Multiplicity (statistical mechanics)

A simplified model of the two-state paramagnet provides an example of the process of calculating the multiplicity of particular macrostate.[1] 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 ${\displaystyle N_{\uparrow }}$ represent the number of dipoles that are aligned with the external field and ${\displaystyle N_{\downarrow }}$ represent the number of anti-aligned dipoles. The energy of a single aligned dipole is ${\displaystyle U_{\uparrow }=-\mu B,}$ while the energy of an anti-aligned dipole is ${\displaystyle U_{\downarrow }=\mu B;}$ thus the overall energy of the system is

$${\displaystyle U=(N_{\downarrow }-N_{\uparrow })\mu B.}$$

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 ${\displaystyle N_{\uparrow }}$ and ${\displaystyle N_{\downarrow }.}$ 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 ${\displaystyle N_{\uparrow }=0,1,2.}$ Since the ${\displaystyle N_{\uparrow }=0}$ and ${\displaystyle N_{\uparrow }=2}$ macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the ${\displaystyle N_{\uparrow }=1,}$ 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 ${\displaystyle N_{\uparrow }}$ aligned dipoles follows from combinatorics, resulting in

$${\displaystyle \Omega ={\frac {N!}{N_{\uparrow }!(N-N_{\uparrow })!}}={\frac {N!}{N_{\uparrow }!N_{\downarrow }!}},}$$

where the second step follows from the fact that ${\displaystyle N_{\uparrow }+N_{\downarrow }=N.}$

Since ${\displaystyle N_{\uparrow }-N_{\downarrow }=-{\tfrac {U}{\mu B}},}$ the energy U can be related to ${\displaystyle N_{\uparrow }}$ and ${\displaystyle N_{\downarrow }}$ as follows:

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

Thus the final expression for multiplicity as a function of internal energy is

$${\displaystyle \Omega ={\frac {N!}{\left({\frac {N}{2}}-{\frac {U}{2\mu B}}\right)!\left({\frac {N}{2}}+{\frac {U}{2\mu B}}\right)!}}.}$$

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.