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Out-of-time-ordered correlators for turbulent fields: a quantum-classical correspondence

Motoki Nakata

Abstract

An extended formulation of out-of-time-ordered correlators (OTOCs), which quantify noncommutative operator growth and information scrambling in quantum many-body systems, is developed for turbulence dynamics as a representative of non-canonical Hamiltonian systems. Based on the Wigner-Weyl transform and the Moyal bracket formalism, the semiclassical limit of OTOC for turbulent plasmas governed by the Hasegawa-Mima equation is derived as an ensemble-averaged squared Lie-Poisson bracket between two chosen functionals of the turbulent fields. The classical-limit OTOC provides a quantitative measure of how a variational perturbation applied to one functional propagates across scales in the turbulent dynamics and how it affects another functional at a later time, thereby capturing scale-dependent or field-dependent transfer processes. In a quasilinear approximation with a strong zonal flow, we derive an inverse-square decay of the OTOC with the time lag, indicating an algebraic suppression of the zonal-mode response to perturbations of the non-zonal fields. This behavior is attributed to zonal-flow shearing, which rapidly scrambles the non-zonal perturbation toward higher wavenumbers, and consequently reduces the low-wavenumber non-zonal content that can feed back onto large-scale zonal modes.

Out-of-time-ordered correlators for turbulent fields: a quantum-classical correspondence

Abstract

An extended formulation of out-of-time-ordered correlators (OTOCs), which quantify noncommutative operator growth and information scrambling in quantum many-body systems, is developed for turbulence dynamics as a representative of non-canonical Hamiltonian systems. Based on the Wigner-Weyl transform and the Moyal bracket formalism, the semiclassical limit of OTOC for turbulent plasmas governed by the Hasegawa-Mima equation is derived as an ensemble-averaged squared Lie-Poisson bracket between two chosen functionals of the turbulent fields. The classical-limit OTOC provides a quantitative measure of how a variational perturbation applied to one functional propagates across scales in the turbulent dynamics and how it affects another functional at a later time, thereby capturing scale-dependent or field-dependent transfer processes. In a quasilinear approximation with a strong zonal flow, we derive an inverse-square decay of the OTOC with the time lag, indicating an algebraic suppression of the zonal-mode response to perturbations of the non-zonal fields. This behavior is attributed to zonal-flow shearing, which rapidly scrambles the non-zonal perturbation toward higher wavenumbers, and consequently reduces the low-wavenumber non-zonal content that can feed back onto large-scale zonal modes.
Paper Structure (13 sections, 66 equations, 1 figure)

This paper contains 13 sections, 66 equations, 1 figure.

Table of Contents

  1. 1 Introduction
  2. 2 Theory of out-of-time-ordered correlators
  3. 2.1 OTOCs in quantum systems
  4. 2.2 Semiclassical limit of OTOC in Weyl-Wigner-Moyal formalism
  5. 2.3 Extension to general non-canonical systems
  6. 3 OTOC for plasma and fluid turbulence
  7. 3.1 Hasegawa-Mima equation for wave-vortex turbulence
  8. 3.2 Classical-limit OTOC in H-M turbulence
  9. 3.3 Scaling of OTOC for zonal-mode response to non-zonal perturbation
  10. 4. Concluding remarks
  11. A. Non-canonical formulation of the H-M equation
  12. B. Weak-form derivation of the functional-generated perturbation
  13. C. Tangent linear equation for nonlinear dynamical system

Figures (1)

  • Figure 1: Schematic diagram of the classical-limit OTOC as a quadratic variational response to initial perturbations.