By Garrett P.

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For example, consider a collection of long chain polymers in chemical equilibrium with a reservoir of monomers. The monomer bath controls the molecular weights of the polymers by establishing a chemical potential, µ – or, alternatively, a fugacity z – per monomeric unit. The ensemble of interest then consists of a distribution of polymers having a variety of molecular weights, and the Boltzmann factor associated with an n-unit polymer contains the factor z n . Random walk statistics are appropriate to the statistical mechanics of this ensemble of polymers.

37) N . 2) n = d/2l + N /2, we have m = d/2l. This means that the likelihood that a walker will end up a distance d from its point of departure is given by 1 d2 exp − . 41), we have divided by the requisite factor of 2 N to arrive at a probability density that is normalized to one. 41) is a Gaussian. We will encounter this ubiquitous form repeatedly in the course of our investigation of random walk statistics. It reflects the consequences of the central limit theorem of statistics (Feller, 1968), as it applies to the random walk process.

Show that d 4 = 3(d N2 )2 . 6 Supplement: method of steepest descents There is a general method for extracting the coefficient of z N in the power series expansion of a function of z that is free of singularities in the vicinity of z = 0. Let’s see how it works. If f (z) is the function of interest, and if it admits the power series expansion ∞ f (z) = CN zN (1:S-1) N =0 then a fundamental theorem of complex analysis tell us that CN = 1 2πi f (z) dz z N +1 (1:S-2) where the closed integration contour encircles the origin and does not enclose any singularity of f (z).