I agree that thermodynamic relations - and Legendre transformations - are fascinating. I don’t think I ever fully understood them though - at least not to the point where they became “intuitive” :-)
Erm sorry again to have implied they were intuitive, all I meant was that it was relatively intuitive --maybe i should have said "retrievable in a high-pressure concept-doodling game" --compared to a wall of text..
I'm pretty sure I don't understand the possible meanings of what you said there either so let's try :)
<layman-ish op-research lingo>
I meant that the tangent to the convex conjugate ("momentum") provides bounds on what the values returned by the dual step in a primal-dual algo should be. I don't know which meaning of "exponential" I should focus on here (the action perhaps? A power set? A probability distribution?), but "implications" seem to refer to a constraint on outputs contingent on the inputs so I will go with that. Delimited continuations seem to be the closest thing I found in the PL lit, aka wikipedia, feel free to suggest something less kooky :)
That makes much more sense than my flash, which had been following a spark in the other direction:
Delimited continuations are functions, and as such (in a world of algebraic types where we can take sums and products of types) exponentials of types, ran^dom.
[in particular, with the substitution of isomorphism for equality they follow the normal K-12 rules: C^(A+B) ~= C^A * C^B, etc.]
I'd just been glancing at https://en.wikipedia.org/wiki/Convex_conjugate#Examples and the pattern by which single-branched f(x) seems to often become a multibranch f(x) reminded me of how logic-reversing functions in general and logical implication in particular "adds branching": if we wish to establish x'<=5 then if x is already <= 5 we may `skip` but otherwise we must calculate x-5 (and then subtract it off); similarly an implication x->y may be satisfied on one branch by not x but on the other requires y.
[and on the general topic: I like to think of temperature as tying together energy and entropy, where positive temperatures yield the familiar relationships but negative temperatures "unintuitive" ones]
This reverse flash might be what could motivate me to make the connection useful.. an exercise in geometric vengeance (and intuition building) for me to use backtracking DCs in optimization problems (engineering => SDP/IPs)? now to find a plug-and-chug example..
Besides negative T occuring in situations where the arrow of t appears reversed..., PG13 "exponentials turn products into sums"
There is also the pun where S stands for both "action" and "entropy" so that's another direction in which to hunt for the Lagrange multiplier/Lagrangian-Hamiltonian connecting unicorn e.g. picking the "most representative", not necessarily the most optimal path.
I don't know if this counts as an indecent flash, because.. well it is a half-formed opinion (malformed intuition?) born of recent experience..
It's hard to describe this personal experience succinctly, nevertheless I can relate it to wizened physicists often marvelling at having terms miraculously cancel when they engage in the voudou popularly known as path integration
