## Phase space, ensemble and Liouville’s theorem

Phase space, Ensemble and Liouville’s theorem.

Topics covered in this lecture

a. Ensemble and average — thermodynamic systems

b. Phase space — a classical system

c. Liouville’s theorem

Ensemble and average in thermodynamic systems

For a given “macrostate” (N, V, E) a statistical system, at any instant of time, t, is likely to be found in any one of an extremely large number of distinct “microstates”.

When time passes, the system evolves into different microstates. In due course of time the system exhibits an average behavior of all microstates it passes through.

We can equivalently depict this behavior by envisaging a large number of mental copies of the system, with the same macrostate as the original system, but all the possible microstates, in which the system can exist, all at once. Such a collection of hypothetical or mental copies of the given system is known as an ensemble.

Thus the average behavior of the ensemble is expected to be identical with the time-averaged behavior of the actual physical system. In fact this is one of the fundamental requirements for statistical mechanics to be valid. No matter which mathematical avenue we prefer to meander through we must in the end reach our unique destination of physical validity.

To understand the deeper aspects of this ensemble theory we need to define what is known as “phase space” of a statistical system.