# Thermodynamics / Statistical Mechanics - Additivity of Entropy for Two Spin

1. Feb 17, 2008

### smithg86

[SOLVED] Thermodynamics / Statistical Mechanics - Additivity of Entropy for Two Spin

1. The problem statement, all variables and given/known data

(This is question 2.5.b in "Thermal Physics" by Kittel/Kromer.)

Consider two spin 1/2 systems. Each has a macroscopic number of particles (spins), $$N_1 \cong N_2 = 10^{22}$$. System 1 has multiplicity function $$g_{1}(N_{1}, s_{1})$$, system 2 has $$g_{2}(N_{2},s-s_{1})$$, where the total $$s=s_{1}+s_{2}$$, and the product $$g_{1} g_{2}$$ is sharply peaked at the equilibrium $$s_{1} = \hat{s_{1}}$$.

Estimate the order of magnitude of the ratio of the total number of micro-states g(N,s) of the combined system to the peak value of their product, $$\sum g_{1}(N_{1},s_{1})g_{2}(N_{2},s_{2})/(g_{1}g_{2})_{max}$$, for $$s=10^{20}$$.

2. Relevant equations

Gaussian Approximation:

$$(g_{1}g_{2})_{max} \equiv g_{1}(\hat{s}_{1})g_{2}(s-\hat{s}_{1}) = g_{1}(0)g_{2}(0) e^{-2s^{2}/N}$$, where

$$g_{1}(0) = g_{1}(N_{1},0)$$

and

$$g(N_{1},s) = \frac{(N_{1})!}{(\frac{1}{2} N_{1} + s_{1})! (\frac{1}{2} N_{1} - s_{1})!} \cong g_{1}(N_{1}, 0) e^{-2s^{2}/N}$$

3. The attempt at a solution

I tried replacing the summation with an integral from negative infinity to infinity, but couldn't do the integral by hand - Maple tells me its either 0 or infinity, so I'm doing something wrong. I know the $$g_{1}(0)g_{2}(0)$$ terms cancel out, but I can't simplify any more. I'm looking for a numerical answer.

2. Feb 18, 2008