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:
“Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
—Jane Heap (c. 1880–1964)
“Much is made of the accelerating brutality of young people’s crimes, but rarely does our concern for dangerous children translate into concern for children in danger. We fail to make the connection between the use of force on children themselves, and violent antisocial behavior, or the connection between watching father batter mother and the child deducing a link between violence and masculinity.”
—Letty Cottin Pogrebin (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)