## Hamiltonian mechanics is conservation of information entropy

TL;DR – Hamiltonian systems are those that conserve information entropy during time evolution. The idea is the following: suppose you have a distribution over position and momentum $\rho(x, p)$. Suppose you evolve it in time and get a final distribution $\hat{\rho}(x, p)$ using Hamiltonian evolution. That is: you take each little area in phase space …