Multiplicity (statistical mechanics)

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 {{mvar|N}} microscopic dipoles {{mvar|μ}} which may either be aligned or anti-aligned with an externally applied magnetic field {{mvar|B}}. Let $N\_\backslash uparrow$ represent the number of dipoles that are aligned with the external field and $N\_\backslash downarrow$ represent the number of anti-aligned dipoles. The energy of a single aligned dipole is $U\_\backslash uparrow\; =\; -\backslash mu\; B,$ while the energy of an anti-aligned dipole is $U\_\backslash downarrow\; =\; \backslash mu\; B;$ thus the overall energy of the system is

$$U\; =\; (N\_\backslash downarrow-N\_\backslash uparrow)\backslash mu\; B.$$

The goal is to determine the multiplicity as a function of {{mvar|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 $N\_\backslash uparrow$ and $N\_\backslash downarrow.$ This approach shows that the number of available macrostates is {{math|N + 1}}. For example, in a very small system with {{math|1=N = 2}} dipoles, there are three macrostates, corresponding to $N\_\backslash uparrow=0,\; 1,\; 2.$ Since the $N\_\backslash uparrow\; =\; 0$ and $N\_\backslash uparrow\; =\; 2$ macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the $N\_\backslash uparrow\; =\; 1,$ 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 $N\_\backslash uparrow$ aligned dipoles follows from combinatorics, resulting in

$$\backslash Omega\; =\; \backslash frac\{N!\}\{N\_\backslash uparrow!(N-N\_\backslash uparrow)!\}\; =\; \backslash frac\{N!\}\{N\_\backslash uparrow!N\_\backslash downarrow!\},$$

where the second step follows from the fact that $N\_\backslash uparrow+N\_\backslash downarrow\; =\; N.$

Since $N\_\backslash uparrow\; -\; N\_\backslash downarrow\; =\; -\backslash tfrac\{U\}\{\backslash mu\; B\},$ the energy {{mvar|U}} can be related to $N\_\backslash uparrow$ and $N\_\backslash downarrow$ as follows:

$$\backslash begin\{align\}$$

N_\uparrow &= \frac{N}{2} - \frac{U}{2\mu B}\\[4pt]

N_\downarrow &= \frac{N}{2} + \frac{U}{2\mu B}.

\end{align}

Thus the final expression for multiplicity as a function of internal energy is

$$\backslash Omega\; =\; \backslash frac\{N!\}\{\; \backslash left(\backslash frac\{N\}\{2\}\; -\; \backslash frac\{U\}\{2\backslash mu\; B\}\; \backslash right)!\; \backslash left(\; \backslash frac\{N\}\{2\}\; +\; \backslash frac\{U\}\{2\backslash mu\; B\}\; \backslash right)!\}.$$

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.

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Category:Statistical mechanics

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