# Statistical mechanics and phase space

1. Nov 3, 2005

### 2bootspizza

it's just not sinking in.. i know a cell in phase space has 6 dimensions, 3 for momentum and the other 3 for position.

but i'd like to understand it(phase space). can someone give me an example maybe or tell me why this constuct is needed?? or a link to a very good description?

2. Nov 3, 2005

### Galileo

The good thing about the phase-space (or configuration space) is that you can specify the entire state of your system by a single point in the state space. As time goes on, the laws of mechanics will change the state of the system, so the point will move in the state space. It's a useful geometrical picture to have.

Take the simple example of a one-dimensional harmonic oscillator. The phase space has 2 dimensions (1 position coordinate, 1 momentum coordinate) which makes it drawable, but any realistic phase-space is so hugely dimensional that it is ofcourse not possible.
Suppose the energy of the system is H. Conservation of energy gives us the trajectory of the point in the phase space:
$$H=\frac{p^2}{2m}+\frac{1}{2}kx^2$$
which is an ellipse. As the particle oscillates, the system point travels along the ellipse in the counterclockwise direction (if you plot p vertically and x horizontally).