scholarly journals Short and long time behavior of the Fokker–Planck equation in a confining potential and applications

2007 ◽  
Vol 244 (1) ◽  
pp. 95-118 ◽  
Author(s):  
Frédéric Hérau
Keyword(s):  
Planck Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Confining Potential ◽  
Fokker Planck Equation ◽  
Long Time ◽  
Fokker Planck

scholarly journals On the Long-Time Behavior of the Quantum Fokker-Planck Equation

2004 ◽  
Vol 141 (3) ◽  
pp. 237-257 ◽  
Author(s):  
C. Sparber ◽  
J. A. Carrillo ◽  
P. A. Markowich ◽  
J. Dolbeault
Keyword(s):  
Planck Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Fokker Planck Equation ◽  
Long Time ◽  
Fokker Planck

scholarly journals Quantitative evaluation of numerical integration schemes for Lagrangian particle dispersion models

2016 ◽  
Author(s):  
Huda Mohd. Ramli ◽  
J. Gavin Esler
Keyword(s):  
Planck Equation ◽  
Particle Dispersion ◽  
Test Problems ◽  
Hermite Function ◽  
Dispersion Models ◽  
Lagrangian Particle ◽  
Integration Schemes ◽  
Fokker Planck Equation ◽  
Long Time ◽  
Fokker Planck

Abstract. A rigorous methodology for the evaluation of integration schemes for Lagrangian particle dispersion models (LPDMs) is presented. A series of one-dimensional test problems are introduced, for which the Fokker-Planck equation is solved numerically using a finite-difference discretisation in physical space, and a Hermite function expansion in velocity space. Numerical convergence errors in the Fokker-Planck equation solutions are shown to be much less than the statistical error associated with a practical-sized ensemble (N = 106) of LPDM solutions, hence the former can be used to validate the latter. The test problems are then used to evaluate commonly used LPDM integration schemes. The results allow for optimal time-step selection for each scheme, given a required level of accuracy. The following recommendations are made for use in operational models. First, if computational constraints require the use of moderate to long time steps it is more accurate to solve the random displacement model approximation to the LPDM, rather than use existing schemes designed for long time-steps. Second, useful gains in numerical accuracy can be obtained, at moderate additional computational cost, by using the relatively simple "small-noise" scheme of Honeycutt.

scholarly journals THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

2012 ◽  
Vol 22 (11) ◽  
pp. 1250034 ◽  
Author(s):  
ANTON ARNOLD ◽  
IRENE M. GAMBA ◽  
MARIA PIA GUALDANI ◽  
STÉPHANE MISCHLER ◽  
CLEMENT MOUHOT ◽  
...  
Keyword(s):  
Steady State ◽  
Large Time ◽  
Planck Equation ◽  
Large Time Behavior ◽  
Positive Density ◽  
Time Behavior ◽  
Spectral Gaps ◽  
Harmonic Oscillator Potential ◽  
Fokker Planck Equation ◽  
Fokker Planck

We consider the linear Wigner–Fokker–Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker–Planck type operators in certain weighted L2-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.

Kinetic theory of a two-dimensional magnetized plasma. Part 2. Balescu-Lenard limit

1972 ◽  
Vol 8 (3) ◽  
pp. 357-374 ◽  
Author(s):  
George Vahala
Keyword(s):  
Kinetic Theory ◽  
Plasma Parameter ◽  
Planck Equation ◽  
Large Integer ◽  
Collision Term ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Long Time ◽  
Time Average ◽  
Fokker Planck

The kinetic theory of a two-dimensional one-species plasma in a uniform d.c. magnetic field is investigated in the small plasma parameter limit. The plasma consists of charged rods interacting through the logarithmic Coulomb potential. Vahala & Montgomery earlier derived a Fokker –;Planck equation for this system, but it contained a divergent integral, which had to be cut-off on physical grounds. This cut-off is compared to the standard cut-off introduced in the two-dimensional unmagnetized Fokker –;Planck equation. In the small plasma parameter limit, it is shown (under the assumption that for large integer n, γn/γn+1 = O(np), with p < 2, where γn = ωn −nΩ. with ωn the nth. Bernstein mode and Q the electron gyro frequency) that the Balescu-Lenard collision term is zero in the long time average limit if one considers only two-body interactions. The energy transfer from a test particle to an equilibrium plasma is discussed and also shown to be zero in the long time average limit. This supports the unexpected result of zero Balescu-Lenard collision term.

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