General form of the partition function for a classical system

So far we have described a system of $N$ particles by using $3N$ cartesian coordinates and $3N$ momenta (or velocities) coordinates to specify the position and the state of motion of each particle. We have also separated the treatment of the centre of mass of the particles from that of the internal degrees of freedom.

More generally, if the system has $\nu$ degrees of freedom it may be convenient to use the $2\nu$ generalised coordinates $q_1,\dots,q_\nu$ and generalised velocities $\dot{q}_1,\dots,\dot{q}_\nu$, where $\dot{q}_\nu = dq_\nu/dt$. In such a description the $\nu$ generalised coordinates are all independent from each other. A simple example to illustrate this idea is the description of a rigid diatomic molecule in which the distance between the two atoms is fixed to be $R$. We could describe its position using the cartesian coordinates of the two atoms in the molecule, ${\bf r}_1 =(x_1,y_1,z_1)$ and ${\bf r}_2 =(x_2,y_2,z_2)$, together with the auxiliary condition specifying the constraint $R^2 = (x_1-x_2)^2+ (y_1-y_2)^2+(z_1-z_2)^2$. Or, more simply, we could use 5 generalised coordinates, choosing for example the cartesian coordinates of atoms 1, ${\bf r}_1$, and the two polar angles $\phi$ and $\theta$ defining the direction of the vector ${\bf r}_2 - {\bf r}_1$, with $R\sin\theta\cos\phi = x_2 - x_1; R\sin\theta\sin\phi=y_2-y_1; R\cos\theta = z_2-z_1$.

The kinetic energy of the system depends on the generalised coordinates and velocities, $K = K(q_1,\dots,q_\nu, \dot{q}_1,\dots,\dot{q}_\nu)$, and so does the potential energy in general (e.g. charged particles in a magnetic field). However, here we will consider only potential energy functions that depend only on the generalised coordinates, $U = U(q_1,\dots,q_\nu)$. An alternative a more convenient way to describe the state of the system is to use the generalised coordinates $q_1,\dots,q_\nu$ and their conjugate momenta $p_1,\dots,p_\nu$, which are defined through the Lagrangian:

$$\begin{aligned}% requires amsmath; align* for no eq. number \ma... ...cal{L}}{\partial \dot{q}_i} ; \quad \quad i = 1,\nu \end{aligned}$$

and the partial differentiation means that all the other $2\nu - 1$ variables are being kept constant. A state of the system is specified by a particular value of the $2\nu$ generalised coordinates, but as discussed in Sec.  in classical mechanics these are continuous variables and so there would be an infinite number of them in any volume of phase space $d\Gamma = dq_1, \dots,dq_\nu$, $dp_1,\dots,dp_\nu$, meaning that a strictly classical description is problematic. We use the quantum mechanics argument that the phase space is granular, divided in cells of volume $h^3$ each of which containing one state and therefore the number of states in a element of volume $d\Gamma$ is:

$$\frac{1}{h^\nu} dq_1, \dots, dq_\nu, dp_1,\dots,dp_\nu.$$

Here is an important principle. As the system is evolved in time according to Newton's equations of motion, all the states with phase space coordinates in $d\Gamma$ at a particular instant in time will have them transformed into a volume $d \Gamma^\prime$ at a future time. Since the number of states remains the same, and each state occupies a volume of phase space $h^3$, it follows that the volume $d \Gamma^\prime$ has the same size as $d\Gamma$, i.e. the volume of phase space is conserved. This is also known as Liouville's theorem.

To write an expression for the partition function we need the total energy of the system, which we define via the Hamiltonian:

$$\mathcal{H}(q,p) = K + U,$$ (4.57)

where we have used the abbreviations $q = q_1,\dots,q_\nu$ and $p = p_1,\dots,p_\nu$. The partition function is the normalisation constant in the Boltzmann probabilities and so we obtain it by summing the Boltzmann factor over all possible states. By turning the sum into an integral we have:

$$Z = \frac{1}{h^\nu} \int dq_1, \dots, dq_\nu, dp_1,\dots,dp_\nu e^{-\beta \mathcal{H}(q,p)},$$ (4.58)

and the integral is extended to the whole phase space. If the system is made of $N$ identical and non-localised subsystems we need to divide it by $N!$. Note that the Hamiltonian $\mathcal{H}$ includes all contributions to the energy of the system.