The random diffusivity approach for diffusion in heterogeneous systems

The two hallmark features of Brownian motion are the linear growth $\langle x^2(t) \rangle = 2 D d t$ of the mean squared displacement (MSD) with diffusion coefficient $D$ in $d$ spatial dimensions, and the Gaussian distribution of displacements. With the increasing complexity of the studied systems deviations from these two central properties have been unveiled over the years. Recently, a large variety of systems have been reported in which the MSD exhibits the linear growth in time of Brownian (Fickian) transport, however, the distribution of displacements is pronouncedly non-Gaussian (Brownian yet non-Gaussian, BNG). A similar behaviour is also observed for viscoelastic-type motion where an anomalous trend of the MSD, i.e., $\langle x^2(t) \rangle \sim t^\alpha$, is combined with a priori unexpected non-Gaussian distributions (anomalous yet non-Gaussian, ANG). This kind of behaviour observed in BNG and ANG diffusions has been related to the presence of heterogeneities in the systems and a common approach has been established to address it, that is, the random diffusivity approach. This dissertation explores extensively the field of random diffusivity models. Starting from a chronological description of all the main approaches used as an attempt of describing BNG and ANG diffusion, different mathematical methodologies are defined for the resolution and study of these models. The processes that are reported in this work can be classified in three subcategories, i) randomly-scaled Gaussian processes, ii) superstatistical models and iii) diffusing diffusivity models, all belonging to the more general class of random diffusivity models. Eventually, the study focuses more on BNG diffusion, which is by now well-established and relatively well-understood. Nevertheless, many examples are discussed for the description of ANG diffusion, in order to highlight the possible scenarios 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 ran- dom diffusivity processes. A general description based on the concept of moment- generating function is initially provided to obtain standard statistical properties of the models. Then, the discussion moves to the study of the power spectral analysis and the first passage statistics for some particular random diffusivity models. A comparison between the results coming from the random diffusivity approach and the ones for standard Brownian motion is discussed. In this way, a deeper physical understanding of the systems described by random diffusivity models is also outlined. To conclude, a discussion based on the possible origins of the heterogeneity is sketched, with the main goal of inferring which kind of systems can actually be described by the random diffusivity approach.BERC.2018-202

A numerical study of fractional diffusion through a Langevin approach in random media.

The study of Brownian motion has a long history and involves many different formulations. All these formulations show two fundamental common results: the mean square displacement of a diffusing particle scales linearly with time and the probability density function is a Guassian distribution. However standard diffusion is not universal. In literature there are numerous experimental measurements showing non linear diffusion in many fields including physics, biology, chemistry, engineering, astrophysics and others. This behavior can have different physical origins and has been found to occur frequently in spatially disordered systems, in turbulent fluids and plasmas, and in biological media with traps, binding sites or macro-molecular crowding. Langevin approach describes the Brownian motion in terms of a stochastic differential equation. The process of diffusion is driven by two physical parameters, the relaxation or correlation time tau and the velocity diffusivity coefficient Dv. An extension of the classical Langevin approach by means of a population of tau and Dv is here considered to generate a fractional dynamics. This approach supports the idea that fractional diffusion in complex media results from Gaussian processes with random parameters, whose randomness is due to the medium complexity. A statistical characterization of the complex medium in which the diffusion occurs is realized deriving the distributions of these parameters. Specific populations of tau and Dv lead to particular fractional diffusion processes. This approach allows for preserving the classical Brownian motion as basis and it is promising to formulate stochastic processes for biological systems that show complex dynamics characterized by fractional diffusion. A numerical study of this new alternative approach represents the core of the present thesis

Identification of transmembrane domains that regulate spatial arrangements and activity of prokineticin receptor 2 dimers

The chemokine prokineticin 2 (PK2) activates its cognate G protein-coupled receptor (GPCR) PKR2 to elicit various downstream signaling pathways involved in diverse biological processes. Many GPCRs undergo dimerization that can modulate a number of functions including membrane delivery and signal transduction. The aim of this study was to elucidate the interface of PKR2 protomers within dimers by analyzing the ability of PKR2 transmembrane (TM) deletion mutants to associate with wild type (WT) PKR2 in yeast using co-immunoprecipitation and mammalian cells using bioluminescence resonance energy transfer. Deletion of TMs 5-7 resulted in a lack of detectable association with WT PKR2, but could associate with a truncated mutant lacking TMs 6-7 (TM1-5). Interestingly, TM1-5 modulated the distance, or organization, between protomers and positively regulated Gαs signaling and surface expression of WT PKR2. We propose that PKR2 protomers form type II dimers involving TMs 4 and 5, with a role for TM5 in modulation of PKR2 function

The random diffusivity approach for diffusion in heterogeneous systems.

164 p.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

Detecting temporal correlations in hopping dynamics in Lennard–Jones liquids

Lennard–Jones mixtures represent one of the popular systems for the study of glass-forming liquids. Spatio/temporal heterogeneity and rare (activated) events are at the heart of the slow dynamics typical of these systems. Such slow dynamics is characterised by the development of a plateau in the mean-squared displacement (MSD) at intermediate times, accompanied by a non-Gaussianity in the displacement distribution identified by exponential tails. As pointed out by some recent works, the non-Gaussianity persists at times beyond the MSD plateau, leading to a Brownian yet non-Gaussian regime and thus highlighting once again the relevance of rare events in such systems. Single-particle motion of glass-forming liquids is usually interpreted as an alternation of rattlingwithin the local cage and cage-escape motion and therefore can be described as a sequence of waiting times and jumps. In this work, by using a simple yet robust algorithm, we extract jumps and waiting times from single-particle trajectories obtained via molecular dynamics simulations. We investigate the presence of correlations between waiting times and find negative correlations, which becomes more and more pronounced when lowering the temperature.European Commission European Commission Joint Research Centre 847693 840195-ARIADNEPolish National Agency for Academic Exchange (NAWA

Single-trajectory spectral analysis of scaled Brownian motion

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, $T\to \infty$. 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 $T\to \infty$ 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 single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.Open Access Publication Fund of Potsdam University