Spontaneous stochasticity and thermal noise in turbulent systems

Bandak, Dmytro

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https://hdl.handle.net/2142/121422 Copy

Description

Title

Spontaneous stochasticity and thermal noise in turbulent systems

Author(s)

Bandak, Dmytro

Issue Date

2023-07-14

Director of Research (if dissertation) or Advisor (if thesis)

Goldenfeld, Nigel D.

Doctoral Committee Chair(s)

Cooper, Lance S.

Committee Member(s)

• Maslov, Sergei
• Leite Noronha, Jr, Jorge

Department of Study

Physics

Discipline

Physics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Turbulence
• spontaneous stochasticity
• thermal noise
• renormalization group
• dissipation range

Abstract

The subject of this thesis is twofold. First, it is about spontaneous stochasticity in turbulent systems, a mechanism through which stochastic behaviour can emerge in formally deterministic systems. Second, it is about the interplay of turbulence and thermal noise, which naturally emerges from the inclusion of the inherent source of stochasticity in molecular turbulent fluids. A brief summary of what we accomplish in each of these areas is as follows. In Part I, Chapter 2 we develop a theoretical approach to “spontaneous stochasticity” in classical dynamical systems that are nearly singular and weakly perturbed by noise. Based upon analogy with statistical-mechanical critical points at zero temperature, we elaborate a renormalization group (RG) theory that determines the universal spontaneous statistics obtained for sufficiently long times after most details of the initial data are forgotten. We propose a toy model of a one-dimensional singular ordinary differential equation that exhibits spontaneous stochasticity, and apply our RG method to solve it exactly. Generalizing and unifying prior results for the model, we obtain the RG fixed points that characterize the spontaneous statistics in the near-singular, weak-noise limit, determine the exact domain of attraction of each fixed point, and derive the universal approach to the fixed points as a singular large-deviations scaling, distinct from that obtained by the standard saddle-point approximation to stochastic path-integrals in the zero-noise limit. We also present numerical simulation results that verify our analytical predictions, propose possible experimental realizations of the toy model, and discuss more generally current empirical evidence for ubiquitous spontaneous stochasticity in Nature. In Part I, Chapter 3 we develop a simple mechanical model of Eulerian spontaneous stochasticity that can be implemented experimentally, which we call the cascading pendulum. The model was inspired by the geometric analogy between the motion of an inviscid fluid and a rigid body proposed by Arnold in 1966, and comes from combining the chaotic double pendulum with scale invariance. In our work we show that, initiated from a resting position with forcing that can be easily implemented experimentally, the cascading pendulum indeed displays spontaneous stochasticity, and as such in the limit does not require any noise or perturbation to behave in an intrinsically indeterministic way. Furthermore, using numerical simulations we show that the cascading pendulum possesses other remarkable properties typically associated with hydrodynamic turbulence such as the dissipative anomaly, cascade of energy and breaking of scale invariance. To conclude we discuss generalizations of the cascading pendulum to higher dimensions and its related linear modification, as well as directions of future investigation such as control and quantization of the cascading pendulum. In Part II, Chapter 4 we revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence, and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation range intermittency predicted by Kraichnan for deterministic Navier- Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an effective field-theory valid below some high-wavenumber cutoff Λ, rather than as a continuum stochastic partial differential equation. In Part II, Chapter 5 we use theoretical estimates and shell model simulations to argue that Eulerian spontaneous stochasticity, a manifestation of the non-uniqueness of the solutions to the Euler equation that is conjectured to occur in Navier-Stokes turbulence at high Reynolds numbers, leads to universal statistics at finite times, not just at infinite time as for standard chaos. We show that thermal noise effects vanish slowly enough with increasing Reynolds number that they are able to trigger spontaneous stochasticity. Thus, turbulent fluid motions are intrinsically stochastic at all scales due to molecular noise. I also show evidence that replica symmetry breaking accompanies spontaneous stochasticity and speculate about its physical interpretation. Our work implies essential indeterminism in the evolution of turbulent flows at scales of practical interest, with far-ranging implications for engineering, geophysics, and astrophysics.

Graduation Semester

2023-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2023 Dmytro Bandak

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