A variational free energy

The variational free energy of the two-spin system whose energy is $E(\bx) = - x_1 x_2$, as a function of the two variational parameters $q_1$ and $q_2$. The inverse-temperature is $\beta=1.4$. % critical point for this system is 1 The function plotted is
  \b \tF = -
   \b  \bar{x}_1 \bar{x}_2
                - H_2^{(e)}(q_1) - H_2^{(e)}(q_2),
where $\bar{x}_n = 2 q_n -1$. Notice that for fixed $q_2$ the function is convex with respect to $q_1$, and for fixed $q_1$ it is convex with respect to $q_2$.
David MacKay
Last modified: Wed Nov 22 20:54:11 2000