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 μ 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: $${\begin{aligned}N_{\uparrow }&={\frac {N}{2}}-{\frac {U}{2\mu B}}\\[4pt]N_{\downarrow }&={\frac {N}{2}}+{\frac {U}{2\mu B}}.\end{aligned}}$$

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.