Multiplicity (statistical mechanics)

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 {\displaystyle \Omega } , it is related to the configuration entropy of an isolated system[2] via Boltzmann's entropy formula S = k B log ⁡ Ω , {\displaystyle S=k_{\text{B}}\log \Omega ,} where  S {\displaystyle S} is the entropy and  k B {\displaystyle k_{\text{B}}} is the Boltzmann constant.

## Example: the two-state paramagnet

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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 ↑ {\displaystyle N_{\uparrow }} represent the number of dipoles that are aligned with the external field and  N ↓ {\displaystyle N_{\downarrow }} represent the number of anti-aligned dipoles. The energy of a single aligned dipole is  U ↑ = − μ B , {\displaystyle U_{\uparrow }=-\mu B,} while the energy of an anti-aligned dipole is  U ↓ = μ B ; {\displaystyle U_{\downarrow }=\mu B;} thus the overall energy of the system is  U = ( N ↓ − N ↑ ) μ B . {\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  N ↑ {\displaystyle N_{\uparrow }} and  N ↓ . {\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  N ↑ = 0 , 1 , 2. {\displaystyle N_{\uparrow }=0,1,2.} Since the  N ↑ = 0 {\displaystyle N_{\uparrow }=0} and  N ↑ = 2 {\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  N ↑ = 1 , {\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  N ↑ {\displaystyle N_{\uparrow }} aligned dipoles follows from combinatorics, resulting in  Ω = N ! N ↑ ! ( N − N ↑ ) ! = N ! N ↑ ! N ↓ ! , {\displaystyle \Omega ={\frac {N!}{N_{\uparrow }!(N-N_{\uparrow })!}}={\frac {N!}{N_{\uparrow }!N_{\downarrow }!}},} where the second step follows from the fact that  N ↑ + N ↓ = N . {\displaystyle N_{\uparrow }+N_{\downarrow }=N.}

Since  N ↑ − N ↓ = − U μ B , {\displaystyle N_{\uparrow }-N_{\downarrow }=-{\tfrac {U}{\mu B}},} the energy U can be related to  N ↑ {\displaystyle N_{\uparrow }} and  N ↓ {\displaystyle N_{\downarrow }} as follows:  N ↑ = N 2 − U 2 μ B N ↓ = N 2 + U 2 μ B . {\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  Ω = N ! ( N 2 − U 2 μ B ) ! ( N 2 + U 2 μ B ) ! . {\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.

## References

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1. ^ a b 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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