The random diffusivity approach for diffusion in heterogeneous systems

Laburpena

The two hallmark features of Brownian motion are the linear growth of the meansquared displacement (MSD) with diffusion coefficient D in d spatial dimensions, andthe Gaussian distribution of displacements. With the increasing complexity of thestudied systems deviations from these two central properties have been unveiledover the years. Recently, a large variety of systems have been reported in which theMSD exhibits the linear growth in time of Brownian (Fickian) transport, however, thedistribution of displacements is pronouncedly non-Gaussian (Brownian yet non-Gaussian, BNG). A similar behaviour is also observed for viscoelastic-type motionwhere an anomalous trend of the MSD is combined with a priori unexpected non-Gaussian distributions (anomalous yet non-Gaussian, ANG). This kind of behaviourobserved in BNG and ANG diffusions has been related to the presence ofheterogeneities in the systems and a common approach has been established toaddress it, that is, the random diffusivity approach.This dissertation explores extensively the field of random diffusivity models. Startingfrom a chronological description of all the main approaches used as an attempt ofdescribing BNG and ANG diffusion, different mathematical methodologies aredefined for the resolution and study of these models.The processes that are reported in this work can be classified in threesubcategories, i) randomly-scaled Gaussian processes, ii) superstatistical modelsand iii) diffusing diffusivity models, all belonging to the more general class of randomdiffusivity models.Eventually, the study focuses more on BNG diffusion, which is by now wellestablishedand relatively well-understood. Nevertheless, many examples arediscussed for the description of ANG diffusion, in order to highlight the possiblescenarios which are known so far for the study of this class of processes.The second part of the dissertation deals with the statistical analysis of randomdiffusivity processes. A general description based on the concept of momentgeneratingfunction is initially provided to obtain standard statistical properties of themodels. Then, the discussion moves to the study of the power spectral analysis andthe first passage statistics for some particular random diffusivity models. Acomparison between the results coming from the random diffusivity approach andthe ones for standard Brownian motion is discussed. In this way, a deeper physicalunderstanding of the systems described by random diffusivity models is alsooutlined.To conclude, a discussion based on the possible origins of the heterogeneity issketched, with the main goal of inferring which kind of systems can actually bedescribed by the random diffusivity approach.