The Relation to Diffusion and Brownian Motion

We have taken the continuous limit of a gaussian equivalent chain for a flexible polymer chain. This procedure leads to a nice physical picture of 

12) the center of force is taken at the center of each bond. This approximation becomes exact as Aj 0. The presence of an external force implies that space is no longer isotropic. Thus, let r = R, and 

As noted in Section II, G[r(j)] contains all the information pertaining to the configurational statistics of the polymer chain. However, this is more information than we can handle or need. A number of reduced distributions are therefore of interest. One of the simplest is the distribution of the end vectors G(R, R L), which is obtained from (3.12) by selecting 

We now demonstrate that G(R, R L) satisfies the diffusion equation for a particle in the external potential K(R). In particular, it is well known that G satisfies 

That (3.21) and (3.22) are obtained should be very clear. We started with a random flight model of a polymer where the contour length represents a timelike variable. For long enough times, random walks, or Brownian motion, can be considered to be diffusion processes. Here the diffusion constant is defined as the mean square displacement per unit time. Using (3.2), we find 

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