The Cluster Expansion

The interacting Gaussian chain model is defined by Eqs. (2.2)-(2.4). As before we consider the generalization to d-dimensional space, and we evaluate expectation values by expanding in powers of 3e. We demonstrate the method with the example of the endpoint Greensfunction G(p n) of a single chain system, which generalizes Gh p n) (Eq, 3.33). 

The partition function and the endpoint distribution function can be expressed in terms of G(p n), the results being identical in form to Eqs. (3.31), (3.35). 

Evaluating the product in Eq. (4.3) we get back the Greensfunction GVjfp n) of the noninteracting model to zero order in 3e, Writing 

(a) A chain configuration, where the interaction of segments jx. j i is taken into account, (b) The corresponding Keynman-diagram. Here for clarity wc indicated the distinguished segments by heavy dots 

Note that these random walks can have any number of intersections, but only the interaction of the pair rjA = rJ2 is taken into account. This situation is represented schematically by the Feynman diagram5 of Fig, 4,2b. The chain is drawn as a straight line, and the two interacting segments are connected by a broken line. This graphic representation generalizes to all orders of the cluster expansion. 

Note that these random walks cmi have any number of intersections, but only the interaction of the pair is taken into account. This situation 

Using the results explained in Appendices A 3.1, A 3.2 we then carry through tlie remaining r-integrals to find 

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The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

The cluster expansions of the correlation functions and potentials of mean force can be found by studying the semi-invariant expansion of logg( n ) and by the use of the linked cluster theorem. The method is a straightforward extension of those given in the preceding section and details can be found in the paper by Allnatt and Cohen.3 The linked cluster method is simpler and... 

Having familiarized ourselves slightly with the cluster expansions let us now look in detail at a more difficult example involving long-range interactions where the quasi-chemical formalism appears less satisfactory. 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

The point of departure of this method is the "cluster expansion of the non-equilibrium distribution functions ... 

The cluster expansion of additively separable and hence extensive) is... 

J. Y. Hsu, Derivation of the density functional theory from the cluster expansion. Phys. Rev. Lett. 91, 133001 (2003). 

Our starting point is a density analogous to that used in [49] in treating the migration of excitons between randomly distributed sites. This expansion is generalization of the cluster expansion in equilibrium statistical mechanics to dynamical processes. It is formally exact even when the traps interact, but its utility depends on whether the coefficients are well behaved as V and t approach infinity. For the present problem, the survival probability of equation (5.2.19) admits the expansion... 

We here define our model and present a self-contained introduction to perturbation theory, deriving the Feynman graph representation of the cluster expansion. To deal with solutions of finite concentration we introduce the grand-canonical ensemble and resum the cluster expansion to construct the loop expansion. We Lhen show that without further insight the expansions can be applied only in the (9-region or for concentrated solutions since they diverge term by term in the excluded volume limit. 

This is a complicated 3(n + l)-fold integral, and there is no hope of an exact evaluation. Our only chance consists in expanding the product in powers of, 9ei a series known as the cluster expansion ... 

Fig. 4.1. Diagrammatic representation of the first few terms of the cluster expansion...

This characteristic energy naturally shows up in the cluster expansion. Note that W can become large even for /3e[Pg.53]

We have argued in the introductory section that the cluster expansion in powers of j3c breaks down as soon as a chain strongly interacts with a lot of others. To overcome this problem we must resum the cluster expansion to arrive at the so-called loop expansion . We first define the number of loops. 

Since each additional interaction vertex in the cluster expansion contributes two segment summations and one momentum integral, with these substitutions a total factor of n2 d = neaccompanies the additional excluded... 

To summarize, d = 4 makes a border line, the upper critical dimensionJ for the excluded volume problem. For d < 4 both the cluster expansion and the loop expansion break down term by term in the excluded volume limit. For d i> 4 the expansions are valid, the leading n- or c-dependence of the results being trivial, however. We may state that for d > 4 the random walk model or Flory-Huggins type mean field theories catch the essential physics of the problem. As will be explained more accurately in Chap. 10 the mechanism behind this is the fact that for d > 4 two nncorrelated random walks in general do not cross. 

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