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Random Splitting of Fluid Models: Ergodicity and Convergence

Andrea Agazzi, Jonathan C. Mattingly, Omar Melikechi

TL;DR

The work introduces random splitting as a general framework to study nonequilibrium fluid-like dynamics by decomposing the original flow into tractable, invariant-preserving components and injecting randomness through exponential dwell times. It proves conditions under which the resulting Markov process has a unique invariant measure and converges to the deterministic dynamics as the mean time step vanishes, and demonstrates these properties for conservative Lorenz-96 and finite-dimensional Galerkin models of 2D Euler and Navier–Stokes, including forcing and dissipation. The results establish ergodicity on invariant level sets and provide both convergence in operator form and almost-sure convergence, with detailed constructions of controllability and spanning to guarantee irreducibility. The framework thus links stochastic perturbations, geometric control (Lie brackets), and conserved quantities to rigorous long-time behavior of complex fluid-like systems, offering insights applicable to nonequilibrium steady states and data assimilation contexts.

Abstract

We introduce a family of stochastic models motivated by the study of nonequilibrium steady states of fluid equations. These models decompose the deterministic dynamics of interest into fundamental building blocks, i.e., minimal vector fields preserving some fundamental aspects of the original dynamics. Randomness is injected by sequentially following each vector field for a random amount of time. We show under general assumptions that these random dynamics possess a unique invariant measure and converge almost surely to the original, deterministic model in the small noise limit. We apply our construction to the Lorenz-96 equations, often used in studies of chaos and data assimilation, and Galerkin approximations of the 2D Euler and Navier-Stokes equations. An interesting feature of the models developed is that they apply directly to the conservative dynamics and not just those with excitation and dissipation.

Random Splitting of Fluid Models: Ergodicity and Convergence

TL;DR

The work introduces random splitting as a general framework to study nonequilibrium fluid-like dynamics by decomposing the original flow into tractable, invariant-preserving components and injecting randomness through exponential dwell times. It proves conditions under which the resulting Markov process has a unique invariant measure and converges to the deterministic dynamics as the mean time step vanishes, and demonstrates these properties for conservative Lorenz-96 and finite-dimensional Galerkin models of 2D Euler and Navier–Stokes, including forcing and dissipation. The results establish ergodicity on invariant level sets and provide both convergence in operator form and almost-sure convergence, with detailed constructions of controllability and spanning to guarantee irreducibility. The framework thus links stochastic perturbations, geometric control (Lie brackets), and conserved quantities to rigorous long-time behavior of complex fluid-like systems, offering insights applicable to nonequilibrium steady states and data assimilation contexts.

Abstract

We introduce a family of stochastic models motivated by the study of nonequilibrium steady states of fluid equations. These models decompose the deterministic dynamics of interest into fundamental building blocks, i.e., minimal vector fields preserving some fundamental aspects of the original dynamics. Randomness is injected by sequentially following each vector field for a random amount of time. We show under general assumptions that these random dynamics possess a unique invariant measure and converge almost surely to the original, deterministic model in the small noise limit. We apply our construction to the Lorenz-96 equations, often used in studies of chaos and data assimilation, and Galerkin approximations of the 2D Euler and Navier-Stokes equations. An interesting feature of the models developed is that they apply directly to the conservative dynamics and not just those with excitation and dissipation.
Paper Structure (25 sections, 39 theorems, 211 equations, 3 figures)

This paper contains 25 sections, 39 theorems, 211 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A class of stochastic models
  3. Two motivating examples
  4. Random splitting in a general setting
  5. $\mathcal{V}$-Orbits
  6. Ergodicity
  7. The Lie bracket condition
  8. Convergence as mean time step goes to zero
  9. Conservative Lorenz-96
  10. Ergodicity
  11. Galerkin approximations of 2D Euler
  12. Constructing the splitting
  13. Conservation and convergence
  14. Ergodicity
  15. Controllability
  16. ...and 10 more sections

Key Result

Theorem 3.1

If there exists $x_*$ in $\mathcal{X}$ such that for all $x$ in $\mathcal{X}$ there is an $m$ in $\mathbb N$ and $t$ in $\mathbb R_+^{mn}$ with $\Phi^m(x,t)=x_*$ and $D_t\Phi^m(x,t):T_{t}\mathbb{R}^{mn}_+\to T_{x_*}\mathcal{X}$ surjective, then $P_h$ has at most one invariant measure on $\mathcal{X}

Figures (3)

  • Figure 1: Representation of the state of the network in a generic initial state (a), after step 1 of the procedure in the proof of Proposition \ref{['p:conteuler']} (b), and after step 2 (c) and after step 3 (d) of the same procedure. In the above pictures, each point corresponds to a mode, i.e., an element of $\mathbb{Z}^2_N$ while the color of each circle represents the real/complex value of the corresponding mode: zero (white, no circle), purely imaginary (red), purely real (blue) or having both nonvanishing real and imaginary parts (green).
  • Figure 2: Orbits $\mathcal{Q}_\iota$ of (\ref{['e:vf']}) (in red) corresponding in (A) to various values of the energy $\mathcal{E}_\iota(q)$ on the sphere of constant enstrophy $E_\iota(q)$ and in (B) to various values of the enstrophy $E_\iota(q)$ on the ellipsoid of constant energy $\mathcal{E}_\iota(q)$. The axes are, sequentially, $q_{\bm k}, q_{\bm j}, q_{\bm \ell}$. The orbit with a degenerate point at the pole of the sphere or ellipsoid corresponds to values of $E_\iota , \mathcal{E}_\iota$ violating (\ref{['e:degcond']}).
  • Figure 3: Ordering of $\mathbb{Z}^2_N$ when $N=4$ .

Theorems & Definitions (88)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • ...and 78 more