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Phase space contraction of degenerately damped random splittings

David P. Herzog, Jonathan C. Mattingly

TL;DR

The paper develops a general probabilistic Lyapunov framework for partially dissipative systems and applies it to randomly split models of Lorenz '96 and Galerkin truncations of the 2D Euler equations. By combining subconservation of an energy-like observable with partial dissipation in a controlled dissipative region and quantified entrance probabilities, the authors prove the existence of stationary measures for the randomly split dynamics and derive rates of return to compact sets (exponential for Lorenz '96 and stretched-exponential for Euler). The main technical contributions include a set of energy-transfer lemmas, a mechanism to bound entrance probabilities to damping regions that is uniform in energy, and a detailed analysis of triadic interactions via random splittings. These results show that minimal damping, when geometrically arranged and injected through random splittings, suffices to stabilize high-dimensional systems and generate statistically steady states, providing insights into energy cascade and dissipation in simplified fluid models. The framework lays groundwork for future work combining random switching with forcing and extending to broader classes of partially damped, high-dimensional systems.

Abstract

When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system to converge to a statistical steady state, the dynamics must transfer the energy from the excited modes to the dissipative directions. The precise mechanisms underlying this transfer are of particular interest and are the topic of this paper. We explore a class of randomly switched models introduced in [2,3] and provide some of the first results showing that minimal damping is sufficient to stabilize the system in a fluids model.

Phase space contraction of degenerately damped random splittings

TL;DR

The paper develops a general probabilistic Lyapunov framework for partially dissipative systems and applies it to randomly split models of Lorenz '96 and Galerkin truncations of the 2D Euler equations. By combining subconservation of an energy-like observable with partial dissipation in a controlled dissipative region and quantified entrance probabilities, the authors prove the existence of stationary measures for the randomly split dynamics and derive rates of return to compact sets (exponential for Lorenz '96 and stretched-exponential for Euler). The main technical contributions include a set of energy-transfer lemmas, a mechanism to bound entrance probabilities to damping regions that is uniform in energy, and a detailed analysis of triadic interactions via random splittings. These results show that minimal damping, when geometrically arranged and injected through random splittings, suffices to stabilize high-dimensional systems and generate statistically steady states, providing insights into energy cascade and dissipation in simplified fluid models. The framework lays groundwork for future work combining random switching with forcing and extending to broader classes of partially damped, high-dimensional systems.

Abstract

When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system to converge to a statistical steady state, the dynamics must transfer the energy from the excited modes to the dissipative directions. The precise mechanisms underlying this transfer are of particular interest and are the topic of this paper. We explore a class of randomly switched models introduced in [2,3] and provide some of the first results showing that minimal damping is sufficient to stabilize the system in a fluids model.
Paper Structure (20 sections, 18 theorems, 254 equations, 2 figures)

This paper contains 20 sections, 18 theorems, 254 equations, 2 figures.

Table of Contents

  1. Introduction and Motivation
  2. General Results for Partially Dissipative Systems
  3. Return Times and Convergence
  4. Random Splittings
  5. Notation
  6. Random Splittings
  7. Random splitting of Lorenz '96
  8. Section overview
  9. Subconservation
  10. Partial dissipation
  11. Entrances to a dissipative region
  12. Entrance probabilities to the dissipative region: the Lorenz '96 Splitting
  13. The key lemmata
  14. Proof of Theorem \ref{['thm:lblor']}
  15. Random Splittings of Galerkin Truncations of 2D Euler
  16. ...and 5 more sections

Key Result

Proposition 2.3

Let $H\colon \mathbf{X} \rightarrow [1,\infty)$ and $p\colon [1,\infty) \rightarrow (0,1]$ be measurable, and let $\alpha \in (0,1)$ and $f_1,f_2\geq 0$ be constants. Suppose for any $x \in \mathbf{X}$ there exists an event $A=A(x) \in \mathcal{F}$ such that $\mathbf{P}_x(A) \geq p(H(x))$ and such for every $x\in \mathbf{X}$. Then the global estimate holds on $\mathbf{X}$.

Figures (2)

  • Figure 1: Phase portrait of solutions of equation \ref{['eqn:trips3']} initialized on the unit sphere in $\mathbf{R}^3$. When started on the interface $E= \mathcal{E}/|\mathbf{k}|^2$ (plotted in red), solutions are not periodic and, in this case, approach one of the equilibria $(0, \pm1, 0)$. Thus the fraction of time spent in the 'thermalized' state where each of the modes $x$, $y$ and $z$ is non-zero, order 1 is of order $1/H$ as $H \rightarrow \infty$. Away from the interface $E= \mathcal{E}/|\mathbf{k}|^2$ (plotted in blue), solutions are periodic. Provided solutions start sufficiently far from this interface (cf. Assumption \ref{['assump:TNS']}), the fraction of time in the 'thermalized' state is at least order $1/|\log H|$ as $H \rightarrow \infty$.
  • Figure 2: Solutions of equation \ref{['eqn:trips3']} plotted over 140 units of time which are progressively closer to the interface $E=\mathcal{E}/|\mathbf{k}|^2$ from left to right and then down. As initial data tend to $E=\mathcal{E}/|\mathbf{k}|^2$, the period of the solutions tends to infinity and the fraction of time spent in the 'thermalized' state where $x$, $y$, and $z$ are all non-zero, order $1$ tends to zero.

Theorems & Definitions (60)

  • Proposition 2.3
  • proof
  • Lemma 2.7
  • Remark 2.9
  • proof : Proof of Lemma \ref{['lem:KBmeas']}
  • Remark 2.11
  • Theorem 2.19
  • Example 2.22
  • Example 2.23
  • Theorem 3.11
  • ...and 50 more