention is given to a propagating soft mode for which a dynamical exponent close to one is obtained.

Numerical study of drift-kinetic evolution of collisional plasmas in tori

Preliminary numerical results for the dynamics of toroidally confined plasmas in the drift-kinetic, Fokker-Planck description are discussed. These solutions were obtained by using the collisional plasma model (CPM). An initial value problem was solved in which collisions and particle dynamics compete in a given magnetic field to set up a quasi-equilibrium. The plasma (guiding center) distribution function and many macroscopic quantities of interest are monitored. Good agreement with corresponding but more approximate theories is obtained over a wide range of collisionality, particularly with regard to the neoclassical particle flux. Confirmation of earlier results for the distribution function is achieved when due account is taken of the differing collisionality of particles with differing energies.

Coherent Transient Study of Velocity-Changing Collisions

The effect of velocity-changing collisions on the optical phase memory of coherently prepared molecular gas samples is examined by the method of Stark-pulse switching. Experiments are interpreted through the solution of a transport equation, which extends the earlier Fokker-Planck description known both in NMR and in Dicke line narrowing. The magnitude of a characteristic velocity jump for binary molecular collisions and its cross section are thereby obtained.