Wave collapse proliferation as a mechanism for anomalous dissipation and spontaneous stochasticity
Abstract
Anomalous dissipation and spontaneous stochasticity are signatures of the roughness of the inviscid limit in fluid turbulence. Spontaneous stochasticity describes the selection of a probability distribution over possible solutions rather than a single deterministic trajectory. We investigate these phenomena in a nonlinear dispersive equation undergoing finite-time blowup, known as wave collapse. Collapse is arrested by two regularization mechanisms, one dissipative and one conservative. Heuristic arguments supported by numerical simulations show that an isolated collapse has a vanishing effect in the inviscid limit and cannot alone account for either phenomenon. However, simulations reveal that the number of collapse events diverges as the regularization vanishes. Their proliferation enables sustained dissipation in the dissipative case and prevents either regularization from selecting a single inviscid solution. We then introduce vanishingly small random perturbations and find numerical evidence for a well-defined statistical limit in which randomness persists. These results identify collapse proliferation as a mechanism connecting anomalous dissipation and spontaneous stochasticity in dispersive waves, allowing a formally deterministic equation to acquire a truly random limiting evolution.
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BibTeX
@misc{ruffenach2026wave,
title = {{Wave collapse proliferation as a mechanism for anomalous dissipation and spontaneous stochasticity}},
author = {Wandrille Ruffenach},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.18788},
eprint = {2607.18788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18788},
}
RIS
TY - UNPB
TI - Wave collapse proliferation as a mechanism for anomalous dissipation and spontaneous stochasticity
AU - Wandrille Ruffenach
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2607.18788
UR - https://arxiv.org/abs/2607.18788
AB - Anomalous dissipation and spontaneous stochasticity are signatures of the roughness of the inviscid limit in fluid turbulence. Spontaneous stochasticity describes the selection of a probability distribution over possible solutions rather than a single deterministic trajectory. We investigate these phenomena in a nonlinear dispersive equation undergoing finite-time blowup, known as wave collapse. Collapse is arrested by two regularization mechanisms, one dissipative and one conservative. Heuristic arguments supported by numerical simulations show that an isolated collapse has a vanishing effect in the inviscid limit and cannot alone account for either phenomenon. However, simulations reveal that the number of collapse events diverges as the regularization vanishes. Their proliferation enables sustained dissipation in the dissipative case and prevents either regularization from selecting a single inviscid solution. We then introduce vanishingly small random perturbations and find numerical evidence for a well-defined statistical limit in which randomness persists. These results identify collapse proliferation as a mechanism connecting anomalous dissipation and spontaneous stochasticity in dispersive waves, allowing a formally deterministic equation to acquire a truly random limiting evolution.
ER -