Ursell functions

The operators 8> are defined starting from the streaming operators in the same way as are the equilibrium Ursell functions starting from the functions ps (with = 1) ... 

It is now possible to derive rather general results on the relaxation of all of the Ursell functions for the particular model just outlined. Let us decompose W(n, t m,0) in the following form... 

The properties relating to decay of correlations have so far been discussed only in terms of Ursell functions, or functions generated from a cumulant expansion. This is obviously not a unique choice. Although there has been no systematic exploration of possible alternative families with, perhaps, better separation of rates of convergence than that expressed in Eqs. (12) and (13), or (17) and (18), it is of some interest to examine the Kirkwood12 approximation form... 

Figure 6.1 Mayer-Montroll expansion for the insertion probability p(0 X"). The notation here is fairly standard (see, for example, Hansen and McDonald, 1976 Andersen, 1977). The solid lines indicate factors of Mayer / functions introduced in Eq. (6.2) and are further discussed as Ursell functions beginning on p. 126. The inclusion-exclusion interpretation for hard-core cases is that the second term - assesses the m molecular volumes excluded to the...

The prescription for determining the functions is that an m-decomposable model should be correct if the system consisted only of m molecules. Formulae such as Eqs. (6.2) and (6.3) involve the Mayer / function that, for a pair decomposable case, is / j) = exp [—/3m( )(0, y)] — 1. A natural accommodation of nonpair-decomposable interactions in this case takes the goal of insuring that successive terms in a virial expansion are ordered by the density. This is the historical approach (Ursell, 1927), and is called an Ursell expansion. In this language, fa (j) is an Ursell function (Stell, 1964 Munster, 1969). Again the idea is to require that the desired m-body Ursell function makes the product of Eq. (6.2) correct if just m molecules are involved. Thus for the case that only two molecules are involved... 

Figure 6.3 Compare with Fig. 6.1. Here again the solid lines indicate factors of Mayer / functions as in Eq. (6.25) and further discussed as Ursell functions beginning on p. 126. The second term shown is / f(r )p6 r r)d r for a simple fluid. The shaded regions with m-l black disks and one white disk represent conditional densities (rj,..., r). That white disk, rightmost here, corresponds to the real atom positioned at r on which the averages are conditioned. The other white disk, leftmost here, corresponds to the test particle.

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

Returning to Eq. (80) we notice that another way to specify the Ursell functions is with the boundary condition W i(l) = %i(l) and the recursion formula... 

The second equation is the same as Leibnitz s rule for the differential operator. The coordinate arguments of the Ursell functions have been suppressed in Eq. (86). 

It is easy to verify that Eq. (84) is consistent with Eqs. (80) and (81), because each term in the Ursell function expansion of Wiv(l,..., N) is distinct from all of the others because the coordinates are all distinguishable. 

Equation (87) is not so trivial as it may first appear The terms in the Ursell function expansion of (Ws) are not all distinct from each other and this leads to the appearance of so-called combinatorial factors. They do not appear in Fig. 6 because they are included in the definitions of the graphs (Section 6.1). 

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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