Physical quantities

The net magnetic moment or ``magnetization' $M$ is given by

$$M=\sum_{i=1}^N s_i.$$ (115)

Usually we are interested in the average $\langle M \rangle$ and the fluctuations $\langle M^2 \rangle - \langle M \rangle ^2$ as a function of the temperature of the system and the applied magnetic field. We can determine the temperature as a function of the energy of the system in two ways. One way is to measure the probability that the demon has an energy $E_d$. An easier way to determine the temperature is to measure the mean demon energy. However, since the values of the demon energy are not continuous for the Ising model, the temperature is not proportional to the mean demon energy as it is for the ideal gas. It can be shown that in the limit of an infinite system, the temperature for $h=0$ is related to $E_d$ by

$$k_BT/J=\frac{4}{\ln{(1+4J/E_d)}}.$$ (116)

This results is obtained by replacing the integrals by sums over all the possible demon energies. Note that in the limit $\vert J/E_d\vert \ll 1$, this reduces to $k_BT=E_d$ as expected.