Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergent Power Series for Anharmonic Chain with Periodic Forcing

Pedro L. Garrido, Tomasz Komorowski, Joel L. Lebowitz, Stefano Olla

TL;DR

The article analyzes energy transport in a finite one-dimensional anharmonic chain driven by a localized, periodic force and damped at the boundaries. By combining a Schauder fixed-point argument with a carefully controlled convergent perturbation series in the anharmonicity strength $ u$, the authors prove the existence and global stability of a unique $ heta$-periodic stationary state when the forcing frequency lies outside the harmonic spectrum and $ u$ is small. The periodic state exhibits exponential spatial localization that is uniform in system size, and the work done by the forcing is shown to be nonnegative and strictly positive under dissipation. The framework extends to higher dimensions and includes a detailed treatment of the harmonic limit, non-uniqueness scenarios for symmetric cases, and an approximation scheme that solidifies the global stability results.

Abstract

We study the propagation of energy in one-dimensional anharmonic chains subject to a periodic, localized forcing. For the purely harmonic case, forcing frequencies outside the linear spectrum produce exponentially localized responses, preventing equi-distribution of energy per degree of freedom. We extend this result to anharmonic perturbations with bounded second derivatives and boundary dissipation, proving that for small perturbations and non-resonant forcing, the dynamics converges to a periodic stationary state with energy exponentially localized uniformly in the system size. The perturbed periodic state is described by a convergent power type expansion in the strength of the anharmonicity. This excludes chaoticity induced by anharmonicity, independently of the size of the system. Our perturbative scheme can also be applied in higher dimensions.

Convergent Power Series for Anharmonic Chain with Periodic Forcing

TL;DR

The article analyzes energy transport in a finite one-dimensional anharmonic chain driven by a localized, periodic force and damped at the boundaries. By combining a Schauder fixed-point argument with a carefully controlled convergent perturbation series in the anharmonicity strength , the authors prove the existence and global stability of a unique -periodic stationary state when the forcing frequency lies outside the harmonic spectrum and is small. The periodic state exhibits exponential spatial localization that is uniform in system size, and the work done by the forcing is shown to be nonnegative and strictly positive under dissipation. The framework extends to higher dimensions and includes a detailed treatment of the harmonic limit, non-uniqueness scenarios for symmetric cases, and an approximation scheme that solidifies the global stability results.

Abstract

We study the propagation of energy in one-dimensional anharmonic chains subject to a periodic, localized forcing. For the purely harmonic case, forcing frequencies outside the linear spectrum produce exponentially localized responses, preventing equi-distribution of energy per degree of freedom. We extend this result to anharmonic perturbations with bounded second derivatives and boundary dissipation, proving that for small perturbations and non-resonant forcing, the dynamics converges to a periodic stationary state with energy exponentially localized uniformly in the system size. The perturbed periodic state is described by a convergent power type expansion in the strength of the anharmonicity. This excludes chaoticity induced by anharmonicity, independently of the size of the system. Our perturbative scheme can also be applied in higher dimensions.

Paper Structure

This paper contains 37 sections, 25 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Microscopic dynamics
  3. Main Results
  4. Definition of norms
  5. A periodic solution for an anharmonic chain
  6. Construction of a periodic solution for an anharmonic chain by perturbative series
  7. Global stability of the unique periodic solution in the anharmonic case
  8. Work done by periodic forcing
  9. Spatial decay of the periodic solution as $N\to\infty$
  10. The case of odd harmonic modes: loss of uniqueness
  11. The case $N=0$: a single anharmonic oscillator with damping
  12. Proof of the convergence of the perturbative series.
  13. Time harmonics of a periodic solution
  14. The case of a harmonic chain
  15. Proof of Theorem \ref{['main:thm']}
  16. ...and 22 more sections

Key Result

Theorem 3.1

Assume $\gamma>0$. For each $\nu\in{\mathbb R}$ there exists a $\theta$-periodic solution ${\bf q}_{\rm p}(t;\nu)=(q_{x,{\rm p}} (t;\nu))_{x\in{\mathbb Z}_N}$ of 012805-24x.

Theorems & Definitions (28)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Remark 3.8
  • Remark 4.1
  • Proposition 5.1
  • ...and 18 more