Reduced density functions

To begin with, we introduce a normalized density function or probability 

Now consider the probability of finding only a certain small group of molecules, such as molecules (2), (4), (5), and (7), in their respective region of phase space. In particular. 

Note that the probability of finding our N molecules in any order is just N times as large, or 

Similarly, for the lower-order function P (r, p, t), when order doesn t matter, it becomes 

In many problems we will only be interested in the configurational part of the density functions, and thus, we define a configurational density function as 

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We are normally interested just in the probability of finding one electron regardless of the location of the remaining electrons. Integration over coordinates X2,.. ., x leads to the reduced density function (xi). 

As will be shown in subsequent chapters, the solution to the Liouville equation for the N -particle density function p r is the basis for determination of the equilibrium and nonequilibrium properties of matter. However, because of the large number of dimensions (6N), solutions to the Liouville equation represent formidable problems Fortunately, as will be shown in Chaps. 4 and 5, the equilibrium and nonequilibrium properties of matter can usually be expressed in terms of lower-ordered or reduced density functions. For example, the thermodynamic and transport properties of dilute gases can be expressed in terms of the two-molecule density function /02(ri,r2, pi,p2,f). In Chap. 3 we will examine the particular forms of the reduced Liouville equation. 

Reduced Density Functions and the Reduced Liouville Equation 55... 

Similarly, we have the reduced density functions in configurational space... 

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, The Molecular Theory of Gases and Liquids, Wiley, New York, 1960. [A classic text on which most of the discussion of reduced density functions given here was based.]... 

Using the reduced density function definitions from Chap. 3... 

Introducing the reduced density function fi the above term becomes... 

Figure 3 Diagrammatic representation of the path integral interactions in a case where the memory length is equal to three time steps. The reduced density functionals that start at / = 0 and t = 3A/ consist of paths specified by the time points shown as red and yellow circles, respectively. The coupling terms in the propagator connecting these two reduced density functionals are shown as cyan lines...

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