Dynamics of the Frenkel–Kontorova model with irrational mean spacing

In this paper we focus on the dynamics of the overdamped limit case of the Frenkel–Kontorova (F–K) model with irrational mean spacing. We extend the results of Baesens and MacKay on rational mean spacing case to irrational case: there exists a depinning force F d ⩾ 0 such that the F–K model possesses an equilibrium with a given mean spacing if the constant driving force F ∊ [0, F d ] or it admits a uniform sliding solution provided F > F d . Moreover, the average velocity of the particles exists and is unique, and it is non-decreasing and continuous with respect to F ∊ [0, + ∞ ). In particular, we prove that the average velocity depends continuously upon the mean spacing.