Ergodic

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 very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

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] ... 

As t becomes large, this should approach the equilibrium value (fi), which for an ergodic system is... 

Arnoid V i and Avez A 1968 Ergodic Problems of Classical Mechanics (New York Ben]amin)... 

A RRKM nnimolecnlar system obeys the ergodic principle of statistical mechanics[H],... 

Sf, spectra of acetylene regular and ergodic regions J. Chem. Phys. 85 453-65... 

Polik W F, Guyer D R, Miller W H and Moore C B 1990 Eigenstate-resolved unimolecular reaction dynamics ergodic character of Sq formaldehyde at the dissociation threshold J. Chem. Phys. 92 3471-84... 

Oref I and Rabinovitch B S 1979 Do highly excited polyatomic molecules behave ergodically Acc. Chem. Res. 12 166-75... 

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. 

P. Walters. An Introduction to Ergodic Theory. Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1981)... 

Figure 8.16 from Frantz, D D, D L Freeman and J D DoU 1990. Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking applications to atomic clusters. The Journal of Chemical Physics 93 2769-2784. 

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