Spontaneous Stochasticity and Anomalous Dissipation

Gregory Eyink
Johns Hopkins University

I will review the notions of anomalous dissipation and spontaneous stochasticity
in turbulence and discuss their relations. Briefly, anomalous dissipation corresponds to the
well-known "zeroth-law of turbulence", or non-vanishing energy dissipation in the inviscid limit.
Spontaneous stochasticity is the statement that Lagrangian particle trajectories for fixed initial
particle locations and for a fixed flow velocity are intrinsically random in the same limit ("God
plays dice for classical dynamics too"). Relations between the two phenomena are known for
turbulent scalar advection, since the late 20th-century work on the Kraichnan model. I present
new results (with T. Drivas) for the Burgers model, which demonstrate similar relations there.
I finally discuss Navier-Stokes turbulence, stating the results known, open questions, and
some conjectures.

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