Thread by @markburgess_osl

1. Sorry, I was busy to reply, but here's my non-pornographic take :). I think physicists (egged on by popularists) have deliberately mythologized the issues around these topics, usually by citing Feynman,
Einstein, etc.

2. The larval Lagrangian, and its adult action (integral of L.dt) has nothing to do with thermodynamics a priori. It is only a prescription for generating equations from a compact algebraic form.

3. The action principle is subtle b/c it looks like, and is often represented as, a dynamical quantity,
but that's misleading. It's really "just" a generating functional. A generating function(al) is an algebraic form
which generates something when you hit it with an operation.

4. You can write action/Lagrangians for all kinds of things, not just T-V. And the reason the generating function works for energy etc is essentially because of differential geometry. The derivative operation seeks out gradient landscapes. Things rolling in potential wells. etc.

5. So, when you hit the action with a variation of the dynamical variable, it generates the equation of motion
for that variable, by deconstructing the shape of how all the energy transactions add up.

6. When you hit the action with a variation of space, it generates momentum. And the continuity of variables in space becomes a proxy for the conservation or continuity of momentum.

7. When you hit the action with a variation of time, it generates the total energy. And the continuity of the variables in time implies the conservation of energy. So variables are not allowed to change discontinuously (e.g. at a boundary) else energy is not conserved.

8. It's beautiful, but not magical. Note that the total energy is not really defined completely, because it's an arbitrary accounting parameter for transactions,

9. Now, this last part gives you a clue about any system, from a single material particle to a gas or bulk thermodynamical system too. And, this brings us to the macroscopic matter of *energy distribution* and histograms.

10. Entropy began with some mysticism as a way of getting energy equations to balance in heat engines. It vaguely represented the amount of energy that had become unavailable to convert into work, because it was too evenly distributed. You need rich | poor boundary to do work.

11. A detail in passing: entropy doesn't have the dimensions of energy and is not a function of variables, so someone found that the temperature acts as an effective integrating factor for entropy to make it into "non-free" energy TdS.Continuity of temperature, gives equilibrium.

12. A brilliant Boltzmann discovered that you could get the right answer for this fraction by counting the various ways the system could move energy around (called degrees of freedom).

12a. Pretty clever, and he invented a new model of entropy based on counting all the bulk states macroscopically--thus inventing statistical mechanics.

13. Later, Shannon (aka information theory) showed that entropy is actually just a summary characteristic and generating function for any probability distribution. and von Neumann found similar uses in QM.

14. Cleverly, if you vary the entropy with respect to probability, and say that the expectation value of some variable is constant, you derive the Boltzmann distribution, where the variables is energy. exp(-E/kT). That's because entropy also acts as a generating function(al).

15. I hope some of this is helpful. It's important to separate statistical averages (foliation) from underlying causal phenomena. They are related, but the statistical odour doesn't wag the dog. Boundaries are the way to do work from statistics: pistons, & wagging puppydog tails.