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We present solutions to the classical Liouville equation for ergodic and completely integrable systems - systems that are known to attain equilibrium. Ergodic systems are known to thermal equilibrate with a Maxwell-Boltzmann distribution and we show a simple derivation of this distribution that also leads to a derivation of the distribution at any time t. For illustrative purposes, we apply the method to the problem of a one-dimensional gravitational gas even though its ergodicity is debatable.arXiv:1311.4594v2 fatcat:oqihmjgbpbb5pds2yxf24btyja