Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Abstract

In this paper we are mainly concerned with existence and modulation of uniform sliding states for particle chains with damping γ and external driving force F. If the on-site potential vanishes, then for each F > 0 there exist trivial uniform sliding states xn(t) = nω + νt + α for which the particles are uniformly spaced with spacing ω > 0, the sliding velocity of each particle is ν = F/γ, and the phase α is arbitrary. If the particle chain with convex interaction potential is placed in a periodic on-site potential, we show under some conditions the existence of modulated uniform sliding states of the form $$x_n(t)=n\omega+\nu t+\alpha+u(n\omega+\nu t+\alpha),$$ where the modulation function u is periodic and unique up to phase. The conditions are that the system is overdamped and the driving force F exceeds some critical value Fd(ω) ≥ 0 depending on mean spacing ω. If \({F\in [0,F_d(\omega)]}\) , the system possesses a set of rotationally ordered equilibrium states for irrational ω, which can be described by a non-decreasing hull function, just as the case γ = F = 0, where Aubry-Mather theory applies to ground states. Meanwhile, we prove that Fd(ω) = 0, which was argued physically much earlier, if the hull function of ground states with irrational rotation number ω for F = 0 is continuous.