Multiplicity (statistical mechanics)

In statistical mechanics, multiplicity (also called statistical weight) refers to the number of microstates corresponding to a particular macrostate of a thermodynamic system. Commonly denoted $$\Omega$$, it is related to the configuration entropy of an isolated system via Boltzmann's entropy formula $$S = k_\text{B} \log \Omega,$$ where $$S$$ is the entropy and $$k_\text{B}$$ is the Boltzmann constant.

Example: the two-state paramagnet
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 $N$ microscopic dipoles $&mu;$ which may either be aligned or anti-aligned with an externally applied magnetic field $B$. 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 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 $U$; 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 $N + 1$. For example, in a very small system with $N = 2$ 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 $U$ 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.