ETD

• anomalous dissipation
• energy cascade
• hierarchical models
• intermittency
• Markov processes
• spontaneous stochasticity
• turbulence

Turbulence remains a paradigmatic example of a strongly non-equilibrium system, where energy is injected, transferred across scales, and dissipated in a statistically stationary yet intrinsically irreversible regime. Capturing such scale-to-scale dynamics within a tractable statistical framework remains a central challenge of non-equilibrium statistical physics.
We introduce a stochastic cascade model defined on a dyadic grid, in which coarse-grained energy variables evolve according to a continuous-time Markov process with self-similar transition rates. The construction is directly rooted in the Kármán–Howarth–Monin–Hill equation, which expresses the conservation of velocity-increment energy across space and scale. Its discretization yields a minimal yet dynamically consistent representation of the turbulent energy cascade, amenable to efficient simulation via the Gillespie algorithm.
Despite its simplicity, the model reproduces three hallmark features of fully developed turbulence: anomalous dissipation, spatial intermittency, and spontaneous stochasticity.
In contrast to classical multiplicative cascade models, intermittency emerges here from the intrinsic stochastic dynamics rather than being imposed a priori. Under steady forcing, the system reaches a non-equilibrium stationary state characterized by non-Gaussian statistics and anomalous scaling of energy moments, reflecting a constant flux of conserved quantities across scales.
At the analytical level, the master equation enables explicit predictions for the statistics of energy fluxes, in good agreement with numerical simulations. The stationary measure is well described by a log-divisible process, highlighting deep connections with multiplicative cascades while extending them to a dynamical, time-resolved setting. Furthermore, the fluctuating direction of energy transfers gives rise to backward propagation of uncertainties from small to large scales, providing a minimal framework to investigate spontaneous stochasticity as an emergent non-equilibrium phenomenon.