Distribution in the bottleneck

The lack of equilibrium between reactant and product zones leads to a distinctly nonequilibrium distribution in the bottleneck, but fortunately it is one that can be expressed easily (11) in terms of the equilibrium distribution and trajectory information. To do this, the equilibrium probability density Peq(p,q) is split into two nonoverlapping parts, Pa(p,g) and Pc(p,c[), the former originating from an equilibrium distribution in A, the latter from an equilibrium distribution in C. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

For the different modules of the product there are different sourcing strategies which can be drawn from whether the product is a strategic, bottleneck, leverage or noncritical product module. The geographic distribution in the sourcing process involves Europe, Asia, North America and Australia. 

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

Fig. 4.18 The denser the quants planned on the resources the more bottlenecks over time arise and the conflict numbers rise. An evolution is shown of three optimizations with permanent improvements. In the upper Gantt chart the conflicts are mostly in the middle time horizon of the Gantt chart. In the middle Gantt chart the conflicts are distributed over the whole time horizon.

Since every phase point (except for uninteresting ones accessible from neither A nor C) satisfies one of the two trajectory conditions above and no phase point satisfies both, the two terms add up to the equilibrium density on the other hand, each term separately represents the situation in which an equilibrium distribution of trajectories attacks the bottleneck from one... 

Suppose one is interested (as Torrie and Valleau were) in the equilibrium probability of an r value, say r=30, outside the observed range alternatively, one may suspect that p(r), the true equilibrium distribution of r, is bimodal, with another peak around r=40, but that a bottleneck around r=30 is preventing this peak from being populated. 

How does the solvent influence a chemical reaction rate There are three ways [1,2]. The first is by affecting the attainment of equilibrium in the phase space (space of coordinates and momenta of all the atoms) or quantum state space of reactants. The second is by affecting the probability that reactants with a given distribution in phase space or quantum state space will reach the dynamical bottleneck of a chemical reaction, which is the variational transition state. The third is by affecting the probability that a system, having reached the dynamical bottleneck, will proceed to products. We will consider these three factors next. 

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