Spin Magnet System

$\Omega (N,s)=\frac{N!}{\left(\frac{1}{2}N+s\right)!\left(\frac{1}{2}N-s\right)!}=\frac{N!}{N_{\uparrow }!N_{\downarrow }!}$

• $\Omega$ is the multiplicity function
• $N$ is the number of magnets
• $s$ is $\frac{N_{\uparrow }-N_{\downarrow }}{2}$, or half the spin-excess

$\Omega (N,s)\approx {\left(\frac{2}{\pi N}\right)}^{1/2}{2}^N{e}^{-2s^{2/N}}$

• For $\frac{s}{N}\ll 1\wedge N\gg 1$

$⟨s^2⟩=\frac{1}{4}N$

• Assume each states is equally likely and that there’s a Gaussian approximation

$\frac{{⟨s^2⟩}^{1/2}}{N}=\frac{1}{2}{N}^{1/2}$

• Fractional fluctuation in $s^2$

$U(s)=-2smB$

• Potential energy for a model spin system with spin excess $2s$, magnetic moment $m$, and magnetic field $B$