Classical statistical mechanics starts with the concept of phase space. This is a multidimensional space whose axes represent the canonical coordinates (or degrees of freedom in the statistical mechanical sense; see Footnote 9 of Chapter 6). For our purposes, these are the Cartesian coordinates specifying the position and velocity of every particle in the system. ^[1]

A point in this space, the phase point, represents one possible set of values of these degrees of freedom. A very simple example is the onedimensional harmonic oscillator, which has two degrees of freedom, and whose phase space is a plane with axes x (displacement) and (velocity). If the oscillator has a constant energy, the phase point describes an ellipse in this plane, as shown in Figure G.1.

However, if the oscillator is in contact with a heat bath at temperature T, its energy is constantly fluctuating, and the phase point moves irregularly in the plane. We then need to know the probability of the phase point being found within a given cell of phase space. Such a cell (illustrated in the top right-hand corner of Figure G.1) is a square of sides dx and d ?, with its corner at the point (x, ?), so that when the phase point is within the cell, the displacement lies between x and x+ dx while the velocity lies between ? and ?+ d ?. By essentially...