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
$S = k_\text \log \Omega,$
where $S$ is the entropy and $k_\text = 1.38\cdot 10^ \, \mathrm$ is Boltzmann's 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 $N_\uparrow=0, 1 \, \mathrm \, 2$. 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
$\Omega = \frac = \frac,$
where the second step follows from the fact that $N_\uparrow+N_\downarrow = N$.

Since $N_\uparrow - N_\downarrow = -U/\mu B$, the energy $U$ can be related to $N_\uparrow$ and $N_\downarrow$ as follows:
$\begin N_\uparrow &= \frac - \frac\\ N_\downarrow &= \frac + \frac. \end$

Thus the final expression for multiplicity as a function of internal energy is
$\Omega = \frac.$

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

Statistical mechanics

{{Thermodynamics-stub