Uniqueness of Steady‐State Solutions to the Fokker‐Planck Equation

1965 ◽  
Vol 6 (4) ◽  
pp. 644-647 ◽  
Author(s):  
A. H. Gray
Keyword(s):  
Steady State ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Steady State Solutions ◽  
Fokker Planck

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.

The lattice Fokker–Planck equation for models of wealth distribution

2020 ◽  
Vol 378 (2175) ◽  
pp. 20190401
Author(s):  
Shaurya Kaushal ◽  
Santosh Ansumali ◽  
Bruce Boghosian ◽  
Merek Johnson
Keyword(s):  
Steady State ◽  
Soft Matter ◽  
Wealth Distribution ◽  
Planck Equation ◽  
Agent Based ◽  
Non Local ◽  
Free Parameters ◽  
Fokker Planck Equations ◽  
Steady State Solutions ◽  
Fokker Planck

Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the ‘inverse problem’ of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the ‘forward problem’ of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

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.

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