Fractional and fuzzy-fractional transport in disordered environments

• Applicant: Professor Dr. Ralf Metzler
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 537528804

Normal Brownian motion in the hydrodynamic limit is characterised by a linear-in-time mean-squared displacement and a Gaussian probability density function. Mathematically, Brownian motion is described by the Wiener process. Deviations from the linear mean squared displacement, typically in the form of a power-law time dependence with scaling exponent alpha, are called "anomalous diffusion" ("subdiffusion" when alpha is smaller than unity, "superdiffusion" otherwise). While a number of stochastic models for anomalous diffusion were devised a considerable time ago, the high complexity of the dynamics revealed in ever more detailed data, e.g., from single-particle tracking in living biological cells or from supercomputing studies, requires increasingly refined models. At the same time many experiments are corrupted by strong measurement noise, by pronounced fluctuations of molecular reaction times, and by strong disorder and/or a rapidly changing environment. In such cases it may be more appropriate to use intrinsically noisy formalisms, such as the concept of fuzzy numbers. In the current project we aim at further developments of anomalous diffusion models and their combination with fuzzy calculus. The result will be physically motivated dynamic models based on dynamic equations with long memory, mathematically conveniently expressed in terms of fractional differential operators, i.e., fractional calculus, and their fuzzy generalisations.

• DFG Programme: Research Grants
• International Connection: Czech Republic
• Partner Organisation: Czech Science Foundation
• Cooperation Partners: Professorin Dr. Irina Perfiljeva; Professor Dr. Zivorad Tomovski