Some properties of the Fokker-Planck equation

1985 ◽  
Vol 33 (2) ◽  
pp. 183-189
Author(s):  
G. J. Lewak ◽  
L. A. Soto
Keyword(s):  
Distribution Function ◽  
Plasma Heating ◽  
Electron Distribution Function ◽  
Planck Equation ◽  
Small Mass ◽  
Mass Ratio ◽  
Physical Conditions ◽  
Fokker Planck Equation ◽  
Velocity Moments ◽  
Fokker Planck

The solution of the Fokker-Planck equation for the distribution function of heavy ions in a background of electrons is studied. It is found that quite broad physical conditions on the distribution function (such as the positive requirement and the existence of all velocity moments) are sufficient to eliminate any ambiguity in the time-independent steady-state solutions and to determine a discrete spectrum of the time-dependent Fokker–Planck operator. The more physical case of a Maxwell-Boltzman electron distribution function is treated using the small mass ratio expansion. First-order mass ratio corrections are calculated. A plasma heating application-is suggested.

Semi-analytical inspection of the quasi-linear absorption of lower hybrid wave in presence of -particles in tokamak reactor

2018 ◽  
Vol 84 (5) ◽  
Author(s):  
A. Cardinali ◽  
C. Castaldo ◽  
R. Ricci
Keyword(s):  
Distribution Function ◽  
Planck Equation ◽  
Current Drive ◽  
Dielectric Tensor ◽  
Lower Hybrid ◽  
Linear Diffusion ◽  
Tokamak Reactor ◽  
Slowing Down ◽  
Fokker Planck Equation ◽  
Fokker Planck

In a reactor plasma like demonstration power station (DEMO), when using the radio frequency (RF) for heating or current drive in the lower hybrid (LH) frequency range (Frankeet al.,Fusion Engng Des., vol. 96–97, 2015, p. 46; Cardinaliet al.,Plasma Phys. Control. Fusion, vol. 59, 2017, 074002), a large fraction of the ion population (the continuously born$\unicode[STIX]{x1D6FC}$-particle, and/or the neutral beam injection (NBI) injected ions) is characterized by a non-thermal distribution function. The interaction (propagation and absorption) of the LH wave must be reformulated by considering the quasi-linear approach for each species separately. The collisional slowing down of such an ion population in a background of an electron and ion plasma is balanced by a quasi-linear diffusion in velocity space due to the propagating electromagnetic wave. In this paper, both propagations are considered by including the ion distribution function, solution of the Fokker–Planck equation, which describes the collisional dynamics of the$\unicode[STIX]{x1D6FC}$-particles including the effects of frictional slowing down, energy diffusion and pitch-angle scattering. Analytical solutions of the Fokker–Planck equation for the distribution function of$\unicode[STIX]{x1D6FC}$-particles with a background of ions and electrons at steady state are included in the calculation of the dielectric tensor. In the LH frequency domain, ray tracing (including quasi-linear damping), can be analytically solved by iterating with the Fokker–Planck solution, and the interaction of the LH wave with$\unicode[STIX]{x1D6FC}$-particles, thermal ions and electrons can be accounted self-consistently and the current drive efficiency can be evaluated in this more general scenario.

Effects of particle trapping on distortion of the distribution function of RF-heated minority ions in a tokamak plasma

1986 ◽  
Vol 35 (1) ◽  
pp. 107-117 ◽  
Author(s):  
D. Anderson ◽  
M. Lisak
Keyword(s):  
Distribution Function ◽  
Planck Equation ◽  
Tokamak Plasma ◽  
Explicit Solutions ◽  
Induced Anisotropy ◽  
Particle Trapping ◽  
Fokker Planck Equation ◽  
Toroidal Geometry ◽  
Fokker Planck

The effect of toroidal geometry upon the distribution function of weakly RF-heated minority ions in a tokamak plasma is considered. An analytic scheme to solve the corresponding bounce-averaged Fokker-Planck equation is developed and explicit solutions are given in the limits of high and low velocities. It is found that trapped-particle effects may significantly reduce the RF-induced anisotropy in the distribution function.

scholarly journals Effect of the Damping on the Magnetization of Antiferromagnetic Nanoparticles in a Presence of an Oblique Magnetic Field

