Reduced coordinate distribution function

The one-body reduced coordinate distribution function is defined similarly... 

Mayer showed that the two-body reduced coordinate distribution function can be expressed as a series of powers and logarithms of the density, with a second type of cluster integrals as coefficients. ... 

Many of the equilibrium properties of such systems can be obtained through the two-body reduced coordinate distribution function and the radial distribution function, defined in Eqs. (27.6-5) and (27.6-7). There are a number of theories that are used to calculate approximate radial distribution functions for liquids, using classical statistical mechanics. Some of the theories involve approximate integral equations. Others are perturbation theories similar to quantum mechanical perturbation theory (see Section 19.3). These theories take a hard-sphere fluid or other fluid with purely repulsive forces as a zero-order system and consider the attractive part of the forces to be a perturbation. ... 

The local functions depend upon at most 3N coordinates of the electron configuration space. This number can be reduced to 3k by integration over the positions of N-k electrons. For example, in the case of spinless reduced density distribution functions [40] ... 

Thus, the allowance for the dependence of the resonance integral on qsk may not be reduced in general to averaging the transition probability over the distribution function in Eq. (102). The function s(qk) plays the role of the distribution function for the coordinates qk in the case of the symmetric transition. In the classical limit, the results of Flynn and Stoneham62 can be obtained from Eq. (103), and in the low-temperature limit, the result of Kagan and Klinger64 can be obtained. 

To construct a diffusion equation for the soft coordinates alone, we consider a reduced distribution function... 

It is then useful to reduce the information content by introducing a projected coordinate. With projection, some information of x is retained, while the others are disregarded. When the projection is implemented with respect to a function P(x), the corresponding distribution functions are generated from the instantaneous distribution given by... 

The right-hand side represents the distribution function of the momentum vector p. The p integration of Ufp, q) reduces to the thermal average of PM q), which gives the probability with which phonons of the state Pstiq) are found at coordinate q,... 

In the second step this makes the Cu(I) ions accessible to water molecules so that further reduction may occur. The corresponding radial distribution function of the final state shows a structure identical to that of a Cu metal foil (Fig. 7). In comparison to the latter the first coordination shell displays a smaller height, which can be explained by a reduced number of nearest copper neighbours in agreement with recent findings in the literature [13]. This interpretation is supported by the comparison of the XANES of the final state (Fig. 6) with that of small copper particles [l4,15]. 

P0 = pressure of a hard-sphere fluid Pc = critical pressure P = Pcr3/c, reduced pressure p< = ith generalized momenta q< = ith generalized coordinate RK = Redlich-Kwong equation RDF — radial distribution function... 

In the energy representation, the interaction energy between the solute and solvent is adopted for the one-dimensional coordinate of the distribution functions, and a functional for the solvation free energy is constructed from energy distribution functions in the solution and reference-solvent systems of interest. The introduction of the energy coordinate for distribution functions is a kind of coarse-graining procedure for reducing the information content of solute-solvent... 

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