CS15: Complex systems III

date: 7/15/2025, time: 14:00-15:30, room: ICS 118

Organizer: Marek Teuerle (Wroclaw University of Science and Technology)

Chair: Agnieszka Wyłomańska (Wroclaw University of Science and Technology)

Spurious (non-)ergodicity and stochastic self-similarity

Aleksei Chechkin (Max Planck Institute of Microstructure Physics, Halle; Wroclaw University of Science and Technology)

Ergodicity and self-similarity are two important concepts widely employed in applications of stochastic processes. We suggest to improve theoretical and experimental analysis of ergodicity used in single particle tracking by incorporating mean squared increments (MSI) for comparison with time-ensemble average mean squared displacement (TEAMSD). Moreover, by using several examples of the generic random motions in complex systems we demonstrate serious drawbacks of the routine analysis based on the comparison of mean squared displacement (MSD) with TEAMSD. Furthermore, we show that the concept of stochastic self-similarity is needed for the description of random motions in complex systems.

Tempered Heavy-Tailed Correlated Noise in Stochastic Optimization Theory

Farzad Sabzikar (Iowa State University)

In modern AI, efficiently optimizing complex non-convex functions is a critical challenge, often demanding vast computational resources and innovative approaches to avoid getting trapped in local minima. Past studies have leveraged fractional Gaussian noise to enhance stochastic gradient descent—demonstrating its ability to overcome obstacles and escape saddle points in intricate landscapes. However, these methods can still fall short in balancing exploration with rapid convergence. To bridge this gap, we analyze a continuous-time stochastic process driven by tempered fractional stable noise (TFSN). This noise model extends the traditional fractional Gaussian framework by integrating heavy-tailed distributions with a tempering mechanism, effectively capturing both gradual long-range dependencies and sudden extreme fluctuations in complex systems. We develop a discretization strategy that leverages the unique properties of TFSN, which underpins the formulation of our new algorithm, tempered fractional perturbed gradient descent (TFPGD). In this study, we construct a theoretical framework for the practical application of TFSN in stochastic optimization. Our theoretical and empirical analyses reveal that TFPGD achieves faster convergence rates, higher success ratios, and more efficient escape from local minima compared to the state-of-the-art fractional version (FPGD).

Mean square displacement analysis for heterogeneous anomalous diffusion

Jakub Ślęzak (Hugo Steinhaus Center, Wrocław University of Science and Technology)

Mean square displacement (MSD) is a fundamental tool in the analysis of diffusion in complex media. When the system is heterogeneous, as much information as possible must be extracted from each single trajectory, which often have short lengths. This is made by time-averaging squared increments, and then the main line of inquiry is fitting the shape of the obtained time averaged MSD values in the log-log space. However, classical regression methods are not suited for this problem due to the correlations introduced during time-averaging and also intrinsic to anomalous diffusion systems. We analyse how to enhance it using custom generalised least squares method (GLS) which does not have such limitations and consequently can reduce the variance and bias of the diffusion parameter estimates, especially for short (\(\approx 100\) points) and ultra short (\(\approx 10\) points) trajectories. The method is mostly automatic and does not require supervision. It also enables us to predict the estimation errors which we use to reconstruct the underlying particle ensemble structure.