Physics: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. Commonly denoted [math]\displaystyle{ \Omega }[/math], it is related to the configuration entropy of an isolated system via Boltzmann's entropy formula [math]\displaystyle{ S = k_\text{B} \log \Omega, }[/math] where [math]\displaystyle{ S }[/math] is the entropy and [math]\displaystyle{ k_\text{B} = 1.38\cdot 10^{-23} \, \mathrm{J/K} }[/math] 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 μ which may either be aligned or anti-aligned with an externally applied magnetic field B. Let [math]\displaystyle{ N_\uparrow }[/math] represent the number of dipoles that are aligned with the external field and [math]\displaystyle{ N_\downarrow }[/math] represent the number of anti-aligned dipoles. The energy of a single aligned dipole is [math]\displaystyle{ U_\uparrow = -\mu B, }[/math] while the energy of an anti-aligned dipole is [math]\displaystyle{ U_\downarrow = \mu B; }[/math] thus the overall energy of the system is [math]\displaystyle{ U = (N_\downarrow-N_\uparrow)\mu B. }[/math]

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 [math]\displaystyle{ N_\uparrow }[/math] and [math]\displaystyle{ N_\downarrow. }[/math] 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 [math]\displaystyle{ N_\uparrow=0, 1, 2. }[/math] Since the [math]\displaystyle{ N_\uparrow = 0 }[/math] and [math]\displaystyle{ N_\uparrow = 2 }[/math] macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the [math]\displaystyle{ N_\uparrow = 1, }[/math] 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 [math]\displaystyle{ N_\uparrow }[/math] aligned dipoles follows from combinatorics, resulting in [math]\displaystyle{ \Omega = \frac{N!}{N_\uparrow!(N-N_\uparrow)!} = \frac{N!}{N_\uparrow!N_\downarrow!}, }[/math] where the second step follows from the fact that [math]\displaystyle{ N_\uparrow+N_\downarrow = N. }[/math]

Since [math]\displaystyle{ N_\uparrow - N_\downarrow = -\tfrac{U}{\mu B}, }[/math] the energy U can be related to [math]\displaystyle{ N_\uparrow }[/math] and [math]\displaystyle{ N_\downarrow }[/math] as follows: [math]\displaystyle{ \begin{align} N_\uparrow &= \frac{N}{2} - \frac{U}{2\mu B}\\[4pt] N_\downarrow &= \frac{N}{2} + \frac{U}{2\mu B}. \end{align} }[/math]

Thus the final expression for multiplicity as a function of internal energy is [math]\displaystyle{ \Omega = \frac{N!}{ \left(\frac{N}{2} - \frac{U}{2\mu B} \right)! \left( \frac{N}{2} + \frac{U}{2\mu B} \right)!}. }[/math]

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.