(adsbygoogle = window.adsbygoogle || []).push({}); [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), [tex]N_1 \cong N_2 = 10^{22}[/tex]. System 1 has multiplicity function [tex]g_{1}(N_{1}, s_{1})[/tex], system 2 has [tex]g_{2}(N_{2},s-s_{1})[/tex], where the total [tex]s=s_{1}+s_{2}[/tex], and the product [tex]g_{1} g_{2}[/tex] is sharply peaked at the equilibrium [tex]s_{1} = \hat{s_{1}}[/tex].

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, [tex]\sum g_{1}(N_{1},s_{1})g_{2}(N_{2},s_{2})/(g_{1}g_{2})_{max}[/tex], for [tex]s=10^{20}[/tex].

2. Relevant equations

Gaussian Approximation:

[tex](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} [/tex], where

[tex]g_{1}(0) = g_{1}(N_{1},0) [/tex]

and

[tex]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}[/tex]

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 [tex]g_{1}(0)g_{2}(0)[/tex] terms cancel out, but I can't simplify any more. I'm looking for a numerical answer.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**