I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of any initial volume element is 0.
From here, how do we say that probability distribution function is constant as we flow in the phase-space?
What's the relation between phase space volume and the density function, which instantaneously tells us the probability of finding the system in a neighborhood in phase-space?
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