Anomalous Time Scaling of the Mean Square Distance
in On-Off Diffusion

Abstract

The mean square distance σ²(t) of the diffusion induced by on-off intermittency is derived based on the continuous-time random walk theory. It obeys a scaling law σ²(t)=2Dt φ(t/τ) with diffusion constant D and characteristic time τ, which is confirmed by the use of numerical iterations of a specific periodic map. The scaling function φ and the power spectrum of the on-off intermittency variable I(ω) are analytically obtained from the distribution function of the laminar duration. Normal diffusion (φ(z)∼ 1) and slow diffusion (φ(z)∝ 1/√z) are observed, respectively, for t>>τ and t<<τ. The former and latter correspond to the flat part (I(ω)∼ const) and the power law (I(ω)∝ 1/\sqrt{ω}), respectively, for the power spectrum.