  Now consider an arbitrary probability density \(\rho(x)\) over this phase space. He shows, by (what is now known as) Liouville's theorem, that \(\rho\) remains constant along a trajectory, i.e., \(\rho(x_0) = \rho(x_t)\). Assuming now for simplicity that all points in a given energy hypersurface lie on a single trajectory, the probability should be a constant over the energy hypersurface. In other words, the only stationary probability with fixed total energy is the microcanonical distribution.