Given two systems of N1 ≈ N2 = 1022 spins with multiplicity functions g1(N1, S1) an g2(N2, s – s1) and g2(N2, s – s1), the product g1g2 as a function of s1 is relatively sharply peaked at s1 = ŝ1. For s1 = ŝ1 + 1012, the product g1g2 is reduced by 10–174 from its peak value. Use the Gaussian approximation to the multiplicity function; the form (17) may be useful.
(a) Compute g1g2/(g1g2)max for s1 = ŝ1 + 1011 and s = 0.
(b) For s = 1020, by what factor must you multiply (g1g2)max to make it equal to ∑s1g1(N1, s1)g2(N2, s – s1); give the factor to the nearest order of magnitude
(c) How large is the fractional error in the entropy when you ignore this factor?