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Periodically forced pinned anharmonic atom chains

Shiva Darshan, Alessandra Iacobucci, Stefano Olla, Gabriel Stoltz

TL;DR

This work investigates energy transport in a periodically forced, pinned, anharmonic chain with $eta$-FPUT interactions, momentum-flip noise, and a forcing amplitude scaled by $1/ olinebreak ext{n}^{1/2}$. It tests a conjectured hydrodynamic limit: a nonlinear diffusion PDE for the macroscopic temperature profile with Dirichlet left boundary and nonlinear Neumann right boundary, coupled to a Green–Kubo–type formula for the limiting energy current. The authors develop a fixed-point framework to establish existence of the limiting temperature profile and validate the GK-based current estimate via extensive simulations across various forcing frequencies, temperatures, and flip rates, observing supratransmission and resonance effects absent in the harmonic case. The results support the diffusion-dominated transport picture in the anharmonic pinned chain and reveal rich frequency-dependent phenomena tied to anharmonicity, with implications for the universality and robustness of hydrodynamic limits in low-dimensional systems.

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 $β$-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.

Periodically forced pinned anharmonic atom chains

TL;DR

This work investigates energy transport in a periodically forced, pinned, anharmonic chain with -FPUT interactions, momentum-flip noise, and a forcing amplitude scaled by . It tests a conjectured hydrodynamic limit: a nonlinear diffusion PDE for the macroscopic temperature profile with Dirichlet left boundary and nonlinear Neumann right boundary, coupled to a Green–Kubo–type formula for the limiting energy current. The authors develop a fixed-point framework to establish existence of the limiting temperature profile and validate the GK-based current estimate via extensive simulations across various forcing frequencies, temperatures, and flip rates, observing supratransmission and resonance effects absent in the harmonic case. The results support the diffusion-dominated transport picture in the anharmonic pinned chain and reveal rich frequency-dependent phenomena tied to anharmonicity, with implications for the universality and robustness of hydrodynamic limits in low-dimensional systems.

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 -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.

Paper Structure

This paper contains 38 sections, 2 theorems, 102 equations, 28 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Model and Expected Behavior in the Hydrodynamic Limit
  3. Presentation of the model
  4. Expected behavior in the hydrodynamic limit
  5. Existence of a solution to \ref{['eq:limite_hydro']}
  6. (Non-)Uniqueness of the Solution to \ref{['eq:limite_hydro']}.
  7. Numerical Results
  8. Numerical setup
  9. Integration of dynamics.
  10. Computation of temperature profiles and bulk energy fluxes.
  11. Estimation of the rate of work.
  12. The harmonic band.
  13. Moderate/large flip rate $\widetilde{\gamma} = 1$
  14. Validity of the Green--Kubo like formula.
  15. Dependence of the work rate on the forcing frequency.
  16. ...and 23 more sections

Key Result

Proposition 1

Let $T_\ell \geq 0$. Suppose that the conductivity $D$ is continuous and uniformly bounded below by a positive constant (i.e there exists $\varepsilon > 0$ such that $D(T) \geq \varepsilon$ for all $T \geq 0$) and that for a fixed forcing $\mathdutchcal{F}$ and period $\theta$, $\mathbb{W}$ is a fun Then, the equation eq:limite_hydro admits a mild solution.

Figures (28)

  • Figure 1: Empirical negative flux observed in simulations (circles) and rate of work estimated using \ref{['eq:finite_size_W']} (squares), both plotted against frequency within the harmonic band, using the same forcing parameters and evaluated at the temperature of the forced atom site.
  • Figure 2: Work rate estimated using \ref{['eq:finite_size_W']} plotted against the forcing frequency $\nu$, with $f_0 = 1$. The harmonic band $\nu \in \left[\widebar{\nu}, \frac{\sqrt{5}}{2\pi}\right]$ is indicated by dashed black lines.
  • Figure 3: Temperature profile obtained by solving the PDE \ref{['eq:limite_hydro']}, compared to the empirical profile computed with the same forcing parameters and flip rate $\widetilde{\gamma} = 1.0$. Error bars around the empirical profile are shown as a blue shaded region. The harmonic temperature profile for the same forcing parameters is plotted in black.
  • Figure 4: Empirical negative flux observed in simulations with flip rate $\widetilde{\gamma} = 0.1$ (circles), and work rate estimated using \ref{['eq:finite_size_W']} (squares), both plotted against frequency within the harmonic band, using the same forcing parameters and evaluated at the temperature of the forced atom site.
  • Figure 5: The work rate estimated using \ref{['eq:finite_size_W']} as a function of the forcing frequency $\nu$ with $f_0 = 1$. The harmonic band of frequencies is marked with dashed black lines.
  • ...and 23 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof