We solved the Frenkel-Kontorova model with the potential V(u)=-λ(u-Int[u]-1/2)2/2 exactly. For given λ>0, there exists a positive integer qc such that the winding number ω of the minimum enthalpy state is locked to rational numbers in the qcth row of Farey fractions. For fixed ω=p/q, there is a critical λc 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.
Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics) 55(3), pp.2628-2631