2021 ◽  
pp. 1-7
Author(s):  
Bachir Ouari ◽  
◽  
Malika Madani ◽  
Mohamed Lagraa ◽  
◽  
...  
Keyword(s):  
Magnetic Field ◽  
Distribution Function ◽  
Relaxation Times ◽  
Planck Equation ◽  
Dynamic Susceptibility ◽  
Analytic Equation ◽  
Fokker Planck Equation ◽  
Oblique Field ◽  
Oblique Magnetic Field ◽  
Fokker Planck

The magnetization of antiferromagnetic nanoparticles is investigated with the Fokker-Planck equation describing the evolution of the distribution function of the magnetization of an nanoparticle. By solving this equation numerically, the relaxation times, and dynamic susceptibility are calculated for dc field orientations across wide ranges of frequencies, amplitude of the fields and damping. Analytic equation for the dynamic susceptibility is also proposed. It is shown that the damping alters the magnetization in the presence of oblique field applied

scholarly journals Generalized Fokker–Planck equation for the distribution function of liquidity accumulation

2019 ◽  
Vol 6 (1) ◽  
pp. 37-43
Author(s):  
B. Hnativ ◽  
◽  
A. Didyk ◽  
M. Tokarchuk ◽  
◽  
...  
Keyword(s):  
Distribution Function ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Spectrally resolved cosmic ray hydrodynamics – I. Spectral scheme

2019 ◽  
Author(s):  
Philipp Girichidis ◽  
Christoph Pfrommer ◽  
Michał Hanasz ◽  
Thorsten Naab
Keyword(s):  
Distribution Function ◽  
Planck Equation ◽  
Cosmic Ray ◽  
Power Laws ◽  
Spectral Evolution ◽  
Astrophysical Plasmas ◽  
Full Spectrum ◽  
Fokker Planck Equation ◽  
Hydrodynamical Simulations ◽  
Fokker Planck

Abstract Cosmic ray (CR) protons are an important component in many astrophysical systems. Processes like CR injection, cooling, adiabatic changes as well as active CR transport through the medium strongly modify the CR momentum distribution and have to be taken into account in hydrodynamical simulations. We present an efficient novel numerical scheme to accurately compute the evolution of the particle distribution function by solving the Fokker-Planck equation with a low number of spectral bins (10 − 20), which is required to include a full spectrum for every computational fluid element. The distribution function is represented by piecewise power laws and is not forced to be continuous, which enables an optimal representation of the spectrum. The Fokker-Planck equation is solved with a two-moment approach evolving the CR number and energy density. The low numerical diffusion of the scheme reduces the numerical errors by orders of magnitude in comparison to classical schemes with piecewise constant spectral representations. With this method not only the spectral evolution of CRs can be computed accurately in magnetohydrodynamic simulations but also their dynamical impact as well as CR ionisation. This allows for more accurate models for astrophysical plasmas, like the interstellar medium, and direct comparisons with observations.

THE ASYMPTOTICS OF COLLISION OPERATORS FOR TWO SPECIES OF PARTICLES OF DISPARATE MASSES

1996 ◽  
Vol 06 (03) ◽  
pp. 405-436 ◽  
Author(s):  
PIERRE DEGOND ◽  
BRIGITTE LUCQUIN-DESREUX
Keyword(s):  
Time Scales ◽  
Planck Equation ◽  
Distribution Functions ◽  
Relaxation Phenomenon ◽  
Mass Ratio ◽  
Homogeneous Case ◽  
Fokker Planck Equation ◽  
Disparate Mass ◽  
Fokker Planck

We analyze the dynamics of a disparate mass binary gas or of a plasma in the homogeneous case, at various time scales, in the framework of the Boltzmann or Fokker–Planck equation. We intend to provide a rigorous foundation to the epochal relaxation phenomenon first pointed out by Grad. From general basic physical hypotheses, we derive the scaling of the equations as a function of the mass ratio of the particles, and we expand the collision operators in powers of this mass ratio. Then, Hilbert or Chapman–Enskog type expansions of the distribution functions allow us to investigate the dynamics of the mixture at various time scales, and we verify that the behavior of the obtained models is coherent with Grad's hypothesis.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck
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