Abstract
We develop a theoretical approach to ``spontaneous stochasticity'' in classical dynamical systems that are nearly singular and weakly perturbed by noise. This phenomenon is associated to a breakdown in uniqueness of solutions for fixed initial data and underlies many fundamental effects of turbulence (unpredictability, anomalous dissipation, enhanced mixing). Based upon analogy with statistical-mechanical critical points at zero temperature, we elaborate a renormalization group (RG) theory that determines the universal statistics obtained for sufficiently long times after the precise initial data are ``forgotten''. We apply our RG method to solve exactly the ``minimal model'' of spontaneous stochasticity given by a 1D singular ODE. Generalizing prior results for the infinite-Reynolds limit of our model, we obtain the RG fixed points that characterize the spontaneous statistics in the near-singular, weak-noise limit, determine the exact domain of attraction of each fixed point, and derive the universal approach to the fixed points as a singular large-deviations scaling, distinct from that obtained by the standard saddle-point approximation to stochastic path-integrals in the zero-noise limit. We present also numerical simulation results that verify our analytical predictions, propose possible experimental realizations of the ``minimal model'', and discuss more generally current empirical evidence for ubiquitous spontaneous stochasticity in Nature. Our RG method can be applied to more complex, realistic systems and some future applications are briefly outlined.
Highlights
Classical dynamical systems [ordinary differential equations (ODEs) and partial differential equations (PDEs)]x = v(x, t ), x(0) = x0 (1.1)with smooth vector fields v have unique solutions of the Cauchy initial-value problem for any fixed initial data x0
Before we present the details of our renormalization group (RG) analysis, we must emphasize that spontaneous stochasticity is qualitatively distinct from the so-called “butterfly effect” or “deterministic chaos” of differentiable dynamics [40,41]
The conclusion of our argument is that the approach to fixed points Q∗(p)(xw, tw ) is the same for all p, independent of P0 and with identical functional form for all Sc and for all regularized velocities v(x) satisfying v (x) 0
Summary
Classical dynamical systems [ordinary differential equations (ODEs) and partial differential equations (PDEs)]. Just as for closely analogous T = 0 critical phenomena such as quantum critical points [38,39] or the order-disorder transition in the 1D Ising model [36], the singular critical point in spontaneous stochasticity can never be strictly achieved but its drastic effects are felt for finite values of and D over a very wide region This is illustrated by the phase diagram in X -Y -parameter plane, Fig. 1, with X = ln(Re) and Y = ln(Pe), where Re is the “Reynolds number” and Pe the “Péclet number,” dimensionless versions of 1/ and 1/D, respectively. As we shall argue below, this is the generic situation in high Reynolds-number turbulent flows and in similar problems with near-singular dynamics in geophysics and astrophysics
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