Ergodic process

Turecek, R N-C bond dissociation energies and kinetics in amide and peptide radicals. Is the dissociation a non-ergodic process J. Am. Chem. Soc. 2003,125, 5954—5963. 

Let the discrete time series X = X(t ), 1 s k s L given at equidistant time steps t < = k-At be a realization of a stationary stochastic ergodic process with zero mean. The time series is divided into M section of equal length of N datapoints, thus L = M-N Then the RV s are... 

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... 

Thus, the well-known concept of stationary reaction rates limitation by "narrow places" or "limiting steps" (slowest reaction) should be complemented by the ergodicity boundary limitation of relaxation time. It should be stressed that the relaxation process is limited not by the classical limiting steps (narrow places), but by reactions that may be absolutely different. The simplest example of this kind is an irreversible catalytic cycle the stationary rate is limited by the slowest reaction (the smallest constant), but the relaxation time is limited by the reaction constant with the second lowest value (in order to break the weak ergodicity of a cycle two reactions must be eliminated). 

The spontaneous emission in atomic problems and the decay of unstable particles are irreversible processes which manifest the ergodicity of these systems. It is therefore interesting to compare the mechanism of irreversibility which is involved to that in the usual many-body systems such as a classical gas. 

Remark. Consider a Markov process that can be visualized as a particle jumping back and forth among a finite number of sites m, with constant probabilities per unit time. Suppose it has a single stationary distribution psn, with the property (5.3). After an initial period it will be true that, if I pick an arbitrary t, the probability to find the particle at n is ps . That implies that psn is the fraction of its life that the particle spends at site n, once equilibrium has been reached. This fact is called ergodicity. For a Markov process with finitely many sites ergodicity is tantamount to indecom-posability. ) In (VII.7.13) a more general result for the times spent at the various sites is obtained. 

There have been a few interesting fundamental papers on ergodicity 42 45 The distinction was emphasized between true equilibrium and quasiequilibrium phenomena, which now are frequently observed in NMR. An isolated finite system should not be expected to become ergodic. Two other fundamentally interesting processes that have been demonstrated with solid-state NMR are dephasing caused by randomization of geometric phase,46 47 and the possibility of the chaotic behaviour of spin systems.48,49... 

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... 

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