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spxtr comments on Entropy and Temperature - Less Wrong
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I posted some plots in the comment tree rooted by DanielFilan. I don't know what you used as the equation for entropy, but your final answer isn't right. You're right that temperature should be intensive, but the second equation you wrote for it is still extensive, because E is extensive :p
You're right. That should be ε, not E. I did the extra few steps to substitute α = E/(Nε) back in, and solve for E, to recover DanielFilan's (corrected) result:
E = Nε / (exp(ε/T) + 1)
I used S = log[N choose M], where M is the number of excited particles (so M = αN). Then I used Stirling's approximation as you suggested, and differentiated with respect to α.
Good show!