scholarly journals Large-Time Asymptotics of Solutions to the Kramers-Fokker-Planck Equation with a Short-Range Potential

2015 ◽  
Vol 336 (3) ◽  
pp. 1435-1471 ◽  
Author(s):  
Xue Ping Wang
Keyword(s):  
Short Range ◽  
Large Time ◽  
Planck Equation ◽  
Large Time Asymptotics ◽  
Asymptotics Of Solutions ◽  
Short Range Potential ◽  
Time Asymptotics ◽  
Fokker Planck Equation ◽  
Range Potential ◽  
Fokker Planck

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.

scholarly journals An interplay between attraction and repulsion in infinite populations

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Yuri Kozitsky
Keyword(s):  
Kinetic Equation ◽  
Short Range ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Infinite Population ◽  
Poisson Law ◽  
Fokker Planck Equation ◽  
Proposed Model ◽  
Fokker Planck

AbstractWe propose and study a model describing an infinite population of point entities arriving in and departing from $$X=\mathbb {R}^d$$ X = R d , $$d\ge 1$$ d ≥ 1 . The already existing entities force each other to leave the population (repulsion) and attract the newcomers. The evolution of the population states is obtained by solving the corresponding Fokker-Planck equation. Without interactions, the evolution preserves states in which the probability $$p(n,\Lambda )$$ p ( n , Λ ) of finding n points in a compact vessel $$\Lambda \subset X$$ Λ ⊂ X obeys the Poisson law. As we show, for pure attraction the decay of $$p(n,\Lambda )$$ p ( n , Λ ) with $$n\rightarrow +\infty $$ n → + ∞ may be essentially slower. The main result is the statement that in the presence of repulsion—even of an arbitrary short range—the evolution preserves states in which the decay of $$p(n,\Lambda )$$ p ( n , Λ ) is at most Poissonian. We also derive the corresponding kinetic equation, the numerical solutions of which can provide more detailed information on the interplay between attraction and repulsion. Further possibilities in studying the proposed model are also discussed.

Spectrum analysis of the linear Fokker–Planck equation

2017 ◽  
Vol 15 (03) ◽  
pp. 313-331 ◽  
Author(s):  
Lan Luo ◽  
Hongjun Yu
Keyword(s):  
Perturbation Theory ◽  
Linear Operator ◽  
Large Time ◽  
Semigroup Theory ◽  
Planck Equation ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Fokker Planck Equation ◽  
Spectrum Structure ◽  
Fokker Planck

In this work, we show the spectrum structure of the linear Fokker–Planck equation by using the semigroup theory and the linear operator perturbation theory. As an application, we show the large time behavior of the solutions to the linear Fokker–Planck equation.

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