An unorthodox analysis of the Frenkel-Kontorova model

The Frenkel-Kontorova model, defined by the Hamiltonian H = Σn=1N (pn2/2 − K cos xn + (xn − xn−1 − λ)2/2), is a well-known simple 1D model of a many-particle system with two competing types of interaction (interacting particles in a spatially periodic substrate), with a peculiar ground-state behavior which is characterized by “devilʹs staircase” functions. The statistical mechanics of the FK model reduces to a relatively simple eigenvalue problem, for which we construct a somewhat unorthodox approximate analytic solution, based on an asymptotic formula for the modified Bessel functions. The predictions of our theory are compared with perturbation theoretical predictions and numerically calculated results.