The Reduced Distributions

The Dirac delta function in (5.15) merely selects out those curves u(j) that satisfy the constraint (5.4). The desired end-to-end vector distribution G(R0 T0) is then to be obtained from (5.15) by some (as yet unspecified) average over U and U,  

The normalization 2[ i s)] is chosen so that C(UU LO) has the usual normalization required to have 

Essentially, Markov s method corresponds to the use of the Fourier representation of the Dirac delta function in (5.15), 

Equations (5.15) and (5.21) are familiar Wiener integrals which are easily evaluated in closed form, provided (5.2a) is not invoked. Before considering this case, it is instructive to recover the results of STY. 

---

The reduced distribution functions at the instant t — 0, f are factorized when the s particles are distributed among several groups separated from each other by a distance greater than the range of the forces. 

The reduced distribution of values for the coordinates alone (e.g., of bond orientations for a linear polymer with constrained bond lengths) is given by the integral... 

We consider a rigid system of / mechanical degrees of freedom in thermal contact with a solvent. As in the discussion of equilibria, p q,p) is the phase space density and /( ) is the reduced distribution for the coordinates alone. Following BCAH, we also define a conditional average (A)p of an arbitrary dynamical variable A with respect to the rapid fluctuations of the momenta and solvent forces, at fixed values of the coordinates q, as... 

We may recover a diffusion equation for the reduced distribution i q) = ldc V Q) by integrating both sides of Eq. (2.175) with respect to the hard coordinates, while approximating the functions and... 

The general solution of this equation is not straightforward except in (a) fast modulation and (b) very slow modulation. In case (a), it is easy to derive from Eq. (286) an equation of motion for the reduced distribution... 

This is represented by the series of diagrams (58), where now i,j, k,. . . = 1, 2,. . N refer only to molecules. For every power of a, the terms referring to identical groups of molecules can be summed and integrated to make the reduced distribution functions appear. It is necessary, however, to distinguish in terms whether the two indices are equal or not, e.g., between i =j and i 7 j. The first few terms are... 

The adsorption kinetics at liquid/liquid interfaces is a more complicated problem, as the transfer of surfactant from one phase to the other has to be taken into account. In the experiments performed by Liggieri and Ravera [197] using the expanded drop method, no preliminary saturation of the oil phase with CjoEOg was made. For this case, instead of Eq. (4.1), the expression (4.94) should be used, where K is the equilibrium distribution coefficient of surfactant between the oil and water phases, and D2 is the surfactant diffusion coefficient in the oil phase. The reduced distribution coefficient defined by = K(D2/Di) is a parameter that reflects quantitatively the adsorption dynamics at such a liquid/liquid interface. 

Figures.11 (left) shows the reduced distribution of the fit of the real interfero-gram with the simulated interferogram for different cut-off wavenumbers. It can be observed that a minimum peak appears around 20 cm Figure3.11 (right) presents...

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. 

The general structure of the reduced distribution f x) is shown in Fig. 1.8 for rf = 3. There is a very strong drop at largex... 

Let us look into what occurs in further detail. Let y/+ denote the mean phase and y/ the relative phase (i.e., y/ -y/ y/2) and Qiy/-,t) the reduced distribution function with respect to the relative phase, or... 

---