Collective transport in the discrete Frenkel-Kontorova model

TL;DR

This paper derives strict bounds on the diffusivity of a harmonic chain in a periodic potential using multiscale analysis, revealing the free energy barrier as a lower bound and accurately predicting non-linear response.

Contribution

It introduces a novel multiscale analysis of the adjoint Fokker-Planck equation to establish bounds on diffusivity in the Frenkel-Kontorova model, linking free energy barriers to migration barriers.

Findings

Bounds on center of mass diffusivity are derived.

Free energy barrier is a lower bound to the migration barrier.

Effective migration potentials accurately predict non-linear response.

Abstract

Through multiscale analysis of the adjoint Fokker-Planck equation, strict bounds are derived for the center of mass diffusivity of an overdamped harmonic chain in a periodic potential, often known as the discrete Frenkel-Kontorova model. Significantly, it is shown that the free energy barrier is a lower bound to the true finite temperature migration barrier for this general and popular system. Numerical simulation confirms the analysis, whilst effective migration potentials implied by the bounds are employed to give a surprisingly accurate prediction of the non-linear response.