Localization and Ballistic Diffusion for the Tempered Fractional Brownian–Langevin Motion
Tóm tắt
This paper discusses the tempered fractional Brownian motion (tfBm), its ergodicity, and the derivation of the corresponding Fokker–Planck equation. Then we introduce the generalized Langevin equation with the tempered fractional Gaussian noise for a free particle, called tempered fractional Langevin equation (tfLe). While the tfBm displays localization diffusion for the long time limit and for the short time its mean squared displacement (MSD) has the asymptotic form
$$t^{2H},$$
we show that the asymptotic form of the MSD of the tfLe transits from
$$t^2$$
(ballistic diffusion for short time) to
$$t^{2-2H},$$
and then to
$$t^2$$
(again ballistic diffusion for long time). On the other hand, the overdamped tfLe has the transition of the diffusion type from
$$t^{2-2H}$$
to
$$t^2$$
(ballistic diffusion). The tfLe with harmonic potential is also considered.
Tài liệu tham khảo
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