Anomalous diffusion and weak nonergodicity

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Abstract
Ergodic behavior of the class of G processes G(t)=$\int${}tmtdu$\hskip0.3em${}K(t,u)$\xi${}(u)-$\int${}tm0du$\hskip0.3em${}K(0,u)$\xi${}(u), $\langle${}$\xi${}(t)$\rangle${}=0, $\langle${}$\xi${}(t)$\xi${}(s)$\rangle${}=$\phi${}(|t-s|) is examined. Ergodicproperties are only G extensions of normal diffusion (K=1) and of Mandelbrot-Van Ness fractional diffusion [K(t,u)=K(t-u), tm$\rightarrow${}-$\infty${}]. Any deviation from these two types results in weak ergodicity breaking which thus is neither exceptional nor limited to some specific events but is typical for much wider class of processes. G processes driven by $\xi${}(t) with nonvanishing correlations are important for describing transport in strongly nonequilibrium systems and may be responsible for peculiarities of diffusion found in biological, glassy, and nanoscale systems.

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