It looks like OP's question underlies a couple of misconceptions. Let me explain:
Schrodinger's equation describes the evolution of the state of an isolated quantum system, which is known to be in the state $\vert \psi (0)\rangle$ at $t=0$. It applies only to isolated quantum systems. While, for a microscopic system, it is sometimes possible to neglect its entanglement with the rest of the universe, this is certainly false for the macroscopic systems described by statistical mechanics (indeed, under normal conditions, it is even false for a dust particle of the size of some $\mu \text m$).
The evolution of the microstate. In the general case, where the system can't be described by a ket, the state (and its evolution) is given by a density matrix $\rho (t)$. However, equilibrium statistical mechanics deals only with the situation in which the system has attained equilibrium and it can be described by a constant canonical density matrix, at least for the purpose of calculating thermodynamic quantities. In particular the concept of "evolution of a microstate" is out of question in statistical mechanics. It would ofcourse be meaningful to ask "how does the system reach the equilibrium", which is the subject of nonequilibrium statistical mechanics, but once you assume that the system is in the grand-canonical ensemble there's no room for evolution.