Connection With Liouville's Theorem
We have
(the curly bracket is Poisson bracket) since is a function of H. Therefore, according to Liouville's theorem (Hamiltonian) we get
In particular, is time-invariant, that is, the ensemble is a stationary one.
Alternatively, one can say that since the Liouville measure is invariant under the Hamiltonian flow, so is the measure .
Physically speaking, this means the local density of a region of representative points in phase space is invariant, as viewed by an observer moving along with the systems.
Read more about this topic: Microcanonical Ensemble
Famous quotes containing the words connection with, connection and/or theorem:
“We live in a world of things, and our only connection with them is that we know how to manipulate or to consume them.”
—Erich Fromm (1900–1980)
“We will have to give up the hope that, if we try hard, we somehow will always do right by our children. The connection is imperfect. We will sometimes do wrong.”
—Judith Viorst (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)