Ergodicity transition state theory

F. Tal and E. Vanden-Eijnden (2006) Transition state theory and dynamical corrections in ergodic systems. Nonlinearity 19, p. 501 31. E. Vanden-Eijnden and F. Tal (2005) Transition state theory Variational formulation, dynamical corrections, and error estimates. J. Chew,. Phys. 123, 184103 T. S. van Erp and P. G. Bolhuis (2005) Elaborating transition interface sampling method. J. Cow,p. Phys. 205, p. 157... 

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)]. 

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. 

Recall the well-known theorem of the standard theory of the ergodic Markov chains one can state the following (e.g. Feller [4]) In any finite irreducible, aperiodic Markov chain with the transition matrix P, the limit of the power matrices/ exists if r tends to infinity. This limit matrix has identical rows, its rows are the stationary probability vector of the Markov chain, y = [v,Vj,...,v,...,v ], that is v = v P, fiuthermore v, >0 ( = 1,...,R) and... 

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