Multiplicity (statistical mechanics)

Short description: Number of microstates for a given macrostate of a thermodynamic system

In statistical mechanics, multiplicity (also called statistical weight) refers to the number of microstates corresponding to a particular macrostate of a thermodynamic system.[1] Commonly denoted Ω, it is related to the configuration entropy of an isolated system [2] via Boltzmann's entropy formula S=kBlog⁡Ω, where S is the entropy and kB=1.38⋅10−23J/K 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.[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 N↑ represent the number of dipoles that are aligned with the external field and N↓ represent the number of anti-aligned dipoles. The energy of a single aligned dipole is U↑=−μB, while the energy of an anti-aligned dipole is U↓=μB; thus the overall energy of the system is U=(N↓−N↑)μB.

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=N!N↑!(N−N↑)!=N!N↑!N↓!, where the second step follows from the fact that N↑+N↓=N.

Since N↑−N↓=−UμB, the energy U can be related to N↑ and N↓ as follows: N↑=N2−U2μBN↓=N2+U2μB.

Thus the final expression for multiplicity as a function of internal energy is Ω=N!(N2−U2μB)!(N2+U2μB)!.

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. ↑ 1.0 1.1 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.

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