Ergodic assumption

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... 

By making an ergodic assumption, the computation of Eq. (26) can be simplified. This assumption is familiar in the theory of stochastic processes (54). The sufficient con-... 

Under suitable ergodicity assumptions (such that in the long-time limit, all distributions converge to the invariant distribution), we have... 

Proof The proof is very simple, and follows almost directly from our ergodicity assumption. If we assume that... 

It is important to realize that the average dissociation rate coefficient for a microcanonical ensemble may or may not accurately eflect the phenomenological behavior of a given molecular system. The assumption that it will accurately reflect the kinetics is of course known as the ergodic assumption. The conditions governing the validity of the ergodic assumption have been thoroughly discussed and documented elsewhere hence we shall not repeat the discussion in any detail here. However, one or two comments are in order. 

The second fundamental assumption is the ergodic hypothesis. Accord-... 

In this section we present a derivation of the FT based on stochastic dynamics. In contrast to deterministic systems, stochastic dynamics naturally incorporates crucial assumptions needed for the derivation, such as the ergodicity hypothesis. The derivation we present here follows the approach introduced by Crooks-Kurchan-Lebowitz-Spohn [38, 39] and includes some results recently obtained by Seifert [40] using Langevin systems. 

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. 

Macroscopic equations are given in Sect. 4 without the assumption of er-godicity. In Sect. 5 we provide comments on the case where ergodicity applies. 

Letting e tend to zero in the sense of stochastic two-scale convergence in the mean we arrive at the homogenized equations. Without the assumptions of ergodicity the fields involved still depend on u) fi. 

In conclusion, it is worth reflecting on a classical trajectory study of neutral ethane [335] in which it was found that there were dynamical restrictions to intramolecular energy transfer among C—H motions and between these and C—C motions. It was pointed out [335] that this non-ergodicity might not produce results observable at present levels of experimental resolution. This is probably the situation in mass spectrometry. QET is a respected theory in mass spectrometry because, proceeding from clearly stated assumptions, it is mathematically tractable and is able to explain the currently available experimental data. 

Perhaps the point to emphasise in discussing theories of translational energy release is that the quasiequilibrium theory (QET) neither predicts nor seeks to describe energy release [576, 720], Neither does the Rice— Ramspergei Kassel—Marcus (RRKM) theory, which for the purposes of this discussion is equivalent to QET. Additional assumptions are necessary before QET can provide a basis for prediction of energy release (see Sect. 8.1.1) and the nature of these assumptions is as fundamental as the assumption of energy randomisation (ergodic hypothesis) or that of separability of the transition state reaction coordinate (Sect. 2.1). The only exception arises, in a sense by definition, with the case of the loose transition state [Sect. 8.1.1(a)]. 

It is on account of this last property that the definition of ergodic systems and the assumption that the gas models are ergodic appear in Boltzmann s investigations (ef. Section 11). 

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 analysis of Krod shows in fact that for t= + p lies closer to the ergodic distribution given in Eq. (30) than at t=ta and that similarly 2(0 lies closer to the corresponding 2 value at < = + >.222 However, it would be a mistake to confound this result, which corresponds to statement (XV) in Section 23d, with the assertion that for t— + < > the ergodic distribution and the corresponding 2 value are approximately attained. The latter corresponds to Gibbs s indispensable statement (XV ). It is precisely for the periodic systems treated by Krod that it is particularly easy to see that the transition from (XV) to (XV ) necessarily invokes an assumption similar to the ergodic hypothesis.224 (Cf. the remarks in Sections 23a and 23b). 

An aspect of the ES FR that has not been fully exploited as yet is the fact that the dissipation function is sensitive to the choice of the distribution function. Therefore, if a system is presumed to have a particular equilibrium distribution function, with an associated dissipation function, then a field is applied, the transient ES FR should be satisfied for all time. This provides a way of testing if a system is equilibrated, for example. If the FR is not satisfied for the presumed Q, then it indicates that the equilibrium state is not what was expected. This fact has recently been used to establish that domains of the nondissipative, nonequilibrium distributions of glassy systems can be described by Boltzmann weights. Apart from the reversibility of the dynamics, the other key assumption in the derivation of the transient ES FR is that the initial distribution and the dynamics are ergodically consistent. In the same paper, Williams and Evans demonstrated that away from the actual glass... 

The microcanonical ensemble may be depleted in the vicinity of the transition state by the absence of trajectories in the reverse direction. This assumption is often referred to as the ergodic approximation, that the microcanonical ensemble is rapidly randomized behind the reaction bottleneck faster that reactive loss can perturb the distribution. 

The key idea that supplements RRK theory is the transition state assumption. The transition state is assumed to be a point of no return. In other words, any trajectory that passes through the transition state in the forward direction will proceed to products without recrossing in the reverse direction. This assumption permits the identification of the reaction rate with the rate at which classical trajectories pass through the transition state. In combination with the ergodic approximation this means that the reaction rate coefficient can be calculated from the rate at which trajectories, sampled from a microcanonical ensemble in the reactants, cross the barrier, divided by the total number of states in the ensemble at the required energy. This quantity is conveniently formulated using the idea of phase space. 

It is important to note that this assumption yields an RRKM rate coefficient, RRKM, that is an upper bound to the ergodic rate coefficient, ergodic, since every reactive trajectory (with xr J) necessarily has a positive velocity through the dividing surface. Thus, RRKM theory may be implemented in a variational manner, with the best approximation to ergodic obtained from the dividing surface S that provides the smallest rrkm-... 

In cases like D2CO or NO2 comparison with experimental data on a state-specific level are ruled out entirely and one has to retreat to more averaged quantities like the average dissociation rate, (fc), and the distribution of rates, Q(k). If the dynamics is ergodic — the basic assumption of all statistical theories — one can derive a simple expression for Q k), which had been established in nuclear physics in order to describe the neutron emission rates of heavy nuclei [280]. These concepts have since developed into the field of random matrix theory (RMT) and statistical spectroscopy [281-283] and have also found applications in the dissociation of energized molecules [121,284-286]. 

The behavior is clearly non-ergodic in the sense that ensemble and time average differ considerably. One direct reason is the fact that the conformational transition can only be measured for a limited time until the molecule is bleached [30,31]. On the other hand, the relaxation rate of 37 classes of single DNA molecules cover relaxation rates between 1 and 55s h This means that under the assumption that all TMR tagged DNA molecules are identical they can switch from one relaxation regime to another which stays constant during the survival time of photobleaching [31]. 

Our objective is to understand the statistical properties of the reactive trajectories in the ensemble (2). We will try to do so under minimum assumptions about the dynamics of x t), but we have to require the following from the start. First we require that the dynamics be Markov, i.e. given x t), its future x t ) . t > t and its past x t ) . t probability density m(x), i.e. given a suitable observable F x) and a generic trajectory x t) —00 [Pg.456]

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