# 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...

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.

Page 55 of Prof. Oliveira's notes: http://cdsagenda5.ictp.trieste.it/askArchive.php?base=agenda&categ=a11165&id=a11165s16t18/lecture_notes

There he partially answers this questions and gives a link to Prof. Ruelle's paper on this issue which might be helpful.

Also see Prof. Sarig's books first 5 pages for more insight into the issue

http://www.wisdom.weizmann.ac.il/~sarigo/TDFnotes.pdf

• Thank you. Prof. Oliveira's notes don't exactly answer this question (actually, I asked him before I posted here), but the answer must lie somewhere into Ruelle's work on lattices... I'll check them. – D. Thomine Aug 11 '13 at 18:05
• The first link is broken. Does anyone have a link to these notes? – Ben CW Dec 15 '15 at 5:44

I think you've essentially found the answer when you say that $P(\varphi, f)$ is called the free energy in physics. That is, the pressure should not be called the pressure, it should be called free energy.

This view is expressed in the paper Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations by van Enter, Fernandez, and Sokal. They define the "pressure" in Definition 2.55 and go on to prove a version of the variational principle for it in Theorem 2.63. But right below this definition, they stated that "this quantity should really be called 'minus the free energy density'".

Now if you want a physical explanation of why the pressure or free energy is called what it's called, I suggest you take a look at the first chapter of Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction by Friedli and Velenik. The authors define temperature, pressure, and free energy from a thermodynamic viewpoint in Sections 1.1.1-1.1.5. These quantities are related to the log of the (canonical or grand canonical) partition function in Section 1.3.1.