Partition function Fourier representation

Edwards approach (Ref. [10]) is based on field-theoretic path-integral representation of the partition function Wn(R0, RN, N) defining the probability density of the fact that end points of an JV-link chain are placed at the points R0 and Rn, respectively, and the chain turns n times around the string (the obstacle). The same problem in a slightly different way was considered by Prager and Frisch by using the combinatorial methods [11] and later by Saito and Chen by employing Fourier analysis [12]. 

Diagrams can also correspond to Fourier transforms, and such a representation is often convenient. We can use it either to calculate +3T(s) or to calculate the Fourier transform of a restricted partition function + J" (r,. . . , rE sly..., sE,S), defined by... 

The DPI representation of the path integral that was developed in the preceding section is not unique. Another path-integral representation is often used that has come to be known as the Fourier representation [33,34,36-42,44,85]. Like the DPI representation, the Fourier representation transforms the path integral into an infinite-dimensional Riemann integral. In this formalism, we consider the paths to be periodic signals that can be represented as a Fourier series. Consider the density matrix p(x, x j8). Since the partition function is the trace of the density matrix, we have... 

In the Fourier representation one need not carry out the normal-mode transformation above or resort to a staging algorithm. The partition function for the particle in a box in 3N dimensions is... 

In Ihe Fourier integral representation of the partition function, the boundary or end conditions of the DNA molecule have corresponded to the first base pair always remaining intact and the Mh base pair always being broken. We now describe more realistic end conditions and show that, for long molecules, the choice of ends makes no difference on log Zjf. 

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