Anomalous Diffusion of Superparamagnetic Walkers with Tailored Statistics

Abstract: From life sciences and ecology to quantum physics and finance, anomalous diffusion appears in countless complex systems as a signature of emergent transport properties beyond Brownian motion. Despite substantial theoretical progress, the experimental study of real-world systems exhibiting anomalous diffusion remains challenging due to an intrinsically elusive ground truth and the limited information contained in typical single trajectories. Here, unlike previous experimental systems, we demonstrate the controlled generation of two-dimensional trajectories with fully tailored statistics spanning the entire spectrum of anomalous diffusion, from subdiffusion to superdiffusion, and over statistically significant temporal and spatial scales (covering at least two decades). We achieve this feat by simultaneously tuning the step-length distribution and, critically, the velocity autocorrelation function of microscopic superparamagnetic colloidal walkers with magnetic fields during extended acquisitions. Supported by theoretical reasoning, fine control of these two quantities combined allows us to generate trajectories compatible with L\'evy walks and fractional Brownian motion with tailored anomalous diffusion exponents. We envisage our approach will offer a robust, controllable experimental framework for validating and advancing theoretical models, analysis techniques, and predictive tools to study anomalous diffusion in real-life phenomena. These include transport in physical and biological systems, animal movement, human ecology, and financial markets.