A Stochastic Model and Associated Fokker–Planck Equation for the Fiber Lay-Down Process in Nonwoven Production Processes

2007 ◽  
Vol 67 (6) ◽  
pp. 1704-1717 ◽  
Author(s):  
T. Götz ◽  
A. Klar ◽  
N. Marheineke ◽  
R. Wegener
Keyword(s):  
Stochastic Model ◽  
Planck Equation ◽  
Production Processes ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

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.

Design of an intelligent stochastic model predictive controller for a continuous stirred tank reactor through a Fokker-Planck observer

2017 ◽  
Vol 40 (10) ◽  
pp. 3010-3022 ◽  
Author(s):  
Ehsan Shakeri ◽  
Gholamreza Latif-Shabgahi ◽  
Amir Esmaeili Abharian
Keyword(s):  
Stochastic Model ◽  
Time Window ◽  
Joint Probability ◽  
Planck Equation ◽  
Stirred Tank ◽  
Continuous Stirred Tank Reactor ◽  
Fokker Planck Equation ◽  
Continuous Stirred Tank ◽  
Fokker Planck

Over the years, different methods have been presented to control continuous stirred tank reactors (CSTRs) in which stochastic behavior of process has rarely been considered. This article uses the stochastic model of CSTR to compute the temperature of coolant as process input in order to control the joint probability density function (PDF) of process concentration and temperature. The computation is carried out based on receding horizon-model predictive control (RH-MPC). Since observer has important role in the determination of process input, we use a nonlinear stochastic Fokker-Planck observer to calculate process PDF. The CSTR model is nonlinear and complex, so the particle swarm optimization (PSO) algorithm is used for simplification of computations and for determination of the optimal value of process input. For this purpose, an MPC problem is described for which the cost function is defined based on the difference between the process PDF and a desired PDF. In this definition, temperature limitation of the coolant and the corresponding Fokker-Planck equation are both assumed as the problem constraints. When this MPC problem is solved by the use of PSO, the process input is calculated for each time window. The existence and uniqueness of our optimal solution is also studied. In the article, the Fokker-Planck equation for CSTR model will be solved by the use of path integral method. In this way, the joint PDF of process concentration and temperature is obtained for any instance of time. The simulation results are also obtained to evaluate the proposed method.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Time-changed fractional Ornstein-Uhlenbeck process

2020 ◽  
Vol 23 (2) ◽  
pp. 450-483 ◽  
Author(s):  
Giacomo Ascione ◽  
Yuliya Mishura ◽  
Enrica Pirozzi
Keyword(s):  
Planck Equation ◽  
Uhlenbeck Process ◽  
Fokker Planck Equation ◽  
Ornstein Uhlenbeck Process ◽  
Fokker Planck

AbstractWe define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process.

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