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.

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.

Richards Growth Model Driven by Multiplicative and Additive Colored Noises: Steady-State Analysis

2020 ◽  
Vol 19 (04) ◽  
pp. 2050032
Author(s):  
Chaoqun Xu ◽  
Sanling Yuan
Keyword(s):  
Steady State ◽  
Stochastic Model ◽  
Growth Model ◽  
Planck Equation ◽  
Steady State Analysis ◽  
Model Driven ◽  
Fokker Planck Equation ◽  
Richards Growth Model ◽  
Fokker Planck ◽  
State Analysis

We consider a Richards growth model (modified logistic model) driven by correlated multiplicative and additive colored noises, and investigate the effects of noises on the eventual distribution of population size with the help of steady-state analysis. An approximative Fokker–Planck equation is first derived for the stochastic model. By performing detailed theoretical analysis and numerical simulation for the steady-state solution of the Fokker–Planck equation, i.e., stationary probability distribution (SPD) of the stochastic model, we find that the correlated noises have complex effects on the statistical property of the stochastic model. Specifically, the phenomenological bifurcation may be caused by the noises. The position of extrema of the SPD depends on the model parameter and the characters of noises in different ways.

Newtonian repulsion and radial confinement: Convergence toward steady state

2021 ◽  
pp. 1-25
Author(s):  
Ruiwen Shu ◽  
Eitan Tadmor
Keyword(s):  
Steady State ◽  
Decay Rate ◽  
Large Time ◽  
Large Time Behavior ◽  
Steady States ◽  
Parameter Family ◽  
Time Behavior ◽  
Comparison Argument ◽  
Algebraic Convergence ◽  
Unique Steady State

We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions.

Sign in / Sign up

Export Citation Format

Share Document