A considerable number of systems have recently been reported in which Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth in time of the mean squared displacement, yet the probability density function of the particle displacement is distinctly non-Gaussian, and often of exponential (Laplace) shape. This apparently ubiquitous behaviour observed in very different physical systems has been interpreted as resulting from diffusion in inhomogeneous environments and mathematically represented through a variable, stochastic diffusion coefficient. Indeed different models describing a fluctuating diffusivity have been studied. Here we present a new view of the stochastic basis describing time-dependent random diffusivities within a broad spectrum of distributions. Concretely, our study is based on the very generic class of the generalised Gamma distribution. Two models for the particle spreading in such random diffusivity settings are studied. The first belongs to the class of generalised grey Brownian motion while the second follows from the idea of diffusing diffusivities. The two processes exhibit significant characteristics which reproduce experimental results from different biological and physical systems. We promote these two physical models for the description of stochastic particle motion in complex environments. uncovered anomalous diffusion of the power-law formwith the anomalous diffusion exponent 0<α<1 and the generalised diffusion coefficient D α [11], for the motion of charge carriers in amorphous semiconductors [12]. With the advance of modern microscopy techniques, in particular, superresolution microscopy, as well as massive progress in supercomputing, anomalous diffusion of the type (3) has been reported in numerous complex and biological systems [13,14]. Thus, subdiffusion with 0<α<1 was observed for submicron tracers in the crowded cytoplasm of biological cells [15][16][17][18][19] as well as in artificially crowded environments [20][21][22][23]. Further reports of subdiffusion come from the motion of proteins embedded in the membranes of living cells [24][25][26]. Subdiffusion is also seen in extensive simulations studies, for instance, of lipid bilayer membranes [27][28][29][30] and relative diffusion in proteins [31]. Superdiffusion, due to active motion of molecular motors, was observed in various biological cell types for both introduced and endogenous tracers [16,17,32,33].Most of the anomalous diffusion phenomena mentioned here belong to two main classes of anomalous diffusion: (i) the class of continuous time random walk processes, in which scale-free power-law waiting times in between motion events give rise to the law (3) [12,34], along with a stretched Gaussian displacement probability density G(x, t) [11,12,34] as well as weak ergodicity breaking and ageing [35,36]. We note that similar effects of non-Gaussianity, weak non-ergodicity, and ageing also occur in spatially heterogeneous diffusion processes [37][38][39][40]. (ii) The secon...

A standard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T  ¥. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T  ¥ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion. We demonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent. We also compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.

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/f 2-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.