Conference Agenda

Experimentally well-established, anomalous diffusion (AD) is a phenomenon observed in many different natural systems belonging to different research fields. In particular, AD has become fundational in living systems after a large use of single-particle tracking techniques in the recent years. Generally speaking, AD labels all those diffusive processes that are governed by laws that differ from that of classical diffusion, namely, all that cases when particles' displacements do not accomodate to the Gaussian density function and/or the variance of such displacements does not grow linearly in time.

We propose an attempt for establishing the physical origin of AD within the classical picture of a test-particle kicked by infinitely many surrounding particles. We consider a stochastic dynamical system where the microscopic thermal bath is the source for the mesoscopic Brownian motion of a bunch of $N$ particles that express the environment of a single test-particle. Physical conservation principles, namely the conservation of momentum and the conservation of energy, are met in the considered particle system in the form of the fluctuation-dissipation theorem for the motion of the surround-particles. The key feature of the considered particle-system is the distribution of the masses of the particles that compose the surround of the test-particle. When the number of mesoscopic Brownian particles $N$ is large enough for providing a crowded environment, then the test-particle displays AD characterised by the distribution of the masses of the surround-particles. More precise, we prove that, in the limit $N\to\infty$, the test-particle diffuses according to a quite general non-Markovian Gaussian process $(Z_t)_{t\geq0}$ characterised by a covariance function \begin{align}\label{eq:Covariance} Cov(Z_t,Z_s)=C(v(t)+v(s)-v(|t-s|)), \end{align} where $v(\cdot)$ is determined by the distribution of the masses of the surround-particles. With a particular choice of distribution of surround-particles, we obtain fractional Brownian motion (fBm) with Hurst parameter $H\in(1/2,1)$ as a special case. In this respect, we remind that the fBm experimentally turned out to be the underlying stochastic motion in many living systems. We present also some distributions of masses of the surround-particles which lead to a mixture of independent fractional Brownian motions with diferent Hurst parameters or to a classical Brownian motion as a limiting process $(Z_t)_{t\geq0}$. Moreover, we present some distributions of masses of the surround-particles leading to the limiting processes which perform a transition from ballistic diffusion to superdiffusion, or from ballistic diffusion to classical diffusion.

Furthermore, the constant $C$ in the above formula for the covariance of the process $(Z_t)_{t\geq0}$ depends on the coupling parameter between the test-particle and the surround. Therefore, if we consider several independent identical copies of the same Brownian surround and immerse into each copy of the surround a single test particle of the same art but with its own individual characteristics (assuming our test-particle is a complex macromolecule with its individual shape, radius, densty e.t.c.), we may obtain different coupling parameters and hence different coefficients $C$ in the covariance of the limiting Gaussian process $(Z_t)_{t\geq0}$ in different copies of the experiment. This fact may serve as a physical basis for the formulation of AD within the framework of the superstatistical fBm, where further randomness is provided by a distribution of the diffusion coeffients associated to each diffusing test-particle and also within the framework of its generalisation called diffusing-diffusivity approach, where the diffusion coefficient of each test-particle is no longer a random variable but a process.

The proof of our result is based on properties of the Ornstein--Uhlenbeck processes that describe the dynamics of the Brownian surround-particles in the exact fashion of the Langevin equation and on a kind of Central-Limit-Theorem arguments which however deal with a worser scaling than the one in the classical CLT. A worser scaling is however compensated by good properties of Ornstein--Uhlenbeck processes.

We study multivariate kinetic-type equations in the general case, which includes a.o. the spatially homogeneous Boltzmann equation with Maxwellian molecules, both with elastic and inelastic collisions. Assuming, that the collisional kernel is of the form, derived by Bassetti et.al. in \cite{1}, we prove the existence and uniqueness of time-dependent solutions with the help of continuous-time branching random walks, under weakest assumptions possible. We further derive an exact representation of stationary solutions, e.g. equilibrium solutions for kinetic-type equations, using the central limit theorem for triangular null arrays. \footnotesize \begin{thebibliography}{1} \bibitem{1} {F. Bassetti, L. Ladelli and D. Matthes} (2015). Infinite energy solutions to inelastic homogeneous Boltzmann equations. {\em Electronic Journal of Probability}. 20:1--34. \end{thebibliography}

We present a new stochastic model for the sulphation process of calcium carbonate at the microscale, focusing on the chemical reaction that leads to the formation of gypsum and to the consequent marble degradation, which is relevant in Cultural Heritage conservation.

The Langevin dynamics of the sulfuric acid particles is described via first order stochastic differential equations (SDEs) of It\^o type, while calcium carbonate and gypsum are modelled as underlying random fields evolving according to random ODEs. Furthermore, particles interact pairwise via a strongly singular potential of Lennard Jones type. The system is finally coupled with a marked Poisson compound point measure for realizing the chemical reactions.

We discuss the well-posedness of the system for a broad class of singular potentials, including Lennard Jones, by proving that, almost surely, particle collisions do not occur in a finite time.

This is a joint work with Daniela Morale, Stefania Ugolini (University of Milano) and Adrian Muntean, Nicklas Javergard (University of Karlstad).