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Non-equilibrium steady state for a three-mode energy cascade model

Zaher Hani, Yao Li, Andrea Nahmod, Gigliola Staffilani

TL;DR

The paper develops a rigorous construction of a nonequilibrium steady state for a three-mode resonant NLS model with forcing and dissipation by introducing a novel Feynman–Kac–Lyapunov framework. It reduces the infinitely complex wave turbulence problem to a tractable SDE system, proves existence and uniqueness of the invariant measure, and establishes a polynomial convergence rate to the NESS via a dual Lyapunov structure augmented by Feynman–Kac pre-factors. The work elucidates how energy can cascade through a minimal mode network without overheating or freezing the interior mode, offering a mathematically solid perspective on cascade statistics that complements wave-kinetic and Kolmogorov–Zakharov theories. It also lays the groundwork for future extensions to longer chains and more natural heat-bath couplings, with potential numerical and theoretical connections to cascade spectra in nonlinear waves.

Abstract

Motivated by the central phenomenon of energy cascades in wave turbulence theory, we construct non-equilibrium statistical steady states (NESS), or invariant measures, for a simplified model derived from the nonlinear Schrödinger (NLS) equation with external forcing and dissipation. This new perspective to studying energy cascades, distinct from traditional analyses based on kinetic equations and their cascade spectra, focuses on the underlying statistical steady state that is expected to hold when the cascade spectra of wave turbulence manifest. In the full generality of the (infinite dimensional) nonlinear Schrödinger equation, constructing such invariant measures is more involved than the rigorous justification of the Kolmogorov-Zakharov (KZ) spectra, which itself remains an outstanding open question despite the recent progress on mathematical wave turbulence. Since such complexity remains far beyond the current knowledge (even for much simpler chain models), we confine our analysis to a three-mode reduced system that captures the resonant dynamics of the NLS equation, offering a tractable framework for constructing the NESS. For this, we introduce a novel approach based on solving an elliptic Feynman-Kac equation to construct the needed Lyapunov function.

Non-equilibrium steady state for a three-mode energy cascade model

TL;DR

The paper develops a rigorous construction of a nonequilibrium steady state for a three-mode resonant NLS model with forcing and dissipation by introducing a novel Feynman–Kac–Lyapunov framework. It reduces the infinitely complex wave turbulence problem to a tractable SDE system, proves existence and uniqueness of the invariant measure, and establishes a polynomial convergence rate to the NESS via a dual Lyapunov structure augmented by Feynman–Kac pre-factors. The work elucidates how energy can cascade through a minimal mode network without overheating or freezing the interior mode, offering a mathematically solid perspective on cascade statistics that complements wave-kinetic and Kolmogorov–Zakharov theories. It also lays the groundwork for future extensions to longer chains and more natural heat-bath couplings, with potential numerical and theoretical connections to cascade spectra in nonlinear waves.

Abstract

Motivated by the central phenomenon of energy cascades in wave turbulence theory, we construct non-equilibrium statistical steady states (NESS), or invariant measures, for a simplified model derived from the nonlinear Schrödinger (NLS) equation with external forcing and dissipation. This new perspective to studying energy cascades, distinct from traditional analyses based on kinetic equations and their cascade spectra, focuses on the underlying statistical steady state that is expected to hold when the cascade spectra of wave turbulence manifest. In the full generality of the (infinite dimensional) nonlinear Schrödinger equation, constructing such invariant measures is more involved than the rigorous justification of the Kolmogorov-Zakharov (KZ) spectra, which itself remains an outstanding open question despite the recent progress on mathematical wave turbulence. Since such complexity remains far beyond the current knowledge (even for much simpler chain models), we confine our analysis to a three-mode reduced system that captures the resonant dynamics of the NLS equation, offering a tractable framework for constructing the NESS. For this, we introduce a novel approach based on solving an elliptic Feynman-Kac equation to construct the needed Lyapunov function.

Paper Structure

This paper contains 25 sections, 32 theorems, 301 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Derivation of the reduced system
  3. Statement of the result
  4. Remarks on the result
  5. Future Perspectives
  6. Organization of the paper
  7. Acknowledgements
  8. Probability preliminary
  9. Proof strategy
  10. Main idea: when Feynman-Kac meets Lyapunov
  11. Mechanism of energy transfer
  12. Dissipation of high energy
  13. Analysis of the infinitesimal generator
  14. Reduced system and proof of assumption (H)
  15. Recovery of low energy
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $X_{t}$ be a stochastic differential equation that satisfies assumptions (A2G) and (A1) with set $C$ given by for some $\gamma \in (e^{-a T/2}, 1)$. Let $\hat{X}_{n} = X_{nT}$ be an embedded chain of $X_{t}$. Then there exists a unique invariant probability measure $\pi$ and constants $r(\gamma) \in (0, 1)$, $\kappa(\gamma) \in (0, \infty)$, such that for all measurable function $f$ with $|

Figures (2)

  • Figure 1: Left: Illustration of Case I and II in the proof of Lemma \ref{['highI2']} and some contour plots of $V( \mathbf{x})$. Middle: Illustration of different subcases of Case I in the proof of Lemma \ref{['highI2']}. Right: Illustration of different cases in the proof of Lemma \ref{['lowI2']}.
  • Figure 2: Illustration of the dynamics of equation \ref{['theta3']} and sets $A_1$, $A_2$ used in the proof.

Theorems & Definitions (56)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Lemma 4.1
  • proof
  • ...and 46 more