# Is microcanonical ensemble invariant under the Hamiltonian time evolution?

Consider a Hamiltonian system with the initial states distributed according to the microcanonical ensemble. That is all the states at the initial moment of time are uniformly distributed over the constant energy surface in the phase space.

If we allow the system to evolve, the states will still lie on that constant energy surface. However, will they still be uniformly distributed?

## 1 Answer

Yes, Liouville's theorem states that the representative points in a phase space distribution move like an incompressible fluid under time evolution. So if the region of phase space they occupy is unchanged and the initial density is uniform (both true of the mircocanonical ensemble), the phase space density does not change in time.

Physically, if the distribution were not invariant in time, it would not represent the equilibrium state of the system.