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Canonical ensemble treatment

The canonical partition function for the ideal gas is much easier to evaluate than the microcanonical partition function. Recall the expression for the canonical partition function Q(N,V,T):

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Note that this can be expressed as

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since the Hamiltonian is completely separable. Evaluating the Gaussian integral gives us the final result immediately:

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The expressions for the energy

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and pressure

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gives rise to the results E=3NkT/2 and PV=NkT just as for the microcanonical ensemble. Note also that the entropy S(N,V,T) given by

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becomes

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which reduces to the microcanonical expression exactly if we use the fact that tex2html_wrap_inline641 :

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Thus, the canonical and microcanonical ensembles gives rise to exactly the same thermodynamics!! Let us now look more carefully at the expression for the entropy.



Mark Tuckerman
Thu Feb 20 00:47:55 EST 2003