I have been reading the book named Entropy, Order Parameters, and Complexity. And here is a question it ask
Liouville’s theorem tells us that the total derivative of the probability density is zero; following the trajectory of a system, the local probability density never changes. The equilibrium states have probability densities that only depend on energy and number. Something is wrong; if the probability density starts nonuniform, how can it become uniform?
The last sentence seems to imply that the probability density $\rho$ for points in phase space can evolve from a non-uniform distribution to an uniform distribution. I wonder why this is true.
In the book, below the referred paragraph, the book then shows that in an isolated system the entropy $S = \int k_B\rho\log\rho\ {\rm d}P{\rm d}Q$ is constant in time from a microscopic description that is from the perspective of points in momentum space P, and configuration space Q, thus entropy increases is an emergent property. I can understand what's going on here, but I don't know if such a fact is related to the last sentence of the referred paragraph.
