Discussion
Consider a system of two Ising spins \(S_{1}\) and \(S_{2}\) taking values ±1 with interaction energy \(E = -JS_{1}S_{2}\). For large temperature T, the average energy of the system varies as \(C/(k_{B}T)\) with C given by :
Answer: A
The constant is \(C = -J^{2}\).
The four configurations have energy -J (both spins equal, two states) or +J (opposite, two states). The partition function and mean energy are
\[ \begin{aligned} Z &= 2e^{\beta J} + 2e^{-\beta J} = 4\cosh\beta J \\ \langle E\rangle &= -\frac{\partial\ln Z}{\partial\beta} = -J\tanh\beta J \end{aligned} \]
At high temperature \(\beta J\) is small and \(\tanh\beta J \approx \beta J\), so
\[ \langle E\rangle \approx -\frac{J^{2}}{k_{B}T} \]
The negative sign reflects the slight preference for the lower-energy aligned states. \(+J^{2}\) has the wrong sign, and \(-2J^{2}\) or \(4J^{2}\) would follow from miscounting the states.
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