20 ms·

More intuitively: that TdS has the same "units" as -PdV suggests that temperature [difference] is a "pressure" (thermodynamic potential) that drives entropy inc

More intuitively: that TdS has the same "units" as -PdV suggests that temperature [difference] is a "pressure" (thermodynamic potential) that drives entropy increase.

kgwgk 2y ago
It has the same “units” (if you mean “energy”) as mc2 as well and that doesn’t suggest anything to me… Your intuition is much better than mine - or it’s informed by what you know about temperature.
- vinnyvichy 2y ago
  Sorry! I meant they have the same form as used in the energy differential (1-form), but I had thought "units" would make more sense. In fact, this comparison was how I came to the intuition, although, as you coyly suggested, I did do a check with my earlier intuitions.. https://physics.stackexchange.com/questions/415943/why-does-the-internal-energys-total-differential-behave-like-this https://physics.stackexchange.com/questions/415943/why-does-...
  - kgwgk 2y ago
    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” :-)
    - vinnyvichy 2y ago
      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..
      - kgwgk 2y ago
        No need to apologize! I was joking, I think I get what you mean.
    - vinnyvichy 2y ago
      If you let me flash you my (still indecent) state of intuition.. "convex conjugates" (more precisely but limited sense "momentum maps") are delimited continuations in a optimization algorithm. https://en.wikipedia.org/wiki/Convex_conjugate https://en.wikipedia.org/wiki/Convex_conjugate https://en.wikipedia.org/wiki/Delimited_continuation https://en.wikipedia.org/wiki/Delimited_continuation
      - 082349872349872 2y ago
        Delimited continuations are to exponentials as convex conjugates are to implications?
        vinnyvichy 2y ago
        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 :) </lol>
        082349872349872 2y ago
        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 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]
shiandow 2y ago
It's also precisely what will show up if you use Lagrange multipliers to maximize entropy given a fixed energy. (though for that to make sense you're no longer looking at a single state, you're optimizing the probability distribution itself)
- vinnyvichy 2y ago
  Yeah I have been ruminating on the strange coincidence in the naming of Lagrange multipliers, Lagrangian, Lagrangian duals.. (See below about my comment on convex conjugates and delimited continuations)

More intuitively: that TdS has the same "units" as -PdV suggests that temperature [difference] is a "pressure" (thermodynamic potential) that drives entropy increase.

kgwgk 2y ago

It has the same “units” (if you mean “energy”) as mc2 as well and that doesn’t suggest anything to me… Your intuition is much better than mine - or it’s informed by what you know about temperature.

vinnyvichy 2y ago
Sorry! I meant they have the same form as used in the energy differential (1-form), but I had thought "units" would make more sense. In fact, this comparison was how I came to the intuition, although, as you coyly suggested, I did do a check with my earlier intuitions.. https://physics.stackexchange.com/questions/415943/why-does-the-internal-energys-total-differential-behave-like-this https://physics.stackexchange.com/questions/415943/why-does-...
- kgwgk 2y ago
  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” :-)
  - vinnyvichy 2y ago
    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..
    - kgwgk 2y ago
      No need to apologize! I was joking, I think I get what you mean.
  - vinnyvichy 2y ago
    If you let me flash you my (still indecent) state of intuition.. "convex conjugates" (more precisely but limited sense "momentum maps") are delimited continuations in a optimization algorithm. https://en.wikipedia.org/wiki/Convex_conjugate https://en.wikipedia.org/wiki/Convex_conjugate https://en.wikipedia.org/wiki/Delimited_continuation https://en.wikipedia.org/wiki/Delimited_continuation
    - 082349872349872 2y ago
      Delimited continuations are to exponentials as convex conjugates are to implications?
      - vinnyvichy 2y ago
        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 :) </lol>
        082349872349872 2y ago
        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 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]

shiandow 2y ago

It's also precisely what will show up if you use Lagrange multipliers to maximize entropy given a fixed energy. (though for that to make sense you're no longer looking at a single state, you're optimizing the probability distribution itself)

vinnyvichy 2y ago
Yeah I have been ruminating on the strange coincidence in the naming of Lagrange multipliers, Lagrangian, Lagrangian duals.. (See below about my comment on convex conjugates and delimited continuations)