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Derivation of the Entropy

To calculate the entropy of a two dimensional hard disk gas, one can begin with the partition function[9]:

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where, tex2html_wrap_inline465 are the position vectors and tex2html_wrap_inline467 are the momentum vectors. This reduces to

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This needs to be divided by N! because the demon does not distinguish individual molecules, so

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In terms of the partition, the entropy is

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For a 2 dimensional gas,

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Solving for the entropy, we get

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Using Sterling's approximation for N!, we get

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All quantities here are either known or arbitrary constants.



Eric H. Neilsen
Mon Jun 16 13:53:44 EDT 1997