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My question stems from the sentence said by my professor "The action is the free energy" which I don't understand. Thinking that probably I'm missing some key concepts, I'd like to know: what are the links between thermodynamic quantities (entropy, temperature, free energy, internal energy...) and lagrangian quantities (action, potential, Lagrangian, Hamiltonian...)?

I'd like to know which ones of the mentioned quantities, if all, can be "matched", and precisely to which one (by "precisely" I mean that, for example, I know that the Hamiltonian is the energy, but which one? Internal? Free? Total?).

Edit: the "action" my professor was talking about is a Euclidean DBI action for a D8-brane. I purposefully didn't mention it in the original question because I wanted to know whether there are general links between thermodynamics and lagrangian mechanics, but from the comments received it seems that more context is needed in order to give an answer. For even more context (if needed) I want to calculate the energy $E$ of a D8-brane. Still, I only know how to calculate its action, so my professor proposed to use the thermodynamic formula $E=F+TS$ where $T$ is the temperature, $S=\partial F/\partial T$ is the entropy and $F$ is the free energy, "which is the same of the action" in the professor's words.

Edit 2: the specification about the branes is most likely crucial because I found that according to the holographic correspondence if $S$ is a gravity action then $F=ST$ is the free energy density of a dual strongly coupled (3+1)-dimensional conformal field theory. Still, I'll leave this question open because I'd like to know whether this claim is supported by a more general thermodynamic-lagrangian match.

Níckolas Alves
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Mauro Giliberti
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  • Related: https://physics.stackexchange.com/q/87306/2451 , https://physics.stackexchange.com/q/35135/2451 – Qmechanic Jul 27 '22 at 16:38
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    I think the question needs more precision, regarding whether the professor speaks from the point of view of QFT (as in linked posts) or whether his statement is done within classical theory - the meaning of statement is not exactly the same in the two cases. – Roger V. Jul 27 '22 at 17:03
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    Your question is not entirely clear. However, in (Euclidean) QFT the partition function is given by a path integral over the exponential of the action. The free energy is the logarithm of the partition function and taking the logarithm of only the integrand of the path intgral then, indeed, gives the action of a system. – horropie Jul 27 '22 at 17:29

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