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

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

Glassy phases of the Gaussian Core Model

We present results from molecular dynamics simulations exploring the supercooled dynamics of the Gaussian Core Model in the low- and intermediate-density regimes. In particular, we discuss the transition from the low-density hard-sphere-like glassy dynamics to the high-density one. The dynamics at low densities is well described by the caging mechanism, giving rise to intermittent dynamics. At high densities, the particles undergo a more continuous motion in which the concept of cage loses its meaning. We elaborate on the idea that these different supercooled dynamics are in fact the precursors of two different glass states

Towards a robust criterion of anomalous diffusion

Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian -- or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.Comment: 13 pages, 6 figures, RevTe