Solution Of Fokker-Planck Equation Research Articles

THE SLOWING DOWN OF FAST PARTICLES PASSING THROUGTH FUSION PLASMA AND THE ENERGY GAIN OF FUSION REACTION

The extended Landau theory about slowing down of electron traversing through matter has been applied to the investigation of slowing down of fast charged particles passing through fusion plasma. It is shown that there are some obvious differences between the solution of Fokker-Planck equation and our results. These differences come from different descriptions for larger energy exchange events in collisions. As a con crete application of the present theory, the energy gain of the fusion reaction produced by mixed deuterium and tritium beam has been calculated and the significance of improving particle confinement time has been shown.

Path integral approach to the solution of Fokker-Planck equations

The classical transport equation of the Fokker-Planck type is transformed into a path integral formulation and solved within the saddle point approximation. The numerical results are compared with the exact solution and with the approximation gained by applying a moment expansion for some simple drift potentials and constant diffusion. The limitations of the present approach as well as general problems connected with the saddle point approximation are discussed.

Some properties of stochastic equations for coupled chemical reactions far from equilibrium

Solutions of master equations for coupled chemical reactions far from equilibrium with one varying molecule species are studied and used for getting information about nonlinear Fokker-Planck equations and slow time-dependent processes such as extinction to an absorbing state and transition between several steady states. The Fokker-Planck equation solution is compared to that of the master equation in a relative sense and it is shown that they agree quite well in some important situations but that in general the cases can deviate considerably, when, e.g., accounting for the mutual importance of two probability maxima.

Solution of Fokker Planck equations with and without manifest detailed balance

Previous work on Fokker Planck equations with manifest detailed balance is generalized to include also the case without manifest detailed balance. The two cases are unified by exhibiting a general time reversal transformation with respect to which any Fokker Planck equation satisfies detailed balance, provided its steady state distribution exists. We also introduce a new method for solving some Fokker Planck equations with nonvanishing steady state drift by analytic continuation of the solution of a hermitian eigenvalue problem.

Quasiadiabatic solutions of Fokker Planck equations with time-dependent drift and fluctuations coefficients

A systematic algorithm for finding solutions of Fokker Planck systems with time-dependent drift and fluctuation coefficients is developed. The solutions lag behind the adiabatic quasistationary distribution. This delay effect vanishes for slowly varying control parameters. In a first example the evolution of the phase transition from a monostable to a bistable system is treated. There is a fluctuation enhancement and the delay of the probability distribution leads to a transient central peak. In a second example a linear model with time-dependently shifted drift is considered. The introduced approximation scheme reproduces the exact solution already in first order.

On the application of truncated generalized Fokker-Planck equations

The Pawula theorem states that the generalized Fokker-Planck equation with finite derivatives greater than two leads to a contradiction to the positivity of the distribution function. Though negative values are inconsistent from a logical point of view, we show that such “distribution functions” with negative values can be very useful from a practical point of view. For a Poisson-process, where the exact solution is known, we compare the solution of the second order Fokker-Planck equation to the solutions of Fokker-Planck equations of finite order. It turns out that for certain parameters the approximations of the distribution function and the moments are much better for some higher order and that the magnitude of negative values may be very small in the relevant region of variables.

Manifolds of equivalent path integral solutions of the Fokker-Planck equation

Path integral solutions of the multi-dimensional Fokker-Planck equation with variable dependent diffusion coefficients are deduced in a simple and exact manner. We show that the Onsager-Machlup function is not defined uniquely but is definable only together with the discretization prescription and the measure in the functional space. We present wide classes of mathematical equivalent path integral representations characterized by nonlinear variable transformationsv(q′, q) and the coefficientsαK of a linear combination, all giving exactly the same solution of the Fokker-Planck equation.

Asymptotic solutions of Fokker-Planck equations for nonlinear irreversible processes and generalized Onsager-Machlup theory

Fokker-Planck equations for nonlinear processes are solved asymptotically in the limit k→0 where k is the Boltzmann constant. It is shown that the leading asymptotic solutions for conditional (two-gate) distribution functions simply correspond to generalizations of the Onsager-Machlup theory to nonlinear processes. The asumptotic solution method used in the paper is similar to the well-known W.K.B. method in quantum mechanics. A stability criterion of nonlinear irreversible processes is also considered and compared with the Glansdorff-Prigogine stability criterion.

Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the master equation

We determine the path integral solution of a stochastic process described by a generalized Langevin equation with coordinate-dependent fluctuating forces and white spectrum. Since such equations do not permit a unique determination of the distribution function but require the Ito or Stratonovich prescription, we first pass over to the corresponding Fokker-Planck equation adopting such a prescription. By means of the one-dimensional case we show that the path integral solutions are not uniquely determined in form but allow for a class of equivalent representations. Adopting an especially suitable representation we then present the path integral solution of the multi-dimensional Fokker-Planck equation. The exponent of the exponential function occurring in that solution can be interpreted as generalization of the Onsager-Machlup function. Finally we give a path-integral solution of the master equation. In the “Fokker-Planck limit” again a generalized Onsager-Machlup function is obtained.

Stationary solution of the Fokker-Planck equation for the continuum mode laser with spatially inhomogeneous dissipation and excitation

An approximation scheme is developed in order to obtain an expression for the stationary solution of the multivariate nonlinear Fokker-Planck equation for the continuum mede laser having spatially inhomogeneous losses and inversion. This Fokker-Planck equation, being valid throughout the threshold region, has been derived in a previous paper. After diagonalizing the linear drift matrix the so-called diagonal diffusion approximation will be introduced, which permits to find the stationary probability distribution for the mode amplitudes within the quasi-plane-wave approximation.

Time-dependent solution of Fokker-Planck equation for alpha-particles and its effect on alpha-particle heating characteristics in a D-T fusion reactor

The transient time history of the energy distribution function of alpha-particles in the D-T fusion reactors is obtained by numerically solving a Fokker-Planck equation, by taking into consideration the possible alpha-particle loss from the core-plasma volume. On the basis of the results obtained, the total amount of energy transfer to background plasma and the efficiency of the alpha-particle heating, together with the partition of energy transfer to electrons and to ions, are calculated. In addition, a new fact has been found, i. e. the effect of an ‘inertial’ property of the alpha-particle distribution function on the transient response of the total amount of energy transfer for a temporal perturbation of the parameters of background D-T plasma, which is important in connection with stability and control problems of D-T reactor-core plasmas.

Solutions of Fokker-Planck Equation with Applications in Nonlinear Random Vibration

In the course of analyzing the dynamic behavior of mechanical systems subjected to random excitations, we investigate the associated Fokker-Planck equation. We also discuss the relationship between the characteristics of the random excitation and the nonlinear intensity coefficients governed by the physical properties of the system. This relationship leads to some simplified methods for solving the response probability density of certain nonlinear systems. We present general solutions to a class of multidimensional problems with desirable constraints. The random motion of a single-mode mechanical oscillator with various nonlinear stiffnesses and a charged particle moving in an electromagnetic field are examples. Cosine-power and sech-power distributions are found to be associated with the steady state response of a tangent stiffness system and a hyperbolic tangent stiff ness system, respectively. When the influencing magnetic vector potential M is irrotational, the stationary probability for the moving particle in the 6-dimensional response phase-space is statistically independent.

Numerical solution of the Fokker-Planck equations for a hydrogen plasma formed by neutral injection

We describe a program for the solution of the time-dependent Fokker-Planck equations for electron and ion distribution functions in velocity space. We consider the formation of a plasma in a magnetic mirror configuration by the injection of energetic neutral atoms. In the equations we have a source of energetic protons and cold electrons, and we include losses due to scattering and charge exchange. The coupled nonlinear partial differential equations are solved by an implicit difference method which is described. Numerical results are presented for two cases of interest.

Plasma potential and energy distributions in high-energy injection machines

Steady state conditions are calculated for the Oak Ridge DCX-2 and similar devices in which a plasma confined by magnetic mirrors is maintained continuously by the injection and trapping of energetic ions. The potential difference from plasma interior to exterior is determined in the course of the numerical solution of Fokker-Planck equations for ion and electron energy distributions. It is assumed that the only electron source is ionization of neutrals and trapped ions are lost by charge exchange with neutrals and by Coulomb scattering out of the mirrors. Justification for neglecting cold ions from ionization is discussed. The plasma potential and also, aside from normalization, the energy distributions are found to depend only on the ion injection energy and the fraction of `burn-out' of the neutrals and not directly on absolute magnitudes of neutral pressure or ion injection current. Results are combined with equations governing neutrals to obtain a new DCX-2 performance curve relating injection current to ion density achieved. Enhancement of the charge-exchange loss-rate due to collisional cooling of the ions on electrons, neglected in earlier calculations, is found to increase by a factor of <2 the critical injection current required to reach densities limited only by mirror losses.