Subdiffusion and ergodicity breaking in heterogeneous environments subject to Lévy noise

The anomalous diffusion and ergodicity breaking of Brownian particles in heterogeneous environments (space-dependent diffusion coefficient D(x)=D0|x|α) subject to Lévy noise are investigated, by both the analytically theory and the numerically simulating schemes. Within the tailored parameter, the competition of the power-law exponent α and the Lévy index μ ensures the long tail of probability density function with the form P(x)∼1|x|μ∕p+1 with p=22−α, falling rapidly when μ∕p>2. Meanwhile, we observe the subdiffusive motion that is characterized by the scale of ensemble averaged mean-square displacement (MSD), 〈x2(t)〉∝tκ with κ=2pμ<1. Furthermore, we also observe the weak ergodicity breaking which means the inequivalence between individual time averaged MSD and the corresponding ensemble averaged MSD. We quantify the weak ergodicity breaking behavior in terms of the dimensionless parameters EB and EB. For 1≤μ<2 (except for μ=2), the ergodicity breaking enhances as the α increases, but weakens as μ or D0. Overall, the EB follows the from, EB∝(Δ∕T)(12−2pμ), in the large time. Additionally, the anomalous diffusion above might be particularly relevant for ultracold atoms diffusion in heterogeneous media.