Abstract
Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f2-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.
Highlights
Diffusive processes came to the attention of the broader scientific community with the experiments on ‘active molecules’ by Brown, who reported the jittery motion of granules of ‘1/4000th to 1/5000th of an inch in length’ contained in pollen grains as well as control experiments on powdered inorganic rocks [1]
Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media
Exact expressions for arbitrary frequency and observation time We investigate the power spectral density (PSD) of an individual trajectory xt encoded in the stochastic dynamics (3) with t ∈ (0, T), 1 ST(f ) = T
Summary
Vittoria Sposini1,2 , Denis S Grebenkov , Ralf Metzler1,7 , Gleb Oshanin and Flavio Seno. Cedex 05, France 5 Interdisciplinary Scientific Center J-V Poncelet (ISCP), CNRS UMI 2615, 11 Bol. Vlassievsky per., 119002 Moscow, Russia 6 INFN, Padova Section and Department of Physics and Astronomy “Galileo Galilei”, University of Padova, 35131 Padova, Italy 7 Author to whom any correspondence should be addressed
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