A simplified model of the two-state
paramagnet provides an example of the process of calculating the multiplicity of particular macrostate. This model consists of a system of microscopic dipoles which may either be aligned or anti-aligned with an externally applied magnetic field . Let N_\uparrow represent the number of dipoles that are aligned with the external field and N_\downarrow represent the number of anti-aligned dipoles. The
potential energy of a single aligned dipole is U_\uparrow = -\mu B, while the energy of an anti-aligned dipole is U_\downarrow = \mu B; thus the overall energy of the system is U = (N_\downarrow-N_\uparrow)\mu B. The goal is to determine the multiplicity as a function of ; from there, the entropy and other thermodynamic properties of the system can be determined. However, it is useful as an intermediate step to calculate multiplicity as a function of N_\uparrow and N_\downarrow. This approach shows that the number of available macrostates is . For example, in a very small system with dipoles, there are three macrostates, corresponding to N_\uparrow=0, 1, 2. Since the N_\uparrow = 0 and N_\uparrow = 2 macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the N_\uparrow = 1, either dipole can be chosen for the aligned dipole, so the multiplicity is 2. In the general case, the multiplicity of a state, or the number of microstates, with N_\uparrow aligned dipoles follows from
combinatorics, resulting in \Omega = \frac{N!}{N_\uparrow!(N-N_\uparrow)!} = \frac{N!}{N_\uparrow!N_\downarrow!}, where the second step follows from the fact that N_\uparrow+N_\downarrow = N. Since N_\uparrow - N_\downarrow = -\tfrac{U}{\mu B}, the energy can be related to N_\uparrow and N_\downarrow as follows: \begin{align} N_\uparrow &= \frac{N}{2} - \frac{U}{2\mu B}\\[4pt] N_\downarrow &= \frac{N}{2} + \frac{U}{2\mu B}. \end{align} Thus the final expression for multiplicity as a function of
internal energy is \Omega = \frac{N!}{ \left(\frac{N}{2} - \frac{U}{2\mu B} \right)! \left( \frac{N}{2} + \frac{U}{2\mu B} \right)!}. This can be used to calculate entropy in accordance with Boltzmann's entropy formula; from there one can calculate other useful properties such as temperature and
heat capacity. == References ==