Function stationary probability distribution

Fig. 5.12 The volume-dependence of the exact stationary probability distribution function of system / and comparison with its Euler-McLaurin approximation (after Ebeling and Schimansky-Geier).

The Lyapunov function < ), (2.13), is both the thermodynamic driving force toward a stable stationary state and determines the stationary probability distribution of the master equation. The stationary distributions (2.33, 2.34) are nonequilibrium analogs of the Einstein relations at equilibrium, which give fluctuations around equilibrium. 

Thus the stationary probability distribution of the master equation in the eikonal approximation is a Lyapunov function, which gives necessary and sufficient conditions of the existence and stability of non-equilibrium stationary states and provides a measure of relative stability on the basis of inhomogeneous fluctuations, (6.17). 

The function yields the stationary probability distribution of a stochastic, birth-death master equation for single variable systems... 

In order to get some ideas about the mechanism of transitions between steady states let us consider a qualitative model, in which fluctuations are described by adding a random force term to the determnistic description. Let us also assume that the matrix of the noise correlation function is completely diagonal with noise strength equal to B, The stationary probability distribution is then... 

The parallel-replica method [5] is perhaps the least glamorous of the AMD methods, but is, in many cases, the most powerful. It is also the most accurate AMD method, assuming only first-order kinetics (exponential decay) i.e., for any trajectory that has been in a state long enough to have lost its memory of how it entered the state (longer than the correlation time icorr, the time after which the system is effectively sampling a stationary distribution restricted to the current state), the probability distribution function for the time of the next escape from that state is given by... 

It is instinctive to first consider isotropic rotational diffusion of an atom, which is a simple but somewhat accurate description of a molecular hquid like liquid methane just below rmeiting, a Situation wherein the centers of mass of the molecules remain stationary, but the molecules rotate freely about their centers of mass. If we use the polar angle — 9, (p) to define the orientation of the vector d, the probability distribution function, G (see equation 12), which... 

An interesting investigation on the influence of multiplicative non-white noise in an analog circuit simulating a Langevin equation of a Brownian particle in a double-well potential has been carried out by Sancho et al. This device allowed them to study the stationary properties as a function of the noise correlation time. Theory in a white-noise limit cannot provide a satisfactory explanation for experimental results such as a relative maximum of the probability distribution and the maximum position in the stationary distribution for noises of weak intensity. 

Doubling the volume fraction of one phase doubles the probability of solute interaction and, consequently, doubles its contribution to retention. There is another interesting outcome from the results of Purnell and his co-workers. Where a linear relationship existed between the retention volume and the volume fraction of the stationary phase, the linear functions of the distribution coefficients could be summed directly, but their logarithms could not. In many classical thermodynamic descriptions of the effect of the stationary-phase composition on solute retention, the stationary-phase composition is often taken into account by including an extra term in the expression for the standard free energy of distribution. The results of Purnell indicate that this is not acceptable, as the solute retention or distribution coefficient is linearly not exponentially related to the stationary-phase composition. The stationary phases of intermediate polarities can easily be constructed from... 

After some iterations (typically 1000) it is possible to find an approximate distribution for the probability of success and also for the type I (the process is considered non-stationary while it is in fact) and II errors. This procedure is applied for each test and for different parameters, thus portraying the sensitivity of the probability distribution as a function of the parameter values. Even for non-parametric tests, such sensitivity can be studied with respect to the size of the window, for instance. 

In this chapter, we use a type of initial condition that is different from the waterbag used in Refs. 15 and 18, and we show that (i) probability distribution functions do not have power-law tails in quasi-stationary states and (ii) the diffusion becomes anomalous if and only if the state is neither stationary nor quasi-stationary. In other words, the diffusion is shown to be normal in quasi-stationary states, although a stretched exponential correlation function is present instead of usual exponential correlation. Some scaling laws concerned with degrees of freedom are also exhibited, and the simple scaling laws imply that the results mentioned above holds irrespective of degrees of freedom. 

A picture of this type, while useful in developing an intuitive feeling for the wave-mechanical equations, must not be taken too seriously, for it is not completely satisfactory. Thus it cannot be reconciled with the existence of zeros in the wave functions for the stationary states, corresponding to points where the probability distribution function becomes vanishingly small. 

It is possible to consider a stationary distribution of the process I, (if it exists) and to optimize the expected value of an objective function with respect to the stationary distribution. Typically, such a stationary distribution cannot be written in a closed form and is difficult to compute accurately. This introduces additional technical difficulties into the problem. Also, in some situations the probability distribution of the random variables D, is given in a parametric form whose parameters are decision variables. We will discuss dealing with such cases later. 

The simple, linear relationship between volume fraction of one component of a binary mixture and the retention volume of a solute, where there is only weak interaction between the individual components, again, is to be expected. The volume fraction of each phase will determine the probability that a given solute molecule will interact with a molecule of that phase, in much the same way as the partial pressure of a solute in a gas, determines the probability that a solute molecule will collide with a gas molecule. For those phase systems that give a linear relationship between retention volume and volume fraction of stationary phase, it is clear that the linear functions of the distribution coefficients could be summed directly, but their logarithms could not. The results of Purnell indicate that when there is little, or only weak, interaction between the components of a binary mixture used as a stationary phase, the solute retention or distribution coefficient is linearly, not exponentially, related to the stationary phase composition. 

All the wavefunctions that will be considered in this book arc functions only of the spatial coordinates of the system they do not contain time as a variable. This means that we are confining our attention to systems where the probability of finding the particle concerned at various points in space does not vary with time. This does not mean that the particle does not move, but merely that the probability distribution associated with its movement does not vary with time. For example, an electron orbiting a hydrogen atom moves around the nucleus, but its measurable properties do not vary with time. Such systems are known as stationary states, and they include all the stable states of atoms and molecules. 

Stationary process Stochastic process in which the probability distribution function is not a function of time. 

What are the advantages of the formulation, (18.8) The term t x) is the net flux of the deterministic kinetics, (18.6), and the derivative of the state function (j> is the species specific affinity, (/tx — for the linear case, or (/Xx — /x x) for the nonlinear case, the driving force for the reaction toward a stationary state. Thus we have a flux-driving force relation. Second, the formulation is symmetric with respect to /+(x) and t (x), which is not the case with other formulations. Third, the state function 4> determines the probability distribution of fluctuations in x from its value at the stationary state, see (2.34). Further, as we shall show shortly, the term D x) is a measure of the strength... 

Major methods used to account for mixing in reactors. Illustrations on statistically stationary field of a velocity component. DNS Direct Numerical Simulation PDF Probability Density Function I Internal distribution function RTD Residence Time Distribution <> macro-scale averaged reactor-scale averaged. 

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