# Why is the topological pressure called pressure?

Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy.

For any probability measure $\mu$ on $X$, one can define the measure-theoretic (or Kolmogorov-Sinai) entropy. Without reference to any measure, one can define the topological entropy, which has the good property of being and invariant under homeomorphism. These two notions are related via a variational principle:

$$h_\mathrm{top} (f) = \sup_{\{\mu\ \mathrm{inv.}\}} h_\mu (f),$$

and are also related to the physical notion of entropy of a system (well, the KS entropy is, at least. The case for the topological entropy is less clear for me, although things behave nicely in the cases I know and which have a physical interest).

Given a continuous potential $\varphi:X \to \mathbb{R}$, one can define the topological pressure $P(\varphi, f)$ by mimicking the definition of the topological entropy (other definitions include the following equation, and some extensions for complex potentials). Then one can get another variational principle:

$$P (\varphi, f) = \sup_{\{\mu \ \mathrm{inv.}\}} \left\{ \int_X \varphi \ d \mu + h_\mu (f) \right\}.$$

The RHS in the variational principle above is the supremum of $\int_X \varphi \ d \mu + h_\mu (f)$, which is, up to a change of sign (1), what is called in physics the free energy of the system. And we try to maximize it, as in physics (modulo the change of sign).

So it would seem logical if, as we have measure-theoretic and topological entropy, we would have measure-theoretic and topological free energy. And I can't find why one would like to call "pressure" what is the maximum of the free energy. I looked at some old works by David Ruelle, but couldn't find how this term was coined, and soon ran into the "not on the Internet nor in the library" wall. It may have something to do with lattices gases, but I emphasize the "may".

So my question is: why is this thing called pressure?

1. The first clue is that the entropy has a positive, and not negative, sign. The second is that we try to maximize the quantity, while in physics one tries to minimize it. Other clues include the fact that, in non-compact cases, a good condition is to have $\lim_\infty \varphi = - \infty$, again in opposition with physics.

Edit: I have asked three people which are familiar with the subject, but none gave me a good answer (actually, I got somewhat conflicting answer). I am starting a bounty to draw some attention, but this might be better suited to MathOverflow...

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docs.google.com/viewer?a=v&q=cache:O1CB20zCU3wJ:www.csm.ro/… may be it help you , but i am not sure –  accounts-manager Mar 17 '13 at 19:29

Perhaps because, in the absence of chemical potentials, the free energy is $pV$ (pressure times volume), and in a probability space, the "volume" (or total measure) is 1. Although, the analogies can't be carried too far, as you have indicated.

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