Probability distribution function, join

Figure 3. (Left) The measured scaling exponents q v(q) (joined by dot-dashed straight lines) of the moments of the displacement Ax, as a function of the order q. The dashed line corresponds to 0.65 q while the dotted line corresponds to q 1.04. (Right) The normalized probability distribution function P(Ax(t)/a) versus Ai = Ax/a (a — exp(ln Ax(/ )) for the three times 0=500 (circles), + — 2 (diamonds) and / = 2/ (squares). The dashed line represents the Gaussian function.

In particular, it appears that the probability distribution of the vector joining the chain ends, for a chain with excluded volume, strongly differs from a Gaussian function, and this fact greatly diminishes the value of the Gaussian chains. 

The form of the probability distribution P(r) of the vector r joining the ends of a long polymer in a good solvent has been much studied12 13 between 1970 and 1974. For a long polymer, the function P(r) appears in scaling form... 

The probability PXqr q ) fo move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated [108] by PIMC or PIMD techniques [17-19] combined with umbrella sampling [77,108,123] of the reaction coordinate centroid variable. In the latter computational technique, a number of windows are set up which confine the path centroid variable of the reaction coordinate to different regions. These windows connect in a piecewise fashion the possible centroid positions in going from the reactant state to the transition state. A series of Monte Carlo calculations are then performed, one for each window, and the centroid probability distribution in each window is determined. These individual window distributions are then smoothly joined to calculate the overall probability function in Eq. (4.11). An equivalent approach is to calculate the centroid mean force and integrate it from the reactant well to barrier top (i.e., a reversible work approach for the calculation of the quantum activation free energy [109,124]). 

As in the case of the statistical mechanics of a fluid, these Boltzmann factors contain more information than is necessary in order to characterize the experimental properties of polymer systems. We therefore focus attention upon reduced distribution functions in order to make contact with the macroscopic observable properties of polymers. In the usual many-body problems encountered in statistical mechanics, the reduced distribution functions are the solutions to coupled sets of integro-differen-tial equations. - On the other hand, because a polymer is composed of several atoms (or groups of atoms) that are sequentially joined together by chemical bonds, these reduced distributions for polymers will obey difference equations. Therefore, by employing the limit in which a polymer molecule is characterized by a continuous chain, these reduced probability distributions can be made to obey differential, instead of difference, equations. This limit of a continuous chain then enables the use of mathematical analogies between polymers and other many-body systems. The use of this limit naturally leads to the use of the technique of functional integration. 

A macrostate of a macromolecule can always be described with the help of the end-to-end distance R ). To give a more detailed description of the macromolecule, one should use a method introduced by the pioneering work reported by Kargin and Slonimskii [1] and by Rouse [2] whereby the macromolecule is divided into N subchains of length M/N. The points at which the subchains join to form the entire chain (the beads) will be labelled 0 to N respectively and their positions will be represented by r°,r, ..., r. If one assumes that each subchain is also sufficiently long, and can be described in the same way as the entire chain, then the equilibrium probability distribution for the positions of all the particles in the macromolecule is determined by the multiplication of N distribution functions of the type of (5)... 

When members join the group according to a Poisson process with rate A and a member s lifetime is exponentially distributed with mean 1 / //. we can compute the expected number of the concurrent spaces, E(S), as a function of the shifting probability p ... 

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