Fluctuation Theorem or Relation?

Some years ago I witnessed a small academic fight between Giovanni Gallavotti and Denis Evans, who are known to have a history of tensions. These outbursts of ego are related to the issue of the paternity of the so-called “Fluctuation Theorem”, a major result that has been understood by many people in a lapse of time and in a variety of forms, and that was hidden in previous literature at times when there was no broad community to notice them. I hardly care about these small discussions, to the point where I embraced the policy of referring to results without calling them by any personal name, to avoid subscribing to a party and incurring into Arnold’s Law – which notoriously was not by Arnold…

I wrote “theorem”, but should have I written “relation”? That was the matter of the very polite but provocative remark that Gallavotti made to Evans. Where, it was understood, Evans provided relations, and Gallavotti proved theorems (I have something to say in this respect as well, but let it be…). Gallavotti and I sat together at the conference dinner, so I asked him about this issue, we had a pleasant conversation, but his answer I can’t remember.

Anyway, this thought kept rumbling in the backyards of my mind, and as I was writing my latest paper on the “marginal Fluctuation Relation/Theorem” it popped up again. In this paper, I prove a theorem that is fair to call a (marginal) Fluctuation Theorem, but the main equation in the theorem is not a “theorem” itself: it’s just a relation! If I called that equation a “Fluctuation Theorem”, then I would get something like: “Theorem: Theorem holds”. An ouroboros. This is fascinating question in logic (one can always use different logic systems whereby the assumptions of the two instances of “Theorem” are not the same), but in our small world this is obviously a logical inconsistency.

So I decided that I will be very careful whenever something is a relation (can be an equation, an inequality, a meaningful plot etc.) or a theorem (which implies that some relation is deducted from some hypothesis). These guys, FR and FT, belong to different logical planes. And taking care of these subtleties is not just a matter of dull precision: it’s a matter of forging our own minds.