Phase Space: a Framework for Statistics

Statistics involves the counting of states, and the state of a classical particle is completely specified by the measurement of its position and momentum. If we know the six quantities

then we know its state. It is often convenient in statistics to imagine a six-dimensional space composed of the six position and momentum coordinates. It is conventionally called "phase space". The counting tasks can then be visualized in a geometrical framework where each point in phase space corresponds to a particular position and momentum. That is, each point in phase space represents a unique state of the particle. The state of a system of particles corresponds to a certain distribution of points in phase space.

The counting of the number of states available to a particle amounts to determining the available volume in phase space. One might preclude that for a continuous phase space, any finite volume would contain an infinite number of states. But the uncertainty principle tells us that we cannot simultaneously know both the position and momentum, so we cannot really say that a particle is at a mathematical point in phase space. So when we contemplate an element of "volume" in phase space

then the smallest "cell" in phase space which we can consider is constrained by the uncertainty principle to be