Ergodicity function

In the Maximum Entropy Method (MEM) which proceeds the maximization of the conditional probability P(fl p ) (6) yielding the most probable solution, the probability P(p) introducing the a priory knowledge is issued from so called ergodic situations in many applications for image restoration [1]. That means, that the a priori probabilities of all microscopic configurations p are all the same. It yields to the well known form of the functional 5(/2 ) [9] ... 

It is possible to limit our choice for stochastic modeling by stationary, linear, nonlinear, and ergodic models in combination with deterministic function. In this case the following well studied models can be proposed for the accepted concept [1] ... 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

If Pm x) is independent of xq then the system is said to be ergodic. In this case, various temporal averages over some function h x) may be conveniently rewritten as spatial averages over Pm x) ... 

With a computer program that evaluates r as a function of time for a given U(r) we can use the ergodic hypothesis (which states that the time average over a long time is equal to the configuration average) and write... 

In order to evaluate the autocorrelation function we again exploit the ergodic hypothesis and replace the average over phase space ( ) by a time average writing,... 

Here, w(xfc) is the weighting factor for any property at a given position on the fcth step xfc. For example, for a constant-temperature molecular dynamics or a Metropolis MC run, the weighting factor is unity. However, we wish to leave some flexibility in case we want to use non-Boltzmann distributions then, the weighting factor will be given by a more complicated function of the coordinates. The ergodic measure is then defined as a sum over N particles... 

Because the velocity u contains the random component u, the concentration c is a stochastic function since, by virtue of Eq. (2.2), c is a function of u. The mean value of c, as expressed in Eq. (2.5), is an ensemble mean formed by averaging c over the entire ensemble of identical experiments. Temporal and spatial mean values, by contrast, are obtained by averaging v ues from a single member of the ensemble over a period or area, respectively. The ensemble mean, which we have denoted by the angle brackets ( ), is the easiest to deal with mathematically. Unfortunately, ensemble means are not measurable quantities, although under the conditions of the ergodic theorem they can be related to observable temporal or spatial averages. In Eq. (2.7) the mean concentration (c) represents a true ensemble mean, whereas if we decompose c as... 

If Gi,..., Gm are all ergodic components of the system, then there exist m independent positive linear functionals b (c),..., b c), such that nd... 

This is the zero-one law for multiscale networks for any l,i, the value of functional b (30) on basis vector d, b (e ), is either close to one or close to zero (with probability close to 1). We already mentioned this law in discussion of a simple example (31). The approximate equality (71) means that for each reagent A e there exists such an ergodic component G of that A transforms when t -> 00 preferably into elements of G even if there exist paths from A to other ergodic components of W. 

Index L is given to GL to distinguish it from the Lyapunov function for closed systems. Strictly speaking, it is not the Lyapunov function, since it cannot be differentiated on the hyperplanes prescribed by the equations 2, = 2 . Therefore, instead of estimating its derivative by virtue of eqn. (152), let us determine its decrease for a finite period of time x. Actually, we will find an ergodicity coefficient [42] for the matrix exp xK... 

A unique feature of SM-FRET experiments is their ability to provide information on the correlations between structure and dynamics. One way of extracting this information is by monitoring the distribution of (R)tw as a function of Tw [30] Correlations between conformational structure and the time scale on which it moves can be obtained from the different values of Tw at which different subsets of the ensemble which are characterized by different values of R reach the ergodic limit. 

A dynamical system is said to be ergodic, if every invariant function, i.e. satisfying /(T(x)w) = f(u) is constant almost everywhere in fi. 

Here, the angular brackets denote an ensemble average, which is the same as time average from the ergodic hypothesis. 8ap and 8,- are Kronecker delta, and 8(f — t ) is Dirac s delta function. 

The fundamental assumption underlying this investigation is the hypothesis that the gas models are ergodic systems (cf. Section 10). With the help of this hypothesis Boltzmann computed the time average of, for instance, the kinetic energy of each atom (the same value is obtained for all atoms ).108 Likewise he calculated the time average of other functions (q, p) which characterize the average distribution of state. 

The corresponding correlation functions are sketched for characteristic temperatures in Fig. 9b. The critical temperature Tc marks the crossover from an ergodic state (liquid) to a nonergodic state (glass). In the liquid state, a two-step process describes the decay of the correlation function, where/is the fraction relaxed by the slow process (a-process) and 1 / is the part decaying due to the... 

In the limit of long times and if the ergodic assumption holds, then we have 7 = (/), where (I) is the ensemble average. As usual, we may generate many intensity trajectories one at a time, to obtain the ensemble-averaged correlation function... 

Figure 9. The probability density function of 7 = T+ /T for the case i t+(t) = / (r) 1 P(I) is a delta function on (/) = 1 /2. In the nonergodic phase, 7 is a random function for small values of a the P(J) is peaked on 7 = 0 and 7=1, indicating a trajectory which is in state off or on for a period which is of the order of measurement time T.

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