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