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Background

The principle of Maximum Entropy

In equilibrium statistical physics, we can get Maxwell-Boltzmann distribution in classical mechanics and Bose-Einstein distribution and Fermi-Dirac distribution in quantum mechanics from the principle of maximum entropy. $$ S=-\rm Tr(\rho\ln\rho) $$ But in nonequilibrium statistical physics or when the particles are interacting, we give up the statistical description of the system as in equilibrium statistical physics and turn to dynamical description. To the best of my knowledge, the most important result in nonequilibrium statistical physics is the fluctuation-dissipation theorem.

The principle of Minimum Energy

When we consider the particles are interacting, we should consider the Gibbs free energy $$ G = H+pV-TS $$

The principle of Least Action

From the principle of least action, we can get Lagrange equation in classical field theory and Maxwell's equations from Maxwell action and Einstein Field Equation from Einstein-Hilbert action. $$ S=\int dtL=\int dx^4\mathcal{L}(\phi,\partial_\mu\phi) $$

Question

Q1: What is the relationship between maximum entropy and minimum energy? Is maximum entropy is a part of minimum energy? Because when $H$, $p$ and $V$ don't change, maximum entropy $S$ just give us minimum energy $G$?

Q2: What is the relationship between minimum energy and least action? Is minimum energy a minimum of hamiltonian $H$ while least action a minimum of Lagrange $L$?

Q3: Can we use the principle of minimum energy or least action in nonequilibrium statistical physics?

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    $\begingroup$ You should try to write single, well-focused questions. Do the following links answer your questions Q1 and Q2 ? physics.stackexchange.com/q/230289 physics.stackexchange.com/q/338613 $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Dec 28, 2022 at 9:37
  • $\begingroup$ Thanks for reply. yeah, these two links are very similar to my questions. But I'm not sure if we can get the conclusion that maximum entropy is a special case of minimum energy and minimum energy is another special case of least action. $\endgroup$
    – lbyshare
    Commented Dec 28, 2022 at 11:03
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    $\begingroup$ Within equilibrium thermodynamics, maxent is not a special case of the minimum energy. They are consistent ways to get the equilibrium conditions under different conditions (fixed energy or fixed entropy). Minimum energy can be seen as a special case of least action, although some careful definition of the range of validity of the two principles would be important. I suggest you reformulate your question in a more focused way around question 2, and separately, question 3. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Dec 28, 2022 at 15:48
  • $\begingroup$ See also the proof at Why are thermodynamic potentials minimised? $\endgroup$
    – Chemomechanics
    Commented Dec 28, 2022 at 16:27
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/47581/2451 , physics.stackexchange.com/q/483580/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Nov 26 at 7:19

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