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Renormalization-group perspective on spontaneous stochasticity

Alexei A. Mailybaev, Luca Moriconi

Abstract

We present a renormalization-group perspective on spontaneous stochasticity in hydrodynamic turbulence, viewed through the lens of multiscale dynamical systems. Building on previously established results for a solvable multiscale Arnold's cat model, we show that spontaneous stochasticity emerges as a universal fixed point of an RG transformation acting on Markov kernels, independent of the microscopic regularization. Classical examples - including the Feigenbaum equation, the central limit theorem, and hierarchical spin models - are reinterpreted within the same framework, placing spontaneous stochasticity alongside other universality phenomena.

Renormalization-group perspective on spontaneous stochasticity

Abstract

We present a renormalization-group perspective on spontaneous stochasticity in hydrodynamic turbulence, viewed through the lens of multiscale dynamical systems. Building on previously established results for a solvable multiscale Arnold's cat model, we show that spontaneous stochasticity emerges as a universal fixed point of an RG transformation acting on Markov kernels, independent of the microscopic regularization. Classical examples - including the Feigenbaum equation, the central limit theorem, and hierarchical spin models - are reinterpreted within the same framework, placing spontaneous stochasticity alongside other universality phenomena.
Paper Structure (23 sections, 3 theorems, 72 equations, 10 figures)

This paper contains 23 sections, 3 theorems, 72 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Spontaneous stochasticity: the physical picture
  3. Fluctuating hydrodynamics
  4. Ideal limit and spontaneous stochasticity
  5. Phenomenology of noise dynamics
  6. Solvable toy model: multiscale Arnold's cat
  7. Ideal model
  8. Regularization, ideal limit, and spontaneous stochasticity
  9. RG framework for the multiscale Arnold's cat
  10. RG relation in the absence of noise
  11. RG relation for Markov kernels
  12. RG fixed point and its stability
  13. Feigenbaum RG equation on the fractal lattice
  14. Representation on the fractal lattice
  15. RG dynamics and its fixed point
  16. ...and 8 more sections

Key Result

theorem 1

For any prescribed initial condition eq:cat_IC, the infinite multiscale system eq:cat_multiscale admits uncountably many distinct solutions defined for all times $t\ge 0$. Specifically, one may choose arbitrary values of $u_n(t)$ at times $t=m\tau_n$, where $m\ge 2$ for $n=0$ and $m\ge 3$ odd for $n

Figures (10)

  • Figure 1: Schematic description of turbulence: energy is injected at the largest scale $L$ and transported through a hierarchy of progressively smaller eddies toward the dissipation scale, while microscopic noise propagates in the opposite direction, from small to large scales.
  • Figure 2: Structure of the multi-scale lattice with the variables $u_n(t)$ corresponding to scales $\ell_n$ and discrete times $t \in \tau_n \mathbb{Z}^+$. Gray arrows represent the Arnold's cat map (shown on the top of the figure), which appear in the coupling relation (\ref{['eq:cat_multiscale']}) and correspond to one turn-over time $\tau_n$. White circles correspond to initial conditions. Red circles denote the variables taking arbitrary values in Theorem \ref{['thm:nonuniqueness']}. Small black circles denote the remaining variables, which are uniquely determined by dynamical relations in terms of the white and red ones.
  • Figure 3: Multiscale system truncated at the cutoff scale $\ell_N$. Only the variables $u_n(t)$ for $n=0,\ldots,N$ are retained, while $u_n(t)\equiv 0$ for $n>N$. At the smallest active scale $\ell_N$, random forcing $\xi(t)$ is injected at discrete times $t=m\tau_N$ (red arrows), modifying the evolution according to Eq. \ref{['eq:cat_multiscale_cutoff']}.
  • Figure 4: Block decomposition underlying the RG relation \ref{['eq:RGdet']}. The unit-time evolution of the cutoff-$(N+1)$ system is represented as two consecutive unit-time evolutions $\varphi^{(N)}$ of the cutoff-$N$ system, followed by an update of the largest scale via the map $f$. Scale alignment is achieved using the shift maps $\sigma_+$ and $\sigma_-$.
  • Figure 5: (a) Probability density functions and (b) mean values for the first component of the random variable $u_1(1) = (x,y)$. The thin green line corresponds to the uniform distribution associated with the ideal limit. The data are based on $10^8$ samples from numerical simulations with initial conditions $a_n = (2^{n/2},\,2^{n/3})$ and normally distributed noise $\xi(t)$ with zero mean and variance $\sigma^2 = 10^{-6}$.
  • ...and 5 more figures

Theorems & Definitions (3)

  • theorem 1: Non-uniqueness of solutions mailybaev2023spontaneously
  • theorem 2: Spontaneously stochastic limit mailybaev2023spontaneously
  • theorem 3: Stochastic RG relation mailybaev2023spontaneous