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Thermodynamics / Statistical Mechanics - Additivity of Entropy for Two Spin

  1. Feb 17, 2008 #1
    [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.
     
  2. jcsd
  3. Feb 18, 2008 #2
    i figured it out. please delete the thread.
     
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