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We consider the Farey fraction spin chain in an external field $h$. Utilisingideas from dynamical systems, the free energy of the model is derived by meansof an effective cluster energy approximation. This approximation is valid fordivergent cluster sizes, and hence appropriate for the discussion of themagnetizing transition. We calculate the phase boundaries and the scaling ofthe free energy. At $h=0$ we reproduce the rigorously known asymptotictemperature dependence of the free energy. For $h \ne 0$, our results arelargely consistent with those found previously using mean field theory andrenormalization group arguments.Comment: 17 pages, 3 figure

Cluster Approximation for the Farey Fraction Spin Chain