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

Abstract

The mean square distance σ 2 ( t ) of the diffusion induced by on-off intermittency is derived based on the continuous-time random walk theory. It obeys a scaling law σ 2 ( t )=2 D t 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)∼ c o n s t ) and the power law (I(ω)∝1/ω−−√), respectively, for the power spectrum.