  We consider the Farey fraction spin chain in an external field $h$. Using
ideas from dynamical systems and functional analysis, we show that the free
energy $f$ in the vicinity of the second-order phase transition is given,
exactly, by $$ f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for}
\quad h^2\ll t \ll 1 . $$
  Here $t=\lambda_{G}\log(2)(1-\frac{\beta}{\beta_c})$ is a reduced
temperature, so that the deviation from the critical point is scaled by the
Lyapunov exponent of the Gauss map, $\lambda_G$. It follows that $\lambda_G$
determines the amplitude of both the specific heat and susceptibility
singularities. To our knowledge, there is only one other microscopically
defined interacting model for which the free energy near a phase transition is
known as a function of two variables.
  Our results confirm what was found previously with a cluster approximation,
and show that a clustering mechanism is in fact responsible for the transition.
However, the results disagree in part with a renormalisation group treatment.
