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Nonlinear fluctuations for a chain of weakly anharmonic oscillators with stochastic perturbation

Kohei Hayashi, Stefano Olla

TL;DR

The paper analyzes fluctuations of two phonon modes in a one-dimensional weakly anharmonic oscillator chain with stochastic momentum-exchange noise. Using a Dynkin martingale decomposition, equipartition of energy, and a second-order Boltzmann-Gibbs principle, the authors show that under diffusive scaling and $ abla_n extstyle{ ext{ε}}_n=n^{-1/2}$ the recentered phonon fields converge to stationary energy solutions of two uncoupled stochastic Burgers equations with a cubic-term–driven nonlinearity and a drift depending on higher-order derivatives of the potential. The energy mode decouples and retains diffusive behavior, while the nonlinear phonon fluctuations emerge only when the cubic term is present, with the nonlinearity coefficient proportional to $ rac{c_3}{8c_2^2}$ and a diffusion drift $D_V= rac{2c_2c_4-c_3^2}{24c_2^3}$. The result provides a rigorous link between weak anharmonic perturbations in lattice dynamics and the KPZ/NLFluctuating-Hydrodynamics universality class, robust under a highly degenerate microscopic noise and supported by a second-order replacement principle and Riemann-Lebesgue-type decoupling estimates.

Abstract

We study the fluctuations of the phonon modes in a one-dimensional chain of anharmonic oscillators where the deterministic Hamiltonian dynamics is perturbed by random exchanges of momentum between nearest neighbor particles. There are three locally conserved quantities: volume, momentum and energy. We study the evolution in equilibrium of the fluctuation fields of the two phonon modes (linear combination of the volume stretch and momentum), on a diffusive space-time scale after recentering on their sound velocities. We show that, weakening the anharmonicity with the scale parameter, the recentered phonon fluctuations fields converge to the stationary solutions of two uncoupled stochastic Burgers equations. The nonlinearity in the Burgers equation depends on the presence of a cubic term in the anharmonic potential (corresponding to the $α$-FPUT dynamics). Main ingredients of the proof, based on a compactness argument for the Dynkin's martingale decomposition, are the second-order Boltzmann-Gibbs principle, as well as equipartition of energy, to characterize the nonlinear term and Riemann-Lebesgue estimates showing that fields with diverging velocity to different directions have no interaction in the limit.

Nonlinear fluctuations for a chain of weakly anharmonic oscillators with stochastic perturbation

TL;DR

The paper analyzes fluctuations of two phonon modes in a one-dimensional weakly anharmonic oscillator chain with stochastic momentum-exchange noise. Using a Dynkin martingale decomposition, equipartition of energy, and a second-order Boltzmann-Gibbs principle, the authors show that under diffusive scaling and the recentered phonon fields converge to stationary energy solutions of two uncoupled stochastic Burgers equations with a cubic-term–driven nonlinearity and a drift depending on higher-order derivatives of the potential. The energy mode decouples and retains diffusive behavior, while the nonlinear phonon fluctuations emerge only when the cubic term is present, with the nonlinearity coefficient proportional to and a diffusion drift . The result provides a rigorous link between weak anharmonic perturbations in lattice dynamics and the KPZ/NLFluctuating-Hydrodynamics universality class, robust under a highly degenerate microscopic noise and supported by a second-order replacement principle and Riemann-Lebesgue-type decoupling estimates.

Abstract

We study the fluctuations of the phonon modes in a one-dimensional chain of anharmonic oscillators where the deterministic Hamiltonian dynamics is perturbed by random exchanges of momentum between nearest neighbor particles. There are three locally conserved quantities: volume, momentum and energy. We study the evolution in equilibrium of the fluctuation fields of the two phonon modes (linear combination of the volume stretch and momentum), on a diffusive space-time scale after recentering on their sound velocities. We show that, weakening the anharmonicity with the scale parameter, the recentered phonon fluctuations fields converge to the stationary solutions of two uncoupled stochastic Burgers equations. The nonlinearity in the Burgers equation depends on the presence of a cubic term in the anharmonic potential (corresponding to the -FPUT dynamics). Main ingredients of the proof, based on a compactness argument for the Dynkin's martingale decomposition, are the second-order Boltzmann-Gibbs principle, as well as equipartition of energy, to characterize the nonlinear term and Riemann-Lebesgue estimates showing that fields with diverging velocity to different directions have no interaction in the limit.
Paper Structure (27 sections, 25 theorems, 217 equations, 1 figure)

This paper contains 27 sections, 25 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and Result
  3. Microscopic Dynamics
  4. Equilibrium Measures
  5. Main Result
  6. Martingale decomposition
  7. Identification of the nonlinear part of the drift
  8. Identification of linear terms
  9. Identification of viscosity terms
  10. Equipartition of Energy
  11. The second-order Boltzmann-Gibbs principle
  12. Riemann-Lebesgue Estimates
  13. Handling fields with wrong velocity
  14. Characterization of Martingale Terms
  15. Tightness
  16. ...and 12 more sections

Key Result

Lemma 2.2

Fix $\beta>0$ and $p,\tau\in\mathbb{R}$. For any $\eta>0$, there exists $C_\eta>0$ and $n_c=n_c(\eta)>0$ such that

Figures (1)

  • Figure 1: Expected fluctuations for the two phonon modes, for each choice of scaling $\alpha_n=n^{\mathfrak a-2} \alpha$ and $\varepsilon_n= n^{-\mathfrak b}$ . We only prove here the case $\mathfrak a = 2$ (diffusive scaling) and $\mathfrak b= 1/2$, but the proof can be extended along the red continuous line up to $\mathfrak b>1/4$ and $\mathfrak a > 7/4$.

Theorems & Definitions (40)

  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3: Equipartition of energy
  • Lemma 4.1
  • proof
  • ...and 30 more