Homogenization of fully overdamped Frenkel-Kontorova models. (English) Zbl 1171.49023
The authors consider the simplest Frenkel-Kontorova model, which is a special and important case of a system of ODEs describing the motion of particles in interaction with their neighbours. They call the considered model “overdamped” since they neglet the acceleration term so the system considered is the following
\[
\frac{d U_i}{dt} = U_{i+1} - 2 U_i + U_{i-1} + \sin (2 \pi U_i) + L \, , \hskip20pt t > 0 \, .
\]
They study the behaviour as the number of particles goes to infinity. One of the main result states that the limiting dynamics is described by a first order Hamilton-Jacobi equation. Some qualitative properties of the limit Hamiltonian are given.
Reviewer: Fabio Paronetto (Padova)
MSC:
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35F20 | Nonlinear first-order PDEs |
| 45K05 | Integro-partial differential equations |
| 47G20 | Integro-differential operators |
| 35B10 | Periodic solutions to PDEs |
| 39A12 | Discrete version of topics in analysis |
Keywords:
particle systems; periodic homogenization; Frenkel-Kontorova models; Hamilton-Jacobi equations; Hull function; cumulative distribution function; Slepčev formulationReferences:
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