Expansion cluster

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

Stell G 1964 Cluster expansions for classical systems In equilibrium The Equilibrium Theory of Classical Fluids ed H L Frisch and J L Lebowitz (New York Benjamin)... 

Chandler D and Andersen H C 1972 Optimized cluster expansions for classical fluids II. Theory of molecular liquids J. Chem. Phys. 57 1930... 

Theoretical investigations of quenched-annealed systems have been initiated with success by Madden and Glandt [15,16] these authors have presented exact Mayer cluster expansions of correlation functions for the case when the matrix subsystem is generated by quenching from an equihbrium distribution, as well as for the case of arbitrary distribution of obstacles. However, their integral equations for the correlation functions... 

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... 

Cluster-expansion and cluster-degradation reactions are a feature of many polyhedral borane species. Examples of cluster-expansion are " ... 

Cluster expansion reactions with diborane provide an alternative route to intermediate closo-carboranes, e.g. ... 

Brueckner, K. A., Phys. Rev. 100, 36, Many-body problem for strongly interacting particles. II. Linked cluster expansion."... 

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

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... 

The way in which cluster expansion occurs is not understood. One suggestion is that radical species such as Fe(CO)4 (triplet) or Co(CO)4 are produced and that attack of these radicals on substrates leads to polyhedral expansion ... 

Os(CO)4L are produced, indicating the formation of Os(CO)4 as an intermediate. The suggestion, therefore, that cluster expansion is brought about by radical attack does not receive support from experimental observation, at least as far as the higher clusters (m > 2) are concerned. 

In order to make the linked cluster expansion it is necessary to remove the excluded-site restriction on the summation by writing... 

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... 

The most important application to be considered under this heading is the calculation of intrinsic defect concentrations in dilute solid solutions. If the solution is so dilute that only the leading terms in the various cluster expansions need be retained then the results required are slight generalizations of those above and follow at once from the notation for the general results. For example, the equilibrium concentration of vacancies in a dilute solution of a single solute, s, is found from Eqs. (74a) and (75) to be... 

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 discussion of the defect distribution functions and potentials of average force follows along rather similar fines to that for the activity coefficient. The formal cluster expansions, Eqs. (90)-(91), individual terms of which diverge, must be transformed into another series of closed terms. This can clearly be done by... 

We should finally comment briefly on the calculation of 0. For a crystal with short-range forces and hence short-range defect interactions a cluster expansion is convenient. Let ba B represent a particular subset, number a, of b sites out of the total of B. 

Friedman, H. L., Ionic Solution Theory Based on Cluster Expansion Methods, Interscience Publishers, New York, 1962. 

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

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. 

In Section II, we summarize the ideas and the results of Bogolubov,3 Choh and Uhlenbeck,6 and Cohen.8 Bogolubov and Choh and Uhlenbeck solved the hierarchy equations and derived two- and three-body generalized Boltzmann operators Cohen used a cluster expansion method and obtained two-, three-, and four-body explicit results which he was able to extend to arbitrary concentrations. 

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

We note that the definition of 8LN (see 5) implies that the only non-vanishing contributions to In start with 8LaiK We specify j = 2 and introduce an indistinguishability factor (N — 1) (because of the p2.. . Pjy integration, the particles 2, 3,.. ., N play the same role). Moreover, we insert in (A.39) the dynamical cluster expansion, giving the streaming operators in terms of the operators and from which the formulae (23) have been deduced. Finally, we use the well-known results ... 

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