CS13: Complex systems I

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

Organizer: Krzysztof Burnecki (Wrocław University of Science and Technology)

Chair: Marcin Magdziarz (Wrocław University of Science and Technology)

Langevin equation in quenched heterogeneous landscapes

Diego Krapf (Colorado State University)

The Langevin equation is a common tool to model diffusion at the single-particle level. In nonhomogeneous environments, such as aqueous two-phase systems or biological condensates with different diffusion coefficients in different phases, the solution to a Langevin equation is not unique unless the interpretation of stochastic integrals involved is selected. We analyse the diffusion of particles in such systems and evaluate the mean, the mean square displacement, and the distribution of particles, as well as the variance of the time-averaged mean-squared displacements \([1]\). We provide solutions as a function of the interpretation parameter \(\alpha\), with particular attention to the Itô, Stratonovich, and Hänggi-Klimontovich interpretations, revealing fundamentally different behaviours. Furthermore, heterogeneous diffusion is also considered when the particle is subjected to an external force. Our analytical results provide a method to choose the interpretation parameter from single particle tracking experiments.

Bibliography

\([1]\) A. Pacheco-Pozo, M. Balcerek, A. Wyłomańska, K. Burnecki, I.M. Sokolov, and D. Krapf. "Langevin equation in heterogeneous landscapes: how to choose the interpretation." Phys. Rev. Lett. , vol. 133, 2024, pp. 067102.

Two-dimensional fractional Brownian motion: construction and analysis

Michał Balcerek (Wrocław University of Science and Technology)

In this presentation, I will focus on a extension of Mandelbrot and van Ness’s representation \([1]\) of fractional Brownian motion to a multivariate case. By utilizing correlated Gaussian noises and matrix-valued Hurst exponents, I propose a natural construction of such a process that can model anisotropic anomalous diffusion in complex systems. I define and analyze two versions of such a process: causal and well-balanced \([2]\). For both of them, I present the covariance and cross-covariance structure, and power-spectral density and its asymptotics. It is an alternative construction to one presented in \([3]\), that describes a general operator fractional Brownian motion.

Bibliography

\([1]\) Benoît B. Mandelbrot and John W. Van Ness. Fractional Brownian Motions, Fractional Noises and Applications. SIAM Review, 10:422, 1968.

\([2]\) Stilian A. Stoev, and Murad S. Taqqu. “How rich is the class of multifractional Brownian motions?”. Stochastic Processes and Their Applications 116.2, 2006, pp. 200-221.

\([3]\) Vladas Pipiras and Murad S Taqqu (2017). Long-range dependence and self-similarity. Cambridge University Press.

A souvenirs collector’s walk: The distribution of the number of steps of a continuous time random walk ending at a given position

Igor M. Sokolov (Humboldt-Universität zu Berlin)

We consider a random walker performing a continuous time random walk (CTRW) with a symmetric step lengths’ distribution possessing a finite second moment and with a power-law waiting time distribution with finite or diverging first moment. The problem we pose concerns the distribution of the number of steps of the corresponding CTRW conditioned on the final position of the walker at some long time t. For positions within the scaling domain of the probability density function (PDF) of final displacements, the distributions of the number of steps show a considerable amount of universality, and are different in the cases when the corresponding CTRW corresponds to subdiffusion and to normal diffusion. We moreover note that the mean value of the number of steps can be obtained independently, and follows from the solution of the Poisson equation with the right-hand side depending on the PDF of displacements only. This approach works not only in the scaling domain, but also in the large deviation domain of the corresponding PDF, where the behavior of the mean number of steps is very sensitive to the details of the waiting time distribution beyond its power-law asymptotics.

Bibliography

I.M. Sokolov, Physical Review E, Accepted 23 May, 2025