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We introduce the concept of Randomly Modulated Gaussian Processes as a unifying framework for modeling, analyzing and classifying anomalous diffusion models in heterogeneous media. This formulation incorporates correlations in the…

Biological Physics · Physics 2026-03-16 Yann Lanoiselée , Denis S. Grebenkov , Gianni Pagnini

Diffusive motion is a fundamental transport mechanism in physical and biological systems, governing dynamics across a wide range of scales -- from molecular transport to animal foraging. In many complex systems, however, diffusion deviates…

Statistical Mechanics · Physics 2026-05-01 Takuma Akimoto , Jae-Hyung Jeon , Ralf Metzler , Tomoshige Miyaguchi , Takashi Uneyama , Eiji Yamamoto

The comb model is a simplified description for anomalous diffusion under geometric constraints. It represents particles spreading out in a two-dimensional space where the motions in the x-direction are allowed only when the y coordinate of…

Computational Physics · Physics 2015-07-21 H. V. Ribeiro , A. A. Tateishi , L. G. A. Alves , R. S. Zola , E. K. Lenzi

Machine learning (ML) has become a versatile tool for analyzing anomalous diffusion trajectories, yet most existing pipelines are trained on large collections of simulated data. In contrast, experimental trajectories, such as those from…

Biological Physics · Physics 2025-12-10 Gongyi Wang , Yu Zhang , Zihan Huang

We study a process of anomalous diffusion, based on intermittent velocity fluctuations, and we show that its scaling depends on whether we observe the motion of many independent trajectories or that of a Liouville-like equation driven…

Statistical Mechanics · Physics 2009-11-07 M. Bologna , P. Grigolini , B. J. West

Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local…

Statistical Mechanics · Physics 2026-01-16 Gabriel Barreiro , Vladimir Pérez-Veloz

Anomalous diffusion is discussed in the context of quantum Brownian motion with colored noise. It is shown that earlier results follow simply and directly from the fluctuation-dissipation theorem. The limits on the long-time dependence of…

Quantum Physics · Physics 2007-05-23 G. W. Ford , R. F. O'Connell

Anomalous diffusion, process in which the mean-squared displacement of system states is a non-linear function of time, is usually identified in real stochastic processes by comparing experimental and theoretical displacements at relatively…

Data Analysis, Statistics and Probability · Physics 2013-05-29 Serge F. Timashev , Yuriy S. Polyakov , Pavel I. Misurkin , Sergey G. Lakeev

The random motion of molecules in living cells has consistently been reported to deviate from standard Brownian motion, a behavior coined as ``anomalous diffusion''. Fluorescence Correlation Spectroscopy (FCS) is a powerful method to…

Quantitative Methods · Quantitative Biology 2024-07-18 Nathan Quiblier , Jan-Michael Rye , Pierre Leclerc , Henri Truong , Abdelkrim Hannou , Laurent Héliot , Hugues Berry

Early detection of anomalies in medical images such as brain MRI is highly relevant for diagnosis and treatment of many conditions. Supervised machine learning methods are limited to a small number of pathologies where there is good…

Computer Vision and Pattern Recognition · Computer Science 2023-12-05 Alexander Frotscher , Jaivardhan Kapoor , Thomas Wolfers , Christian F. Baumgartner

In this paper we deal with high-order corrections for the Fractional Derivative approach to anomalous diffusion, in super-diffusive regime, which become relevand whenever one attempts to describe the behavior of particles close to normal…

Statistical Mechanics · Physics 2007-06-15 M. Marseguerra , A. Zoia

Fractional Brownian motion (fBm) is a ubiquitous diffusion process in which the memory effects of the stochastic transport result in the mean squared particle displacement following a power law, $\langle {\Delta r}^2 \rangle \sim…

Applied Physics · Physics 2020-10-06 Raviteja Vangara , Kim Ø. Rasmussen , Dimiter N. Petsev , Golan Bel , Boian S. Alexandrov

The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due…

Statistical Mechanics · Physics 2024-07-10 Michał Balcerek , Agnieszka Wyłomańska , Krzysztof Burnecki , Ralf Metzler , Diego Krapf

In recent years, research and development in nanoscale science and technology have grown significantly, with electrical transport playing a key role. A natural challenge for its description is to shed light on anomalous behaviours observed…

Populations and Evolution · Quantitative Biology 2025-04-10 Sara Bernardi , Paolo Begnamino , Marco Pizzi , Lamberto Rondoni

The task of detecting anomalous data patterns is as important in practical applications as challenging. In the context of spatial data, recognition of unexpected trajectories brings additional difficulties, such as high dimensionality and…

Fractional Brownian motion, a stochastic process with long-time correlations between its increments, is a prototypical model for anomalous diffusion. We analyze fractional Brownian motion in the presence of a reflecting wall by means of…

Statistical Mechanics · Physics 2018-02-21 Alexander H. O. Wada , Thomas Vojta

A generalization of the Drude model is studied. On the one hand, the free motion of the particles is allowed to be sub- or superdiffusive; on the other hand, the distribution of the time delay between collisions is allowed to have a long…

Condensed Matter · Physics 2009-10-30 Hermann Schulz-Baldes

We investigate the dynamics of a single tracer exploring a course of fixed obstacles in the vicinity of the percolation transition for particles confined to the infinite cluster. The mean-square displacement displays anomalous transport,…

Statistical Mechanics · Physics 2015-03-17 Markus Spanner , Felix Höfling , Gerd Schröder-Turk , Klaus Mecke , Thomas Franosch

From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such…

Statistical Mechanics · Physics 2021-01-04 E G Kostadinova , J L Padgett , C D Liaw , L S Matthews , T W Hyde

The motion of self-propelled particles is modeled as a persistent random walk. An analytical framework is developed that allows the derivation of exact expressions for the time evolution of arbitrary moments of the persistent walk's…

Soft Condensed Matter · Physics 2015-07-28 Zeinab Sadjadi , M. Reza Shaebani , Heiko Rieger , Ludger Santen