Rough roads to randomness

by André Freitas

A common way to talk about turbulence is to say that it is chaotic: tiny differences grow quickly, and after some time prediction becomes impossible. This is certainly part of the story, but it is not the whole story. By turbulence one usually means the irregular, swirling motion seen in atmospheric winds, ocean currents, smoke plumes, river rapids, or the wake behind an aircraft. What makes these flows hard to describe is that they contain motion on many scales at once: large eddies break into smaller ones, which in turn interact with still smaller ones.
The standard mathematical model for the motion of an ordinary incompressible fluid is given by the Navier–Stokes equations,

∂tu + u · ∇u = −∇p + ν∆u,   ∇ · u = 0

where u(x,t) is the velocity field, p(x,t) is the pressure, and ν is the kinematic viscosity. The term u · ∇u expresses the nonlinear transport of momentum by the flow itself, while the viscous term ν∆u acts to smooth velocity variations. Much of fluid dynamics is governed by the competition between these two effects: nonlinear transport tends to generate complicated multiscale motion, while viscosity tends to damp it out. This balance is summarized by the Reynolds number, a dimensionless quantity that compares inertial effects with viscous effects. When the Reynolds number is small, viscosity is strong enough to smooth the motion efficiently, and the flow is often relatively regular. When the Reynolds number is large, viscosity is comparatively weak, and this is the regime in which fully developed turbulence is usually observed. In ordinary deterministic chaos, there is nevertheless a unique future associated with each exact initial condition. The practical difficulty is simply that one never knows that initial condition perfectly. Turbulence at high Reynolds number may be different in a deeper sense. In that limit, one is led to consider the possibility that the relation between very small perturbations and the resulting motion is itself singular. This is one of the settings in which the idea of spontaneous stochasticity appears.

A useful place to begin is with a puzzle. In many physical problems, if a dissipative coefficient becomes smaller and smaller, then the dissipative effect it produces also becomes smaller and smaller. Since the viscous term ν∆u is multiplied by ν, one would naturally expect viscous dissipation to disappear as the viscosity ν tends to zero. In turbulent flows, however, this expectation may fail. The mean rate at which kinetic energy is dissipated can remain finite even in a vanishing-viscosity limit. This phenomenon is called anomalous dissipation. Its importance is that it signals a singular limit: solving the equations with a very small viscosity and then letting ν → 0 is not the same thing as setting ν = 0 at the start. The limiting flow may retain effects that are absent from the perfectly smooth inviscid picture. Onsager’s 1949 analysis [1] made this point especially clear. If anomalous dissipation persists in the inviscid limit, then the limiting velocity field cannot remain smooth in the classical sense. The flow must become rough enough that some of the most familiar deterministic intuitions are no longer automatically valid.
One of those intuitions concerns the motion of individual particles. There are two standard ways to describe a flow. In the Eulerian description, one looks at the velocity field at fixed points in space, as a meteorologist might do on a weather map. In the Lagrangian description, one follows individual fluid particles as they move, more like tracking a leaf carried by a river. If the velocity field u(x, t) is sufficiently regular, then the path X(t) of such a particle is determined by

dX/dt = u(X(t),t),

together with its initial position X(0). In that familiar setting, one starting point gives one trajectory. But once the velocity field becomes rough enough, this uniqueness can fail. Mathematically, this is a fact about differential equations: existence of solutions does not by itself guarantee uniqueness unless the vector field is sufficiently regular [2]. Physically, it means that in the limiting rough flow, fixing the starting point of a tracer particle may no longer determine a single path.
This is precisely where the role of noise becomes subtle. If the limiting deterministic dynamics were still unique, then one would expect a tiny stochastic perturbation to disappear harmlessly as its amplitude tends to zero. But if the limiting deterministic problem is not unique, the noise may do something more interesting: it may select among several admissible trajectories. In that case, even though the perturbation becomes arbitrarily weak, the limiting object need not collapse to a single deterministic path. It may remain probabilistic. That persistence of randomness in the vanishing-noise limit is what is meant by spontaneous stochasticity.

