Legendre transforms in the Ising model

The Legendre transforms of the logarithm of the partition function for the Ising model are considered. In the language of the first transform ($\Phi$) the magnetization is found by means of a variational principle: $\Phi$ plays the role of the varied functional whose stationarity points correspond to the desired values of the magnetization. Equations of motion are derived for $\Phi$, and their iterative solution is described (diagrams). The diagram expansion of $\Phi$ is equivalent to the high-temperature expansion of the logarithm of the partition function (free energy) in thetemperature – magnetizationvariables (instead of the usual temperature – external field variables). The possibility of using a diagram expansion for the first Legendre transform for the approximate calculation of the critical indices is discussed. The main advantage of the method is that it is equally applicable both above and below $T_c$.