The singlet distribution function

All the results obtained in this section are very simple and almost self-evident. Nevertheless, we have taken the liberty to make the subject matter appear more complicated than is really necessary. The reason is that we would like to stress some subtle points which are particularly clear in the context of this simple case, but are more difficult to grasp in other cases that will appear throughout the book. 

The system under consideration consists of N rigid particles at a given temperature and contained in a vessel of volume V, The basic probability density for such a system is (see Section 1.5) 

In some cases, we use a special symbol, such as F , for an average quantity. However, we refrain from using this notation whenever the meaning of that quantity as an average is quite evident. 

This relation can be rewritten in a somewhat more complicated form, which will turn out to be useful for later applications. 

we have exploited the fact that all the particles are equivalent hence, the sum over the index i produces N integrals having the same magnitude. We may therefore select one of these integrals, say / = 1, and replace the sum by a product of N equal integrals. 

In this and the following chapter, we shall always start with a one-component system, then generalize for multicomponent mixtures. This is done mainly for 

The symbol e means belongs to. Hence, A,(R S) is unity whenever R, is within S and zero elsewhere. The quantity MXV, S) can be expressed as 

FIGURE 5.1. A region S within the system. A shaded circle indicates a particle whose center falls within S. 

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Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

Since the system is uniform, the potential energy is a function of relative coordinates only so that Z can be written as the product of the integral in the numerator times the volume of the system V. It follows that the singlet distribution function is simply the number of particles or molecules in the fluid N per unit volume V, that is. 

By integrating/ (rj, T2) over dr2, that is, over the volume of the system, one obtains the singlet distribution function ... 

In a similar fashion, we can define the singlet distribution function for location and orientation, which by analogy to (2.14) is defined as... 

The singlet distribution function for the species A is defined in complete analogy with the definition in the pure case (section 2.1),... 

The second extreme case occurs when pA —> 0. Note, however, that we still have two A s at fixed positions (R1, R"), but otherwise the solvent (here in the conventional sense) is pure B. We have the case of an extremely dilute solution of A in pure B. Note also that at the limit pA —> 0, both the pair and the singlet distribution functions of A tend to zero, i.e.,... 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

In conclusion, it appears that the application of recent theories of nonequilibrium statistical mechanics to transport in dense media confirms Eyring s theory and provides in addition a convenient framework for possible extensions and refinements for instance, Allen et al. have recently combined the original PNM model with an approximate kinetic equation for the singlet distribution function and obtained a stilt better agreement with experiment. 

Sects. 2-5 deal primarily with notation and definitions that are needed throughout the remainder of the presentation. The main physical result is the general equation of change presented in Sect. 3. Special cases of this general equation are then featured in the next five sections, in Sects. 6-9, where the four main conservation equations of fluid dynamics are obtained, with expressions for the fluxes as by-products, and in Sect. 10, where the equation for the singlet distribution function is developed. The latter is then used in Sect. 11 to obtain the equation of motion for the beads/ in which expressions emerge for the Brownian and hydrodynamic forces. Up to this point the development is relatively free from assumptions. 

In the first line the Dirac delta function of a vector argument represents the product of the delta functions of the three Cartesian components (r) = (x) (3t) (z). In the second hne we have mtroduced a shorthand for the product of Dirac delta functions each with a vector argument The interpretation of the singlet distribution function is ... 

Equations for the Time-Evolution of the Singlet Distribution Functions (DPL, Sect. 17.5)... 

Once the singlet distribution function has been found, we are in a position to evaluate the various contributions to the fluxes that depend on (see Table 1). In this section we discuss the contnbutions to the stress tensor, and in the next two sections the contnbutions to the mass and heat flux vectors. In these sections, for illustrative purposes, we restrict ourselves to the Rouse bead-spring chain and the Hookean dumbbell models, for which we can use the singlet distribution functions , given in Eqs. (13.5) and (13.8). 

We confine our attention here to dilute solutions of several polymer speaes m a solvent. According to Sect. 7, the stress tensor is a sum of four contnbutions, the first three of which involve the singlet distribution function (s), whereas the fourth involves the doublet distribution fiinction(4) ... 

In order to make use of the flux expressions in Sects. 6, 7, and 8, it is necessary to have the singlet distribution function and - unless the short-range force assumption is used - the doublet distribution function as well. Virtually nothing is known about the doublet distribution function. If we knew how to make a reasonable guess of this function (possibly obtainable from molecular or Brownian dynamics), then we could estimate the contributions to the fluxes in Table 1 that involve the molecule-molecule interactions. 

Fig. 2.43 The singlet distribution function xcn K) for the system of non-spherical particles, with parameters given in (2.6.26). The HB energies are indicated next to each curve.

Let us now make a specific choice of two components for water, which seems to be the most useful one for interpreting the thermodynamic behavior of aqueous solutions. We use the singlet distribution function xcn(K) based on coordination number (CN), and define the two mole fractions by... 

Clearly, we may refer to L and H as the components of relatively low (L) and high (H) local densities. Another useful choice could be based on the singlet distribution function for molecules with a different number of hydrogen bonds. With... 

Equation (35) can be regarded as a generalized kinetic equation. The relationship of this equation to the Boltzmann equation is well under-stood. Using linear response theory, one can show that the linear deviation of the singlet distribution function from equilibrium, for a system initially held in constrained equilibrium, is proportional to... 

Therefore, one cannot make a precise analysis of the heat capacity by studying only the singlet distribution function Nb v),... 

We have a mixture of ions of density pi, charge e and diameter tr. /)j(z)is the number density profile of i at a distance z from the electrode, which is always assumed to be flat and perfectly smooth. The singlet distribution function is... 

Thus in order to obtain Pr(5i = Of), we take the probability distribution of the entire system, fix the state of a specific unit, say z = 1, at of, and sum over all possible states of all other units. The resulting probability distribution is called the singlet distribution function. 

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