Spontaneous stochasticity refers to the emergence of intrinsic randomness in
deterministic systems under singular limits, a phenomenon conjectured to be
fundamental in turbulence. Armstrong and Vicol \citep{AV23,AV24} recently
constructed a deterministic, divergence-free multiscale vector field
arbitrarily close to a weak Euler solution, proving that a passive scalar
transported by this field exhibits anomalous dissipation and lacks a selection
principle in the vanishing diffusivity limit.
{\it This work aims to explain why this passive scalar exhibits both
Lagrangian and Eulerian spontaneous stochasticity.}
Part I provides a historical overview of spontaneous stochasticity, details
the Armstrong-Vicol passive scalar model, and presents numerical evidence of
anomalous diffusion, along with a refined description of the Lagrangian flow
map.
In Part II, we develop a theoretical framework for Eulerian spontaneous
stochasticity. We define it mathematically, linking it to ill-posedness and
finite-time trajectory splitting, and explore its measure-theoretic properties
and the connection to RG formalism. This leads us to a well-defined {\it
measure selection principle} in the inviscid limit. This approach allows us to
rigorously classify universality classes based on the ergodic properties of
regularisations. To complement our analysis, we provide simple yet insightful
numerical examples.
Finally, we show that the absence of a selection principle in the
Armstrong-Vicol model corresponds to Eulerian spontaneous stochasticity of the
passive scalar. We also numerically compute the probability density of the
effective renormalised diffusivity in the inviscid limit. We argue that the
lack of a selection principle should be understood as a measure selection
principle over weak solutions of the inviscid system.

Dieser Artikel untersucht Zeitreisen und deren Auswirkungen.

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