Spin Magnet System

#Physics

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

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

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

• For $\displaystyle \frac{s}{N}\ll 1 \land N\gg 1$

$\displaystyle {\left\langle{s^{2}}\right\rangle}=\frac{1}{4}N$

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

$\displaystyle \frac{{\left\langle{s^{2}}\right\rangle}^{1/2}}{N}=\frac{1}{2}N^{1/2}$

• Fractional fluctuation in $\displaystyle s^{2}$

$\displaystyle U(s)=-2smB$

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