Ergodic theorem

A similar theorem is called the quasi-ergodic theorem by Brennan. See footnote 5. 

In its extreme form the ergodic hypothesis is clearly untenable. Only probability statements can be made in statistical mechanics and these have nothing to do with sequences in time [117]. Not surprisingly, a totally convincing proof of the ergodic theorem in its many guises has not been formulated. The current concensus still is that an axiomatic basis, completely independent of ergodic theory should be chosen [115] for the construction of statistical mechanics. 

Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

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... 

This is a simple analogy of Birkhoff s ergodic theorem for dynamical systems, see A.I. Khinchin, Mathematical Foundation of Statistical Mechanics (Dover, New York 1949) L.E. Reichl, A Modern Course in Statistical Physics (University of Texas Press, Austin, TX 1980) ch. 8. 

Abstract The aim of this contribution is to derive macroscopic equations describing flow of two-ionic species electrolytes through porous piezoelectric media with random, not necessarily ergodic, distribution of pores. Under assumption of ergodi-city the macroscopic equations simplify and are obtained by using the Birkhoff ergodic theorem. 

Here K denotes the Lebesgue measure of K. Of crucial importance is the Birkhoff ergodic theorem which states that for / // (fi), a > 1,... 

The ergodic theorem states the equivalence of the average of a molecular ensemble with the time average of a single molecule. We have, therefore, compared the values for the conformational transition of our standard molecule. 

The existence of Lyapunov exponents is proved, under a general condition, by the multiplicative ergodic theorem of Oseledec [8], However, the convergence of the exponents is found to be quite slow (algebraically) in time for a generic dynamical system [9], due to its nonhyperbolicity. 

A fundamental hypothesis of statistical mechanics is the ergodic theorem. Basically it says the system evolves so quickly in the phase space that it visits all of the possible phase points during the time considered. If the system is eigodic, the ensemble average is equivalent to the time average over the trajectory for the time period. The ergodidty of a system depends on the search procedure, force... 

The ergodic theorem of statistical mechanics (see also Section 1.4.2) states that, for realistic systems, these two kinds of averaging, Eqs (5.10) and (5.11) yield identical results. As example of an application of this theorem consider the total kinetic energy of the system. The corresponding dynamical variable is... 

Therefore, under the ergodic theorem, Eq. (5.11) implies that... 

The stationary nature of our system and the ergodic theorem (see Section 1.4.2) imply that time and ensemble averaging are equivalent. This by no means implies that the statistical information in a row of the table above is equivalent to that in a column. As defined, the different systems j = 1,..., 7 7 are statistically independent, so, for example, statistically independent so that (ni(Zi)ni(Z2)) 7 (ni(Zi))time series provide information about time correlations that is absent from a single time ensemble data. The stationary nature of our system does imply, as discussed in Section 6.1, that ( (Zl) (Z2)> depends only on the time difference Z2 — Zi. 

We also note the equality of time and ensemble averages which is a fundamental tenet of statistical mechanics (ergodic theorem)... 

The validity of this assumption (whose content is not equivalent to the content of the usual ergodic theorem) is, of course, difficult to assess, and represents, therefore, an additional postulate in the framework of our theory. 

The actual evaluation of the energy loss is based on explicit use of the ergodic theorem. The quantity A, is defined as the energy lost per unit time by the particle during its traversal in the region of the well ... 

In contrast to the effective harmonic prescription for centroid-based dynamics, in CMD the force is a unique function of the system. That is, the force on a centroid trajectory at some time and position in space is not different from the force experienced by a different centroid trajectory at that same point in space but at a different time. Furthermore, the centroid trajectories are derived from the same effective potential as the one giving the exact centroid statistical distribution so that a centroid ergodic theorem will hold. The CMD approach satisfies this condition, while the analytically continued optimized LHO theory may not. Finally, CMD recovers the exact limiting expressions for globally harmonic potentials and for general classical systems. 

We have already seen two kinds of ergodic theorems, the principal theorem (Section IIID.2) and the first result in Section VIE, Here is another such result, which also generalizes to a quantum situation, a well-known classical theorem. 

Oseledek, V. I. (1968). A multiplicative ergodic theorem characteristic Lyapunov exponents of dynamical systems, Trudy Mask. Mat. (Obsc.), 19, 179-210. 

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