Farey fractions and the Frenkel-Kontorova model

We solved the Frenkel-Kontorova model with the potential V(u)=-λ(u-Int[u]-1/2[formula presented]/2 exactly. For given λ>0, there exists a positive integer [formula presented] such that the winding number ω of the minimum enthalpy state is locked to rational numbers in the [formula presented]th row of Farey fractions. For fixed ω=p/q, there is a critical [formula presented] when a first order phase transition occurs. This phase transition can be understood as the dissociation of a large molecule into two smaller ones in a manner dictated by the Farey fractions.