Anomalous Transport and Anomalous Diffusion

Scientific Programme

Lectures

• Luisa Beghin - Anomalous Diffusion, Lévy Processes, and Fractional Calculus

  The aim of the course is to provide a comprehensive introduction to the mathematical foundations of anomalous diffusion and its modeling through fractional calculus.

  As a first step, we will review Markov processes and introduce Lévy processes as stochastic models characterized by stationary and independent increments. We will then explore the fundamental concepts of fractional operators (e.g. Riemann-Liouville, Caputo, Riesz derivatives and integrals), highlighting their inherent non-locality and application in modeling physical memory effects. We also analyze the connection between the microscopic dynamics—specifically the Continuous-Time Random Walk (CTRW) model—and its macroscopic limit via the Stable Central Limit Theorem, which generates the stable Lévy distributions. Finally, the course will cover the derivation and solution of fractional diffusion equations both in the time- and the space-fractional cases, as tools for modeling sub- and super-diffusions.

• Zhen-Qing Chen and Takashi Kumagai - Anomalous subdiffusion and time-fractional differential equations

  Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells, to the foraging behavior of animals, to the travel times of contaminants in groundwater. This short course will consist of two parts.

  The first part will discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. We will then present some recent results in this area, in particular on the probabilistic representation to the solutions of time fractional equations with source terms. An interesting feature of the latter is that they involve two fundamental solutions. The second part will discuss applications of anomalous sub-diffusions to scaling limits of trap models, and the two-sided estimates of the fundamental solutions arise in the time-fractional parabolic equations.

• Tomasz Komorowski and Stefano Olla - Heat superdiffusion in one-dimensional oscillators chains

  This course provides an in-depth study of anomalous diffusion and energy transport in classical models of statistical mechanics, with particular emphasis on one-dimensional chains of interacting oscillators. Such systems originate in Debye’s early twentieth-century model of energy propagation in solids and achieved renewed significance through the seminal Fermi–Pasta–Ulam–Tsingou numerical experiments of the 1950s, which investigated the mechanisms by which oscillator chains approach thermal equilibrium. One-dimensional systems are especially compelling, as they generically exhibit anomalous—rather than classical diffusive—energy transport.
  After introducing the physical motivation and historical context, the course develops the hydrodynamic limits associated with conserved quantities in stochastically perturbed oscillator chains. Under hyperbolic scaling, in which space and time are rescaled at the same rate, anharmonic chains with stochastic perturbations give rise to macroscopic Euler-type equations. On longer time scales, the focus shifts to harmonic chains. In the presence of momentum-conserving stochastic perturbations, energy transport becomes superdiffusive and is governed by a heat equation involving a fractional Laplacian.
  The course concludes with an analysis of kinetic limits for both closed systems, which evolve in isolation, and open systems interacting with external heat reservoirs or external forcing. Particular attention is paid to the emergence of macroscopic boundary conditions and their role in the resulting transport equations.

Seminars

• Giada Basile - TBA

  TBA

• Krzysztof Bogdan - TBA

  TBA

• Tomasz Grzywny - TBA

  TBA

• Jozsef Lorinczi - TBA

  TBA

• Michal Ryznar - TBA

  TBA

• Marielle Simon - TBA

  TBA

• Bruno Toaldo - TBA

  TBA

• Vanja Wagner - TBA

  TBA