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?
