INSTRUCTOR'S SOLUTIONS MANUAL (to accompany Elementary by MILTON LOYER

By MILTON LOYER

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This will certainly be true in the case of the generating function of ordinary random walks. Finally, as we will soon see, there are cases in which the generating function is the quantity of ultimate interest. 1) to derive the expression for P2 (x). 1) to verify that equal to zero if n = 0. 3 Making use of the result 1 −1 √ 1 + yz 1 = √ ln √ yz 1 − yz y 2 − 2yx + 1 z 2 − 2zx + 1 dx find 1 −1 for arbitrary integers, m and n. 4 The integral order Bessel functions satisfy ∞ e(z−1/z)ρ/2 = Jn (ρ)z n n=−∞ Show that Jn (ρ) = 1 2π π ei(ρ sin θ−nθ) dθ −π Statistical mechanics: the generating function as the ultimate goal of an investigation Generating functions play an important role in statistical mechanics, though they are not usually identified as such.

Consider the integral ∞ e−ax 2 +bx dx (2:S1-3) −∞ Expanding (2:S1-3) in powers of b, we see that the coefficient of bn /n! is the nth moment of the Gaussian. This means that (2:S1-3) is a generating function for 2 moments of the Gaussian e−ax . Given this useful fact, let’s evaluate the integral in (2:S1-3). We do this by completing squares in the exponent. Writing ax 2 − bx = a x − b 2a 2 − b2 2a (2:S1-4) 42 Generating functions I we have ∞ e−ax 2 +bx −∞ = ∞ e−a(x−b/2a)+b −∞ = eb 2 = eb 2 /4a ∞ 2 /4a dx e−ay dy 2 −∞ /4a π a (2:S1-5) performing the expansion and comparing terms, we see that ∞ n −ax 2 x e −∞ dx = n!

21) −∞ We can do this in a variety of ways. The simplest is to expand the exponent with respect to the quartic term. 22) The error we have made is, apparently, unimportant as long as the ratio b/a 4 is small compared to N , and we are interested in what happens when N is very large. A more systematic investigation supports the conclusion that we just reached. In the limit N 1 the number of walks between two points is given in terms of a Gaussian. 22) the Gaussian form for C(N √ ; x, y) holds under the assumption that r = √|x − y| is not large compared to N .

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