A particularly transparent setting in which to see this is turbulent mixing. Imagine a scalar quantity carried by the flow: dye in water, smoke in air, heat in a room, or a pollutant in the atmosphere. Denote it by c(x, t). A standard equation for its evolution is the advection–diffusion equation

∂_t c + u · ∇c = D∆c,

dX(t) = u(X(t), t) dt + √(2D) dW(t),

There is also a more specific physical picture behind this discussion, which goes back to Richardson’s theory of turbulent dispersion [3]. The central idea is that the effective diffusivity for the separation of two tracer particles is itself scale-dependent. When two particles are very close, they are affected mainly by small eddies; once they have separated, larger eddies come into play and drive them apart more rapidly. This leads to Richardson’s law,

〈r²(t)〉 ∼ ε t³,

where r(t) is the particle separation and ε is the mean energy dissipation rate per unit mass. The point is not the exact prefactor, but the mechanism: separation accelerates as separation grows. In that sense turbulent dispersion is not merely diffusive, but explosive. Very small uncertainties can be carried to observable scales in finite time, and in singular limits this loss of memory can be strong enough that particles released from the same point no longer correspond to a unique deterministic trajectory. This is why spontaneous stochasticity is stronger than the usual picture of chaos. The issue is not only sensitivity to initial data, but the possibility that the limiting relation between initial conditions and trajectories is no longer single-valued.

One may then ask whether this phenomenon concerns only tracer particles, or whether something similar can happen at the level of the velocity field itself. This is the question behind Eulerian spontaneous stochasticity. Here the issue is no longer the motion of particles in a given rough velocity field, but the possible non-uniqueness of the fluid evolution itself in a singular high-Reynolds-number limit. One way to formulate the question is to begin with a fluid model perturbed by a very weak microscopic noise, for instance in fluctuating hydrodynamics, where the Navier–Stokes equations are supplemented by random stresses representing thermal fluctuations, and then ask what happens when both viscosity and noise amplitude are sent to zero. One may again write a path-integral description for the probability of whole space-time histories of the velocity field. In a regular situation, one would expect this measure to concentrate onto a single most likely solution. But if the limiting inviscid dynamics admits more than one admissible solution for the same initial data, that concentration need not occur. The zero-noise limit may then remain stochastic with a non-trivial (non-delta) distribution. There is substantial evidence for such behaviour in shell models and in several idealized or singular systems, while for ordinary Navier–Stokes turbulence important aspects remain under active investigation. Even so, the conceptual shift is clear: in a singular turbulent limit, the natural object may be a probability measure on velocity histories rather than one deterministic history.

This also changes, at least slightly, the way one thinks about randomness in turbulence. Usually randomness is treated as a sign of ignorance. We do not know the initial state exactly, we do not resolve all relevant scales, and so we resort to statistics. That point of view remains useful, but it may not be complete. In singular limits, the statistical description may not merely compensate for missing information. It may be the more faithful limiting description of the dynamics itself. In that sense, spontaneous stochasticity is interesting not only for turbulence, but more broadly for the way it forces one to distinguish between two kinds of randomness: randomness due to incomplete knowledge, and randomness that survives as part of the limiting dynamics when classical deterministic well-posedness breaks down.

At sufficiently high Reynolds number, the appropriate goal may therefore be not the prediction of one unique future, but the characterization of an ensemble of possible futures. Singular limits of this kind are familiar elsewhere in physics: when a control parameter approaches a critical value, the limiting description may cease to select one state uniquely and instead be given by a measure over several competing states (multiple equilibria [4]). The suggestion here is that turbulence may have to be understood in a similar way.
Spontaneous stochasticity is not limited to turbulence, even though it provides perhaps the clearest and richest setting in which it can occur. More generally, it may arise in multiscale systems where microscopic perturbations are both strongly amplified and rapidly transferred across scales.

Further reading. The perspective developed above is strongly influenced by the review by Eyink and Goldenfeld [5], as well as by the stimulating lectures given by Greg Eyink in Rome in the autumn of 2024.

References
[1] Lars Onsager. Statistical hydrodynamics. Il Nuovo Cimento (1943-1954), 6(Suppl 2):279–287, 1949.
[2] Sara Daneri, Eris Runa, and Laszlo Szekelyhidi. Non-uniqueness for the euler equations up to
onsager’s critical exponent. Annals of PDE, 7(1):8, 2021.
[3] Lewis Fry Richardson. Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 110(756):709–737, 1926.
[4] Giorgio Parisi. Nobel lecture: Multiple equilibria. Reviews of Modern Physics, 95(3):030501, 2023.
[5] Gregory L. Eyink and Nigel Goldenfeld. Beyond chaos: fluctuations, anomalies and spontaneous stochasticity in fluid turbulence, 2025.