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 $Ω$ , it is related to the configuration entropy of an isolated system via Boltzmann's entropy formula $S = k_{B} \log Ω,$ where $S$ is the entropy and $k_{B}$ 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 $N_{↑}$ represent the number of dipoles that are aligned with the external field and $N_{↓}$ represent the number of anti-aligned dipoles. The potential energy of a single aligned dipole is $U_{↑} = - μ B,$ while the energy of an anti-aligned dipole is $U_{↓} = μ B;$ thus the overall energy of the system is $U = (N_{↓} - N_{↑}) μ B .$

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 $N_{↑}$ and $N_{↓} .$ 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 $N_{↑} = 0, 1, 2.$ Since the $N_{↑} = 0$ and $N_{↑} = 2$ macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the $N_{↑} = 1,$ 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 $N_{↑}$ aligned dipoles follows from combinatorics, resulting in $Ω = \frac{N!}{N_{↑}! (N - N_{↑})!} = \frac{N!}{N_{↑}! N_{↓}!},$ where the second step follows from the fact that $N_{↑} + N_{↓} = N .$

Since $N_{↑} - N_{↓} = - \frac{U}{μ B},$ the energy $U$ can be related to $N_{↑}$ and $N_{↓}$ as follows: $\begin{matrix} N_{↑} & = \frac{N}{2} - \frac{U}{2 μ B} \\ N_{↓} & = \frac{N}{2} + \frac{U}{2 μ B} . \end{matrix}$

Thus the final expression for multiplicity as a function of internal energy is $Ω = \frac{N!}{(\frac{N}{2} - \frac{U}{2 μ B})! (\frac{N}{2} + \frac{U}{2 μ B})!} .$

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

1 2 Schroeder, Daniel V. (1999). An Introduction to Thermal Physics (First ed.). Pearson. ISBN 9780201380279.
↑ Atkins, Peter; Julio de Paula (2002). Physical Chemistry (7th ed.). Oxford University Press.

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[schroeder-1] 1 2 Schroeder, Daniel V. (1999). An Introduction to Thermal Physics (First ed.). Pearson. ISBN 9780201380279.

[2] Atkins, Peter; Julio de Paula (2002). Physical Chemistry (7th ed.). Oxford University Press.