1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

Abstract

On–off intermittency occurs in nonequilibrium physical systems close to bifurcation points, and is characterised by an aperiodic switching between a large-amplitude ‘on’ state and a small-amplitude ‘off’ state. Lévy on–off intermittency is a recently introduced generalisation of on–off intermittency to multiplicative Lévy noise, which depends on a stability parameter α and a skewness parameter β. Here, we derive two novel results on Lévy on–off intermittency by leveraging known exact results on the first-passage time statistics of Lévy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary Lévy noise parameters ( α , β ) for the first time, by a heuristic method, extending previous results. The predictions are verified using numerical solutions of the fractional Fokker–Planck equation. Second, we derive the power spectrum S(f) of Lévy on–off intermittency, and show that it displays a power law S ( f ) ∝ f κ at low frequencies f, where κ ∈ ( − 1 , 0 ) depends on the noise parameters α , β . An explicit expression for κ is obtained in terms of ( α , β ) . The predictions are verified using long time series realisations of Lévy on–off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.

Beta Brownian motion

“…Then, the section will return to the generalized WK theorem for the velocities of the beta model. The spectral results established in [3] and here join a substantial body of research on the intersection of anomalous diffusion and 1/f p noise [145][146][147][148][149][150][151][152][153][154][155][156][157][158].…”

Spatial extreme values of vorticity and velocity gradients in two-dimensional turbulent flows

“…Such extreme fluctuations can govern the transitions between 2D-3D flows in thin-layer turbulence and in other turbulent transitions, [19][20][21]. The intermittency properties of the growth rate of the perturbations will be governed by the statistics of the underlying flow structures, as seen in recent model studies [22,23]. In turbulent flows where sub-critical branches, multi-branch solutions are found, large amplitude perturbations in vorticity or shear could decide the solution branch chosen by the system, [24,25].…”