Path probability

K. Masuda-Jindo, R. Kikuchi and R. Thomson, "Theory and Applications of the Cluster Variation and Path Probability Method" (Plenum, 1996) in press. 

For the PPM, corresponding to the free energy of the CVM is the Path Probability Function (hereafter PPF), P t t -t- At), which is an explicit function of time and is defined as the product of three factors Pj, P2 and P3. Each factor is provided in the following in the logarithmic expression. 

T. Mohri, Y. Ichikawa, T. Nakahara and T. Suzuki, Theory and Applications of Cluster Vajiation and Path Probability Methods, ed. by J. M. Sanchez et al. Plenum, New York, in press. 

The present information does not allow discrimination between these two routes. The high-pH path probably involves concerted attack of OH and Mn04 upon neutral amine because the possibility of attack of oxidant upon C6H5CH2NH or C6H5CHNH2 is too remote in view of known acidities of amines. 

Then the path probability from (7.3) consists of a product of such delta functions. Due to the singular nature of such a path probability it is more convenient to view the entire deterministic trajectory as represented by its initial state z0. In this case the transition path ensemble from (7.10) reduces to a distribution of initial conditions z0 yielding pathways connecting srf with 2%... 

The dynamical randomness of the nonequilibrium process can be characterized by the decay of the path probabilities as defined by the entropy per unit time [12-14] ... 

The first line is known as the Shannon-McMUlan-Breiman theorem [1], the second is its extension for stationary states which are not time-reversal symmetric. The entropy per unit time h characterizes the dynamical randomness of the process. The faster the decay of the path probabilities, the larger the proliferation of these paths as time increases. Therefore, the larger the entropy per unit time h, the higher the temporal disorder of the time evolution. The time-reversed entropy per unit time characterizes the decay of the time reversals of the typical paths in a similar way, and it thus characterizes the dynamical randomness of the backward paths. 

Termolecular collision were also studied. Such collisions may be regarded as a sequence of two binary collisions. In the first, a single Ar atom collides with a benzene molecule and in the second the binary collision complex collides with an additional Ar atom. The beginning and the end of each collision was determined by FOBS. The starting distance between the centers of mass of the binary complex, BAr and the second atom Ar , R n, of the second collision is chosen randomly Irom the Iree paths probability density function... 

In Fig. 12, the average similarity //20 and the average division time are plotted for 50 randomly chosen reaction networks as a function of the path probability p. Roughly speaking, the networks with //2o > 0.9 belong to phase 2, and those with H20 < 0.4 belong to (1), empirically. Hence, for p > 0.2, phase 1 is observed for nearly all the networks (e.g., 48/50), while for lower path rates, the fraction of phase 2 or phase 3 increases. The value p 0.2 gives the phase boundary in this case. 

The probability (density) to observe a given path x T) depends on the probability of its initial condition and the specific dynamics of the system. For Markovian processes, i.e., processes for which the probability to move from Xt to Xt+At after one time step At depends only on xt and not on the history of the system prior to f, the total path probability can be written as the product of single time step transition probabilities p xt Xt At))... 

Since we want to focus on reactive pathways we now restrict the path ensemble to those pathways that start in region A, the reactant region, and end in region B, the product region. This is achieved by multiplying the path probability V[x T) with the appropriate characteristic functions ... 

The general transition path sampling formalism can be applied to various ensembles of pathways differing both in the distributions of initial conditions as well as in the particular transitions probabilities. The specific form of the path probability depends on the process one wants to study and is not imposed by the transition path sampling technique itself. In the following we discuss several path probabilities that frequently occur in the study of condensed matter systems. 

In this expression, all terms stemming from the forward segment have cancelled because the generation probability of the forward shot is identical to the d mamical path probability. Such a cancellation of terms does not occur for the backwards segment, because the corresponding generation probability p x x ) is not a priori related in any simple way to the corresponding dynamical path probability. The acceptance probability (29), however, simplifies considerably if p and p obey a microscopic reversibility condition with respect to the distributions of initial conditions ... 

The combination of Co Kbpy)2 -O2 with these substrates yields product profiles that are similar, which indicates that the reaction path probably involves the same reactive intermediate (species 2, Scheme 6-2). Scheme 6-2 outlines catalytic cycles for the demethylation of N-methyl anilines, and the dehydrogenation of benzylamine and benzyl alcohols. In the case of the latter catalyst [Con(bpy> ] is inactive and must be neutralized with one equivalent of HO-... 

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