Critical branching probability

With somewhat greater generality we may let a represent the actual branching probability determined by the extent of condensation irrespective of whether or not the critical point has been exceeded, while a is the lowest root of Eq. (20) for the same value of If then a = and according to Eqs. (44) and (45) Ws = l and Wg O. If a>acj then a>a and In the trifunctional case, for example,... 

When extensive branching occurs the system can gel. This occurs when —c . Under these conditions the critical branching probability, p, is related to the extent of reaction of the phenolic groups at the gel point, o-c, by... 

Each binary fork is attached to a branch of the preceding fork and is conditioned by the success or failure represented by that branch. Thus, evei7 fork, represents conditional probability. Each limb of the HRA event tree is described or labeled, in shorthand. Capital letters (A) represent I ailure lower case letters (a) represent success. The same convention applies to Greek letters, which represent non-human error events, such as equipment failures. The letters S and F are exceptions to this rule in that they represent system success and failure respectively, in practice, the limbs may be labeled with a short description of the error lo eliminate the need for a legend. The labeling format is unimportant the critical task in developing HRA event trees is the definition of the events themselves and their translation to the trees. 

The development of the HRA event tree is one of the most critical parts of the quantification of human error probabilities. If the task analysis lists the possible human error events in the order of ihcir potential occurrence, the transfer of this information to the HRA event tree is fadlitutcd. Each potential eiTor and success is represented as a binary branch on the HRA event tiec. with subsequent errors and successes following directly from the immediately preceding ones. Cure should be taken not to omit the errors that are not included in the task analysis table but might affect the probabilities listed in the table. For example, administrative control errors that affect a task being performed may not appear in the task analysis table but must be included in the HRA event tree. 

The exact method by which fracture occurs is not known, although it is suggested by I>r r i i 51 that the compressive force produces small flaws in the material. If the energy concentration exceeds a certain critical value, these flaws will grow rapidly and will generally branch, and the particles will break up. The probability of fracture of a particle... 

However at a critical distance from equilibrium, the system must choose between two possible pathways, represented by the bifurcation point Ac. The continuation of the initial pathway, indicated by a broken hne, indicates the region of instability. The concentration of the species A and the value of A assume quite different values, and the more so, the further from equilibrium. An important point is that the choice between the two branching directions is casual, with 50 50 probability of either. The critical point Ac has particular importance because beyond it, the system can assume an organized structure. Here the term self-organization is introduced as a consequence of the dissipative structures, dissipative in the sense that it results from an exchange of matter and energy between system and environment (we are considering open systems). 

Parallel Universes—Chapter 3 discussed parallel universes and the many-worlds interpretation of quantum mechanics. Readers interested in a lively and critical discussion of this topic should consult Professor Victor Stenger s The Unconscious Quantum (Prometheus Books, 1995). For example, he doubts very much that the parallel universes (in the many-worlds interpretation) all simultaneously exist. He also does not believe that all branches taken by the universe under the act of measurement are equally real. Stenger discusses other approaches such as the alternate histories theory that suggests every allowed history does not occur. What actually happens is selected by chance from a set of allowed probabilities. 

The point in the reaction at which gelation occurs has been deduced by Flory. 4 The gel point is developed in terms of the branching coefficient, a, which is the probability that a given functional group on the multifunctional monomers leads, via a chain that can contain any number of bifunctional units, to another multifunctional monomer. The critical value of the branching coefficient, denoted by ac, at which gelation occurs is... 

The first term in this quadratic equation is the initiation reaction rate based on the inflow concentration of the reactant. The coefficient for the term in [X]ss has something of the character of the previous net branching factor. The above equation has a single positive solution for any set of rate constants, residence time and inflow concentration a typical variation of [.ATJss with [A]o is shown in Fig. 5.3(b) and shows a rapid increase in the vicinity of some critical concentration [A]o,cr-The behaviour can be quantified if we make the approximation of ignoring the (probably small) initiation terms, setting ki =0. The steady-state condition can then be written in the form... 

Other segments may also be present with one or no branch units on their ends. Gel formation will occur when at least one of the if—1) segments radiating from the end of the segment of the type shown is in turn connected to another branch unit. The probability that this will occur is l/(f—1). Thus the critical value of a for gelation is... 

The notion of the critical dilution is in harmony with our intuition, suppose a branching process molecular growth can take place only through the intermo-lecular linkages, whereas with increasing dilution the intermolecular reaction is suppressed. When the dilution reaches the well defined point, yc = 1, corresponding to Dc = 1, the probability of an infinite molecule emerging suddenly vanishes hence, the critical dilution. 

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