Claude BAESENS 

Dept of Applied Maths & Theoretical Physics, 
University of Cambridge, Silver Street, Cambridge CB3 9EW
UK 
c.baesens@damtp.cam.ac.uk

Gradient dynamics of tilted Frenkel-Kontorova models

Abstract : The archetype of a Frenkel-Kontorova model is a chain of balls and springs in a periodic potential. Equilibrium states of such a system are in one-to- one correspondence with the orbits of an area-preserving twist map on the cylinder, via a Legendre transform.

Here we consider the gradient dynamics of Frenkel-Kontorova chains in a tilted periodic potential. If the tilt is sufficiently strong then there are no equilibria but there is a globally attracting periodic orbit, in the sense of a configuration which repeats itself after a certain time shifted in space by one period. This result is proved in the following spaces of configurations: finite chains, spatially periodic configurations, androtationally ordered configurations of irrational mean spacing.

A fundamental tool in our analysis is monotonicity of the gradient flow with respect to a partial order on configurations, directly analogous to the use of the maximum principle for parabolic PDEs. A second tool is Birkhoff recurrence theorem.