Collective transport in the discrete Frenkel-Kontorova model
A TSM-CDT researcher has recently introduced a novel extension of multiscale analysis to non-linear many-body systems, providing new results for an "industry standard" model in Physical Review E.
The Frenkel-Kontorova (FK) chain is a set of balls and springs sitting in a periodic "washboard" landscape (see picture). Despite this apparent simplicity, when the FK chain is heated and pushed it exhibits a wide range of behaviour, from stick-slip motion to soliton nucleation. This heady combination of phenomenological variety along with computational efficiency has resulted in the FK chain being used in dislocation theory, polymer dynamics, molecular combustion, Josephson junctions, spin chains, earthquakes and many other areas for over eighty years.
However, whilst various special cases have been investigated theoretically, there are have been no rigorous statements on the thermodynamic transport properties of the FK chain, such as the drift velocity under an applied force or the diffusivity, which are often the key properties of interest for the materials science community.
To address this problem, in Phys. Rev E 88, 012135 Tom Swinburne from the TSM-CDT extended the methods of multiscale analysis, first used by Hilbert to investigate hydrodynamic limits of microscopic collision equations, to produce strict mathematical bounds on these transport quantities for the FK chain.
Significantly, it is found that the Gibbs free energy barrier, often used to calculate finite temperature migration properties, is always lower than the true finite temperature migration barrier, as entropy maximisation does not always have time to occur in the dynamical process of diffusion. These ideas are now being extended to better understand the migration processes of many-body systems.