Spontaneous stochasticity and renormalization group in discrete multi-scale dynamics

TL;DR

This paper develops a theoretical framework for understanding the inviscid limit in multi-scale, discrete-time systems, revealing conditions under which the limit becomes spontaneously stochastic, with applications to digital turbulence.

Contribution

It introduces a renormalization group approach to analyze the inviscid limit in infinite-dimensional, scale-invariant systems, highlighting the phenomenon of spontaneous stochasticity.

Findings

Inviscid limit can be characterized as an attractor of the renormalization group.

Spontaneous stochasticity occurs when the attractor is a nontrivial probability kernel.

Models demonstrate the emergence of digital turbulence and phase interactions.

Abstract

We introduce a class of multi-scale systems with discrete time, motivated by the problem of inviscid limit in fluid dynamics in the presence of small-scale noise. These systems are infinite-dimensional and defined on a scale-invariant space-time lattice. We propose a qualitative theory describing the vanishing regularization (inviscid) limit as an attractor of the renormalization group operator acting in the space of flow maps or respective probability kernels. If the attractor is a nontrivial probability kernel, we say that the inviscid limit is spontaneously stochastic: it defines a stochastic (Markov) process solving deterministic equations with deterministic initial and boundary conditions. The results are illustrated with solvable models: symbolic systems leading to digital turbulence and systems of expanding interacting phases.