Ergodic approximation

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. 

In the first case, the limit (for t- co) distribution for the auxiliary kinetics is the well-studied stationary distribution of the cycle A A , +2, described in Section 2 (ID-QS), (15). The set A j+], A . c+2, , n is the only ergodic component for the whole network too, and the limit distribution for that system is nonzero on vertices only. The stationary distribution for the cycle A i+] A t+2. ., A A t+i approximates the stationary distribution for the whole system. To approximate the relaxation process, let us delete the limiting step A A j+] from this cycle. By this deletion we produce an acyclic system with one fixed point, A , and auxiliary kinetic equation (33) transforms into... 

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. 

If one discards the ergodic hypothesis, or tries to retain it in a modified form,117 then for the time being one lacks any criterion to decide whether Eq. (34) is still valid or even whether it represents a somehow useful approximation. 

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). 

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-... 

If the classical dynamics is ergodic and intrinsically RRKM, one might expect that the classical rate constant approximates the average rate of the quantum mechanical state-specific rates. That is indeed the case for the dissociation of HO2 (Fig. 12 of Ref. 60) the classical rate is only slightly smaller than the average quantum mechanical rate. The same holds also... 

A Markov chain is ergodic if it eventually reaches every state. If, in addition, a certain symmetry condition - the so-called criterion of detailed balance or microscopic reversibility - is fulfilled, the chain converges to the same stationary probability distribution of states, as we throw dice to decide which state transitions to take one after the other, no matter in which state we start. Thus, traversing the Markov chain affords us with an effective way of approximating its stationary probability distribution (Baldi Brunak, 1998). 

These results constitute the first major steps in formalizing statistical theories of reaction dynamics and relating statistical molecular behavior to ergodic theory. Specifically, they demonstrate that by invoking a mixing condition on a well-chosen R we obtain an analytically soluble model for P(t) which is asymptotically well approximated by exponential decay with rate K. The rate of decay is directly affected by the relaxation time t and equals ks(R) in the limit t - 0. A similar approach can be used46 to provide an ergodic theory basis for product distributions. 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

Relaxor ferroelectrics differ from the ferroelectric materials described previously in many ways. Of fundamental importance is the fact that the relative permittivity shows a wide diffuse peak which is approximately the same magnitude as in a normal ferroelectric. The broad peak is relatively temperature insensitive, but the transition is frequency dependent, with a maximum value at a temperature T, rather than a sharp transition at a temperature (Figure 6.18a and b). The region under the curve represents a state called the ergodic relaxor (ER) state (see following text). 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

Mattingly, J., Stuart, A., Higham, D. Ergodicity for SDEs and approximations locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101, 185-232 (2002). doi 10.1016/S0304-4149(02)00150-3... 

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