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Quenched correlation decay for random splittings of some prototypical 3D flows including the ABC flow

Nianci Jiang, Weili Zhang

TL;DR

The paper develops and analyzes a random splitting framework for prototypical 3D flows on $\mathbb{T}^3$, including the ABC flow, to establish two core chaotic indicators: almost-sure positivity of the top Lyapunov exponent and almost-sure quenched correlation decay. By proving uniform geometric ergodicity for the one-point chain, and then for the projective and two-point chains, it obtains rigorous control over long-time dynamics and stochastic forcing. These ergodic properties enable almost-sure, exponential decay of correlations and exponential mixing for passive scalars, and allow the construction of an almost-sure ideal dynamo via a random velocity field $u_{\underline{\tau}}$. The results bridge operator-splitting ideas with stochastic dynamical systems, showing convergence to the deterministic flow as $h\to 0$ while providing robust, quantitative chaotic and mixing behavior in a stochastic setting, with potential implications for dynamo theory and turbulent transport analyses. Overall, the work provides a rigorous, stochastic mechanism to induce and quantify chaotic transport and magnetic field growth in complex 3D flows, including the ABC flow.

Abstract

For the long-time dynamical challenges of some prototypical 3D flows including the ABC flow on $\mathbb{T}^3$, we apply a random splitting method to establish two fundamental indicators of chaotic dynamics. First, under general assumptions, we establish that these random splittings exhibit Lagrangian chaos, characterized by a positive top Lyapunov exponent. Furthermore, we demonstrate the almost-sure quenched correlation decay of these random splittings, which is a stronger property than the almost-sure positivity of Lyapunov exponents alone. This framework is then applied to construct ideal dynamo in kinematic dynamo theory and to establish exponential mixing of passive scalars.

Quenched correlation decay for random splittings of some prototypical 3D flows including the ABC flow

TL;DR

The paper develops and analyzes a random splitting framework for prototypical 3D flows on , including the ABC flow, to establish two core chaotic indicators: almost-sure positivity of the top Lyapunov exponent and almost-sure quenched correlation decay. By proving uniform geometric ergodicity for the one-point chain, and then for the projective and two-point chains, it obtains rigorous control over long-time dynamics and stochastic forcing. These ergodic properties enable almost-sure, exponential decay of correlations and exponential mixing for passive scalars, and allow the construction of an almost-sure ideal dynamo via a random velocity field . The results bridge operator-splitting ideas with stochastic dynamical systems, showing convergence to the deterministic flow as while providing robust, quantitative chaotic and mixing behavior in a stochastic setting, with potential implications for dynamo theory and turbulent transport analyses. Overall, the work provides a rigorous, stochastic mechanism to induce and quantify chaotic transport and magnetic field growth in complex 3D flows, including the ABC flow.

Abstract

For the long-time dynamical challenges of some prototypical 3D flows including the ABC flow on , we apply a random splitting method to establish two fundamental indicators of chaotic dynamics. First, under general assumptions, we establish that these random splittings exhibit Lagrangian chaos, characterized by a positive top Lyapunov exponent. Furthermore, we demonstrate the almost-sure quenched correlation decay of these random splittings, which is a stronger property than the almost-sure positivity of Lyapunov exponents alone. This framework is then applied to construct ideal dynamo in kinematic dynamo theory and to establish exponential mixing of passive scalars.

Paper Structure

This paper contains 18 sections, 40 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Results and discussions
  3. Uniform geometric ergodicity for the one-point Markov chain
  4. Markov chain preliminaries
  5. Proof of Theorem \ref{['Uniformly geometrically ergodic']}
  6. Positivity of the top Lyapunov exponent
  7. Consequences of $3\lambda_1=\lambda_{\Sigma}$
  8. Proof of Theorem \ref{['Positivity of the top Lyapunov exponent']}
  9. Ideal dynamo
  10. Uniform geometric ergodicity for the projective Markov chain
  11. Proof of Corollary \ref{['Ideal dynamo']}
  12. Geometric ergodicity for the two-point Markov chain
  13. Topological irreducibility and small set property
  14. Projective and twisted semigroups
  15. Lyapunov-Foster drift condition
  16. ...and 3 more sections

Key Result

Theorem 2.1

Fix $t > 0$. Then for any $\varepsilon > 0$, where $\underline{\tau}=(\tau_1, \tau_2, \dots)$, and $\{\tau_i\}_{i=1}^{\infty}$ is a sequence of independent exponential random variables with mean $t/m^2$.

Theorems & Definitions (75)

  • Theorem 2.1: Almost sure convergence
  • Theorem 2.2: Uniformly geometrically ergodic
  • Theorem 2.3: Positivity of the top Lyapunov exponent
  • Corollary 2.4: Ideal dynamo
  • Theorem 2.5: Quenched correlation decay
  • Corollary 2.6: Exponential mixing
  • Corollary 2.7: Random splitting of ABC flow
  • Theorem 3.1: BCZG, Theorem 2.3
  • Lemma 3.2
  • proof
  • ...and 65 more