Periodically forced pinned anharmonic atom chains

TL;DR

This paper numerically investigates the hydrodynamic limit of periodically forced anharmonic atom chains with pinning, providing evidence supporting conjectured PDEs for temperature profiles and energy flux, and analyzing the effect of momentum flip rates.

Contribution

It offers the first numerical validation of the hydrodynamic limit conjecture for anharmonic chains with forcing and pinning, extending prior harmonic case results.

Findings

Numerical support for the conjectured PDEs governing temperature and energy flux.

Green--Kubo formula applicability in the anharmonic setting.

Analysis of energy current dependence on forcing period and momentum flip rates.

Abstract

Recent works proved a hydrodynamic limit for periodically forced atom chains with harmonic interaction and pinning, together with momentum flip. When energy is the only conserved quantity, one would expect similar results in the anharmonic case, as conjectured for the temperature profile and energy flux in arXiv:2212.00093. However, outside the harmonic case, explicit computations are generally no longer possible, thus making a rigorous proof of this hydrodynamic limit difficult. Consequently, we numerically investigate the plausibility of this limit for the particular case of a chain with $\beta$-FPUT interactions and harmonic pinning. We present our simulation results suggesting that the conjectured PDE for the limiting temperature profile and Green--Kubo type formula for the limiting energy current conjectured in arXiv:2212.00093 are correct. We then use this Green--Kubo type formula to investigate the relationship between the energy current and period of the forcing. This relationship is investigated in the case of significant rate of momentum flip, small rate of momentum flip and no momentum flip. We compare the relationship observed in the anharmonic case to that of the harmonic case for which explicit formulae are available.