Kirkwood distribution functions

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Equation (95) is obtained from the virial expansion of the equation of state for rigid spheres for higher densities the rigid-sphere equation of state obtained from the radial distribution function by Kirkwood, Maun, and Alder has to be used (K10, Hll, p. 649). When Eq. (95) is substituted in Eqs. (92), (93), and (94) one then obtains the rigorous expressions for the coefficients of viscosity, thermal conductivity, and selfdiffusion of a gas composed of rigid spheres. 

Semi-classical approximations. In classical formulae, quantum effects may be accounted for to low order. For example, the the Wigner-Kirkwood expansion of the pair distribution function may be used [136, 302],... 

Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

In order to obtain better agreement with experimental results, the concept of a distribution of correlation times was introduced in nuclear magnetic relaxation. Different distribution functions, G(i c), such as Gaussian, and functions proposed by Yager, Kirkwood and Fuoss, Cole and Cole, and Davidson and Cole (asymmetric distribution) are introduced into the Eq. (13), giving a general expression for... 

Comparing now this exact equation with equation (9.1.50) we see that the Kirkwood approximation takes only a few terms into account. Next we must express equation (9.1.59) in terms of the single particle densities C (A = 0, A, B) and pair correlation functions F (r). We see that if a rate R is infinite, the corresponding distribution function p(2) is zero and... 

Note that the only approximation made in the derivation of Eq. (177) is the use of the Kirkwood superposition approximation for the triplet distribution function of the liquid [21]. In a dense liquid at low temperature (near its triple point), this is not a bad approximation [21],... 

We note that the values of the hydrodynamic interaction tensor (2.6) averaged beforehand with the aid of some kind of distribution function, are frequently used to estimate the influence of the hydrodynamic interaction, as was suggested by Kirkwood and Riseman (1948).4 For example, after averaging with respect to the equilibrium distribution function for the ideal coil and taking the relation (1.23) into account, the hydrodynamic interaction tensor (2.6) assumes the following form... 

We note that the second term in (108) is the familiar Kirkwood expression for the stress tensor in terms of the n-particle distribution function. 

The radial distribution function was obtained by Pople,103 c.f., Harris and Alder,110 Haggis, Hasted, and Buchanan.111 Pople showed that Kirkwood s assumption of complete hydrogen bonding in the first shell with none in the second was oversimplified. The first shell dipoles are bonded to the second via bent hydrogen bonds of bending... 

Kirkwood, J. G. and Salsburg, Z. W., The statistical mechanical theory of molecular distribution functions in liquids. Disc. Faraday Soc. 15, 28-34 (1953). 

With the above, a formal set of equations Is given, the elaboration of which requiring a solution for the problem that the recurrent relationships p p - p p, . .. diverge. Relatively simple densities, or distribution functions, are converted into more complex ones. A "closure" is needed to "stop this explosion". A number of such closures have been proposed, all involving an assumption of which the rigour has to be tested. Most of these write three-body interactions in terms of three two-body Interactions, weighted in some way. A well known example is Kirkwood s superposition closure, which reads ... 

Above we have given the basic strategy for the statistical thermodynamics of the electrical double layer. We shall not discuss the various elaborations In detail, but note that in addition to the BBGYK hierarchy to solve the distribution functions several other methods have been developed. In the Kirkwood hierarchy a coupling constant Is introduced to avoid the spatial Integration... 

The theory of non-Newtonian viscosity for ellipsoidal particles was first explicitly stated by Kuhn and Kuhn (1945), using Peterlin s distribution function (Peterlin, 1938) and Jeffery s hydrodynamic treatment (Jeffery, 1922-1923) [Eq. (10)]. More elegant treatments have recently been developed by Saito (1951), using the same ellipsoidal model, and also by Kirkwood and his co-workers (Kirkwood, 1949 Kirkwood and Auer, 1951 Kirkwood and Plock, 1956 Riseman and Kirkwood, 1956) for rodlike particles. The equivalence of the three theories has also been demonstrated by Saito and Sugita (1952). The general solution of Eq. (10) for the viscosity increment, v, can be expressed in the form... 

More amenable expressions can be obtained if the asymmetric distribution function 0 (Zj, r) in [2.4.6] can in some way be related to the corresponding distribution function for the bulk fluid, p (r)- Basically such a relation must exist because the two functions are both based on the same interaction energies. Kirkwood and Buff (loc. cit.) elaborated such a procedure. Its level of mathematical abstraction is beyond that of FICS. In their theory higher order densities pjj appear in the surface layer. These pj s apply to a subset of h molecules, taken from the whole and are studied in terms of the Bom-Green theory for liquids It turns out that pj ean be related to by a set of complicated integro-... 

In the next section we shall present a simplified expansion theorem of osmotic pressure which was first obtained by McMillan and Mayer. This cluster expansion theory will be further extended in Section 3 to distribution functions, and medn results of Kirkwood and Buff will be recovered. A new and simple derivation of the cluster expansion of the pair distribution function is also given. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 shows how the general solution theory may be applied to compact macromolecules. Finally, Section 6 deals with the second osmotic virial coefficient of flexible macromolecules and is followaJ in Sa tion 7 by concluding remarks. 

The difficulty in the Kirkwood formalism based on consideration in the Riemanian space is in the precise specification of all the segment positions. We have already seen in Section 4 that the sin et segment distribution function w(r) is important in describing viscosity phenomena. We shall now find out what changes result in w (r) due to a laminar flow. 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

In the above expression, ga/i is the radial distribution function between species a and / , r is the distance between the centers of molecules a and / , and and Ai23 are the following combinations of the Kirkwood-Buff integrals... 

Another method suggested by the authors for predicting the solubility of gases and large molecules such as the proteins, drugs and other biomolecules in a mixed solvent is based on the Kirkwood-Buff theory of solutions [18]. This theory connects the macroscopic properties of solutions, such as the isothermal compressibility, the derivatives of the chemical potentials with respect to the concentration and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. This theory allowed one to extract some microscopic characteristics of mixtures from measurable thermodynamic quantities. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in ternary [19] and multicomponent [20] mixtures with respect to the mole fractions. These expressions for the derivatives of the activity coefficients were used to predict the solubilities of various solutes in aqueous mixed solvents, namely ... 

The KB theory of solution (Kirkwood and Buff, 1951) connects the macroscopic properties of solutions, such as the isothermal compressibility, the concentration derivatives of the chemical potentials, and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. 

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

Statistical mechanics gives relationships between the distribution functions and the bulk properties of fluids. The total internal energy of a fluid is given by the energy equation, the pressure is given by the virial equation, and the isothermal compressibility is given by the compressibility equation, see e. g.. Ref. 11. Through the Kirkwood-Buff formulas (0,... 

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