ε expansion of the stationary probability distribution of two-dimensional Fokker-Planck equations

1991 ◽  
Vol 44 (4) ◽  
pp. 2450-2456 ◽  
Author(s):  
Hu Gang ◽  
Zheng Ren-rong
Keyword(s):  
Probability Distribution ◽  
Stationary Probability ◽  
Two Dimensional ◽  
Stationary Probability Distribution ◽  
Fokker Planck Equations ◽  
Fokker Planck

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.

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (03) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

STATISTICAL PROPERTIES OF INTENSITY FLUCTUATION OF SATURATION LASER MODEL DRIVEN BY CROSS-CORRELATED ADDITIVE AND MULTIPLICATIVE NOISES

2010 ◽  
Vol 24 (14) ◽  
pp. 2175-2188 ◽  
Author(s):  
PING ZHU ◽  
YI JIE ZHU
Keyword(s):  
Probability Distribution ◽  
Cross Correlation ◽  
Laser Intensity ◽  
Laser System ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Model Driven ◽  
Multiplicative Noises ◽  
Laser Model ◽  
The Stability

Statistical properties of the intensity fluctuation of a saturation laser model driven by cross-correlation additive and multiplicative noises are investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution Pst(I), the relaxation time Tc, and the normalized variance λ2(0) of the system. By numerical computation, we discussed the effects of the cross-correlation strength λ, the cross-correlation time τ, the quantum noise intensity D, and the pump noise intensity Q for the fluctuation of the laser intensity. Above the threshold, λ weakens the stationary probability distribution, speeds up the startup velocity of the laser system from start status to steady work, and attenuates the stability of laser intensity output; however, τ strengthens the stationary probability distribution and strengths the stability of laser intensity output; when λ < 0, τ speeds up the startup; on the contrast, when λ > 0, τ slows down the startup. D and Q make the relaxation time exhibit extremum structure, that is, the startup time possesses the least values. At the threshold, τ cannot generate the effects for the saturation laser system, λ expedites the startup velocity and weakens the stability of laser intensity output. Below threshold, the effects of λ and τ not only relate to λ and τ, but also relate to other parameters of the system.

scholarly journals Statistical Mechanics of Unconfined Systems: Challenges and Lessons

2021 ◽  
Vol 3 (1) ◽  
pp. 8
Author(s):  
Bruno Arderucio Costa ◽  
Pedro Pessoa
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
A Priori ◽  
Ideal Gas ◽  
Stationary Probability ◽  
De Sitter ◽  
Stationary Probability Distribution ◽  
Local Measurements ◽  
Priori Information

Motivated by applications of statistical mechanics in which the system of interest is spatially unconfined, we present an exact solution to the maximum entropy problem for assigning a stationary probability distribution on the phase space of an unconfined ideal gas in an anti-de Sitter background. Notwithstanding the gas’ freedom to move in an infinite volume, we establish necessary conditions for the stationary probability distribution solving a general maximum entropy problem to be normalizable and obtain the resulting probability for a particular choice of constraints. As a part of our analysis, we develop a novel method for identifying dynamical constraints based on local measurements. With no appeal to a priori information about globally defined conserved quantities, it is therefore applicable to a much wider range of problems.

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