Periodically forced pinned anharmonic atom chains

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 $β$-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.

💡 Research Summary

The paper investigates, through extensive numerical simulations, the hydrodynamic limit of a one‑dimensional chain of atoms that are (i) pinned by a harmonic on‑site potential, (ii) coupled by a β‑FPUT (quadratic + quartic) nearest‑neighbour interaction, (iii) driven at the rightmost site by a time‑periodic external force, and (iv) subject to stochastic momentum‑flip noise in the bulk. The leftmost particle is attached to a Langevin thermostat at temperature Tℓ. The forcing amplitude is scaled as n⁻¹/² with the system size n, a scaling that prevents the temperature of the forced atom from diverging as n→∞ and ensures a finite average energy current in the thermodynamic limit.

Two conjectures, originally formulated for the harmonic case and extended heuristically to the anharmonic setting in