Theorems pertaining to Fokker–Planck statistical equilibrium for multidimensional stochastic systems

1986 ◽  
Vol 27 (5) ◽  
pp. 1387-1390
Author(s):  
Gerald Rosen
Keyword(s):  
Stochastic Systems ◽  
Statistical Equilibrium ◽  
Fokker Planck

Simulating Drift-Diffusion Processes With Generalized Interval Probability

2012 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Path Integral ◽  
Stochastic Systems ◽  
Diffusion Processes ◽  
Epistemic Uncertainty ◽  
Planck Equation ◽  
Integral Approach ◽  
Drift Diffusion ◽  
Interval Probability ◽  
Fokker Planck Equation ◽  
Fokker Planck

The Fokker-Planck equation is widely used to describe the time evolution of stochastic systems in drift-diffusion processes. Yet, it does not differentiate two types of uncertainties: aleatory uncertainty that is inherent randomness and epistemic uncertainty due to lack of perfect knowledge. In this paper, a generalized Fokker-Planck equation based on a new generalized interval probability theory is proposed to describe drift-diffusion processes under both uncertainties, where epistemic uncertainty is modeled by the generalized interval while the aleatory one is by the probability measure. A path integral approach is developed to numerically solve the generalized Fokker-Planck equation. The resulted interval-valued probability density functions rigorously bound the real-valued ones computed from the classical path integral method. The new approach is demonstrated by numerical examples.

A Fokker–Planck control framework for stochastic systems

2018 ◽  
Vol 5 (1) ◽  
pp. 65-98 ◽  
Author(s):  
Mario Annunziato ◽  
Alfio Borzi
Keyword(s):  
Stochastic Systems ◽  
Control Framework ◽  
Fokker Planck

scholarly journals On the construction of a family of anomalous-diffusion Fokker–Planck−Kolmogorov’s equations based on the Sharma–Taneja–Mittal entropy functional

2021 ◽  
Vol 51 ◽  
pp. 74-95
Author(s):  
Aleksandr Vladimirovich Kolesnichenko
Keyword(s):  
Stochastic Systems ◽  
Probability Density Distribution ◽  
Convexity Property ◽  
Logical Scheme ◽  
Entropy Functional ◽  
Fpk Equation ◽  
Ansatz Approach ◽  
The Ansatz Approach ◽  
Non Equilibrium ◽  
Fokker Planck

A logical scheme for constructing thermodynamics of anomalous stochastic systems based on the nonextensive two-parameter (κ, ς) -entropy of Sharma–Taneja–Mittal (SHTM) is considered. Thermodynamics within the framework (2 - q) -statistics of Tsallis was constructed, which belongs to the STM family of statistics. The approach of linear nonequilibrium thermodynamics to the construction of a family of nonlinear equations of Fokker−Planck−Kolmogorov (FPK), is used, correlated with the entropy of the STM, in which the stationary solution of the diffusion equation coincides with the corresponding generalized Gibbs distribution obtained from the extremality (κ, ς) - entropy condition of a non-additive stochastic system. Taking into account the convexity property of the Bregman divergence, it was shown that the principle of maximum equilibrium entropy is valid for (κ, ς) - systems, and also was proved the H - theorem determining the direction of the time evolution of the non-equilibrium state of the system. This result is extended also to non-equilibrium systems that evolve to a stationary state in accordance with the nonlinear FPK equation. The method of the ansatz- approach for solving non-stationary FPK equations is considered, which allows us to find the time dependence of the probability density distribution function for non-equilibrium anomalous systems. Received diffusive equations FPК can be used, in particular, at the analysis of diffusion of every possible epidemics and pandemics. The obtained diffusion equations of the FPK can be used, in particular, in the analysis of the spread of various epidemics and pandemics.

scholarly journals Interacting Particle Solutions of Fokker–Planck Equations Through Gradient–Log–Density Estimation

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 802
Author(s):  
Dimitra Maoutsa ◽  
Sebastian Reich ◽  
Manfred Opper
Keyword(s):  
Stochastic Systems ◽  
Numerical Solutions ◽  
Particle System ◽  
Particle Density ◽  
Stochastic Simulations ◽  
Density Estimator ◽  
Analytical Treatment ◽  
Interacting Particle ◽  
Fokker Planck Equations ◽  
Fokker Planck

Fokker–Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker–Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker–Planck equations in low and moderate dimensions. The proposed gradient–log–density estimator is also of independent interest, for example, in the context of optimal control.

Equivalent Stochastic Systems

1988 ◽  
Vol 55 (4) ◽  
pp. 918-922 ◽  
Author(s):  
Y. K. Lin ◽  
G. Q. Cai
Keyword(s):  
Probability Distribution ◽  
Stochastic Systems ◽  
Initial Conditions ◽  
Weak Sense ◽  
Wide Sense ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Probability Densities ◽  
Fokker Planck Equations ◽  
Fokker Planck

Equivalent stochastic systems are defined as randomly excited dynamical systems whose response vectors in the state space share the same probability distribution. In this paper, the random excitations are restricted to Gaussian white noises; thus, the system responses are Markov vectors, and their probability densities are governed by the associated Fokker-Planck equations. When the associated Fokker-Planck equations are identical, the equivalent stochastic systems must share both the stationary probability distribution and the transient nonstationary probability distribution under identical initial conditions. Such systems are said to be stochastically equivalent in the strict (or strong) sense. A wider class, referred to as the class of equivalent stochastic systems in the wide (or weak) sense, also includes those sharing only the stationary probability distribution but having different Fokker-Planck equations. Given a stochastic system with a known probability distribution, procedures are developed to identify and construct equivalent stochastic systems, both in the strict and in the wide sense.

Sign in / Sign up

Export Citation Format

Share Document