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In statistical thermodynamics, the canonical, $Z(\beta)$, and microcanonical, $\Omega(E)$, partition functions are related by Laplace tranform. $$ Z(\beta) = \int_0^\infty\text dE \ \Omega(E)\ \text e^{-\beta E} $$ The thermodynamic variables corresponding to both ensembles are obtained as logarithm of their respective partition functions. $$ S = k_B\ln[\Omega(E)] \qquad -\frac{F}{T} = k_B\ln[Z(\beta)] $$

Now, the quantities $S$ and $-F/T$ are Legendre transforms of each other.

A logarithmic map on Laplace transform pairs gives a legendre transform pair. What is the underlying reason for this ? I have learnt from this question that integral transforms are like basis transformation. Now with regards to this question, where does this interpretation take us ?

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These papers will give you some answers: (https://www.andrew.cmu.edu/course/33-765/pdf/Legendre.pdf) and (https://arxiv.org/pdf/0806.1147), the very clear (https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/) and, at a higher level (http://www.math.u-psud.fr/~rumin/recherche/entropy-POVM.pdf).

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