If your system is ergodic, time average=ensemble average. Hence expected constraints can be estimated via following your dynamical system over time
If your system follows the second law, then entropy increases subject to the constraints
So the system converges to the maxent invariant distribution subject to constraint, which is why langevin dynamics converges to the Boltzmann distribution, and you can estimate equilibrium energy by following the particle around
In particular, we often use maxent to derive the prior itself (=invariant measure), and when our system is out of equilibrium, we can then maximize relative entropy w.r.t our maxent prior to update our distribution
Nice, some connections with why are maximum entropy distributions so ubiquitous:
If your system is ergodic, time average=ensemble average. Hence expected constraints can be estimated via following your dynamical system over time
If your system follows the second law, then entropy increases subject to the constraints
So the system converges to the maxent invariant distribution subject to constraint, which is why langevin dynamics converges to the Boltzmann distribution, and you can estimate equilibrium energy by following the particle around
In particular, we often use maxent to derive the prior itself (=invariant measure), and when our system is out of equilibrium, we can then maximize relative entropy w.r.t our maxent prior to update our distribution