  It is a consequence of Liouville's Theorem that this entropy is constant under Hamiltonian flows. While this guarantees entropy is non-decreasing, it presents a problem for providing an account of the appearance of entropy increase. The standard approach to dealing with this is called coarse graining. Coarse graining replaces the probability distribution over each macrostate, p(X3N, P3N | i), with a ‘smoother’ probability distribution, p′(X3N, P3N | i), Typically the ‘uniform’ distribution: