Ursell cluster functions

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

These expressions are analogous to the series expansions of the equilibrium distribution functions in terms of the activity in which appear, in the coefficients, the integrals of the Ursell cluster functions Us (see, for example, ref. 30). 

The Ursell cluster functions are usually defined by their relationship with the grand canonical distribution functions pi l.K)... 

The most compact graphical expansion we have is for the logarithm of S, so it is more convenient to consider a distribution function based upon a functional derivative of logH. This leads naturally to the Ursell cluster... 

The exponential cluster expansion has a long history in statistical physics, where it is known as the Ursell and Mayer linked-cluster expansion for the partition function [68]. A very general argument for the exponential form of the exact wave function can be found in a paper on the origins of the CC method by Kiimmel [69]. Nonetheless,... 

Boltzmann factors have the product property, and the Ursell functions the cluster property. 

J. Cizek, ]. Chem, Phys, 45, 4256 (1966). On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wave Function Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. J. Cizek, Adv, Chem. Phys., 14, 35 (1966). On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules. 

These semiclassical approaches have their origin in the studies of the second and third virial coefficients of a gas of QHS [139-143]. Further developments involving cluster expansions of in terms of Ursell functions led to the conclu-... 

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