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We can prove that gibbs entropy is preserved from Liouville's theorem. We can also prove that its conserved from Noether's theorem and the time reversal symmetry of the Hamiltonian.

But it seems intuitively also that we can prove Liouville's theorem from gibbs entropy conservation, and hence from Noether's theorem. In fact, just setting a uniform distribution seems to me to directly give you Liouville's theorem from gibbs entropy conservation.

Is my understanding correct?

user56834
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  • Noerther's theorem requires an action formulation. What action formulation do you have in mind? – Qmechanic Jul 07 '20 at 06:42
  • @qmechanic, my understanding is that we can always construct a lagrangian from a hamiltonian by taking the legendre transform, and that this is the standard thing to do, and hence in general we don't have to explicitly say what action formulation we want to use when talking at this level of abstraction. – user56834 Jul 10 '20 at 10:40
  • Consider to define all quantities & notions to make sure we are all on the same page. – Qmechanic Jul 10 '20 at 13:49
  • @user56834 Noether's theorem has a Hamiltonian formulation, so would you really even need a Lagrangian? – rschwieb Aug 13 '20 at 17:07
  • Related: https://physics.stackexchange.com/q/611682/2451 – Qmechanic Feb 01 '21 at 15:14

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