the situation in classical statistical mechanics. Given some macrocondition M* at , what are the constraints on the system’s state at a later time ? We can think of M* as being defined in terms of 5(j)—that is, we can think of M* as being a macrocondition that’s picked out in terms of some neighborhood of the state space of S that satisfies 5(j).
By Liouville’s Theorem, we know that the total density ρ of states is constant along any trajectory through phase space. That is:
However, as Albert (2000) points out, this only implies that the total phase space volume is invariant with respect to time. Liouville’s theorem says absolutely nothing about how that volume is distributed; it only says that all the volume in the initial macrocondition has to be accounted for somewhere in the later macrocondition(s). In particular, we have no reason to expect that all the volume will be distributed as a single path-connected region at : we just know that the original volume of M* must be accounted for somehow. That volume could be scattered across a number of disconnected states, as shown in Figure 5.2.