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Energy growth for systems of coupled oscillators with partial damping

Dmitry Dolgopyat, Bassam Fayad, Leonid Koralov, Shuo Yan

TL;DR

This work analyzes energy growth in a two-oscillator system with stochastic forcing and partial damping, showing that under diffusive scaling the energy of one component grows diffusively while the inter-node coupling becomes asymptotically negligible. The authors develop short-time expansions and rigorous bounds to separate large-energy averaging from small-energy deviations, ultimately proving that the rescaled velocity $|r_1(tT)|/\, ext{sqrt}(T)$ converges to a Brownian motion reflected at the origin. The approach combines averaging for large energies with large-deviation controls at small energies, and the main contribution is a blueprint for extending to more complex networks of coupled oscillators under forcing and dissipation. The results underscore a robust energy-growth mechanism where dissipation cannot prevent diffusion-driven growth due to the Hamiltonian coupling structure. The analysis provides a rigorous probabilistic framework for understanding energy transfer in simple nontrivial Hamiltonian systems with noise and friction, with potential extensions to multi-particle settings.

Abstract

We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles are asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies.

Energy growth for systems of coupled oscillators with partial damping

TL;DR

This work analyzes energy growth in a two-oscillator system with stochastic forcing and partial damping, showing that under diffusive scaling the energy of one component grows diffusively while the inter-node coupling becomes asymptotically negligible. The authors develop short-time expansions and rigorous bounds to separate large-energy averaging from small-energy deviations, ultimately proving that the rescaled velocity converges to a Brownian motion reflected at the origin. The approach combines averaging for large energies with large-deviation controls at small energies, and the main contribution is a blueprint for extending to more complex networks of coupled oscillators under forcing and dissipation. The results underscore a robust energy-growth mechanism where dissipation cannot prevent diffusion-driven growth due to the Hamiltonian coupling structure. The analysis provides a rigorous probabilistic framework for understanding energy transfer in simple nontrivial Hamiltonian systems with noise and friction, with potential extensions to multi-particle settings.

Abstract

We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles are asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies.
Paper Structure (12 sections, 16 theorems, 100 equations)

This paper contains 12 sections, 16 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Expansions
  3. Bounds on r2
  4. Behavior of r1 away from the origin
  5. Short time behavior
  6. Moderate time behavior
  7. Long time behavior
  8. Excursions to small values of r1
  9. Length of one excursion: Proof of Proposition \ref{['prop:time_near_zero']}
  10. Number of excursions: Proof of Proposition \ref{['prop:number_excursions']}
  11. Tightness
  12. Proof of the main result

Key Result

Theorem 1.1

For each initial distribution $\mu$ of the processes in eq:the_system, the process ${|r_1(t\cdot T)|}/{\sqrt{T}}$ converges weakly, as $T\to\infty$, to a Brownian motion starting and reflected at the origin.

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • Corollary 4.3
  • ...and 20 more