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Quibbler II (talk | contribs) Signs of $\lambda$ where incorrectly flipped halfway through the calculation. Signs are fixed, results obviously didn't change. |
m It was said that the partition function Z is constant, but this is not correct, as it clearly varies with the exponent lambda_2, which in turn varies with U. I added a parenthetical aside after the identification of lambda_2 with 1/T that explicitly acknowledges this, and offers a brief explanation of why the additional dependencies don't matter. |
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To obtain <math> \lambda_2 </math>, we differentiate <math> S </math> with respect to the average energy <math> U </math> and apply the [[first law of thermodynamics]], <math> dU = T dS - P dV </math>:
<math display="block">\frac{dS}{dU} = \lambda_2 \equiv \frac{1}{T} .</math>
(Note that <math> \lambda_2 </math> and <math> Z </math> vary with <math> U </math> as well; however, using the chain rule and
<math display="block"> \frac{d}{d\lambda_2} \ln(Z) = - \frac{1}{k_\text{B}} \sum_i \rho_i E_i = - \frac{U}{k_\text{B}}, </math>
one can show that the additional contributions to this derivative cancel each other.)
Thus the canonical partition function <math> Z </math> becomes
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