Liouville's theorem states that the total time-derivative of the distribution function is zero along a system trajectory in phase-space. Where the system follows a trajectory that satisfies the Hamilton's equations of motion. I have a hard time getting an inuitive understanding of this statement. For instance, what does this theorem tell me about a free-falling particle in a gravitational field? Edit: I assume that the distribution function for a free falling particle would be proportional to a product of delta functions.