Lennard Kossmann

Superballisticity in anomalous diffusion [PDF]
finished 2026-05
supervised by Michael Schmiedeberg
in the group of Prof. Vasily Zaburdaev

Abstract

Studying the motion of biological cells is an inherently ambiguous endeavor due to the random nature of cell motility. Comparing the different motility patterns of cells is mostly based on estimating the displacement 𝑥𝑟 − 𝑥𝑠 between times 𝑠 2, which is based on motility experiments of NK cells in biological tissue. This type of scaling suggests an underlying acceleration mechanics in the cell motility, which indicates active propulsion or a complex force field in the surrounding matter. In contrast, both the experiment and my calculations showed, that the TAMSD scales at most ballistic, i.e. 𝛼𝑇𝐴 = 2, in these cases. One important tool for understanding the scaling of these two MSDs is the velocity autocorrelation function (VACF) 𝐶𝑣 (𝑟, 𝑠) = ⟨𝑣𝑟 𝑣𝑠 ⟩. I investigate the properties of the VACF from a more theoretical perspective and show how it influences the scaling of both EAMSD and TAMSD directly and indirectly. A special role is given to stationary VACFs 𝐶𝑣 (𝑟, 𝑠) = 𝐶(𝑟 − 𝑠), where the time – dependence lies solely in the lagtime 𝑟 − 𝑠. The EAMSD and TAMSD agree for such systems, but the possibility of superballistic scaling is very narrow. More general, non-stationary VACFs show a greater variability in their EAMSD scaling, but their TAMSD does not cross the superballistic threshold. The underlying reason for this seems too complex to be answered in full generality, but for the cases of this thesis I demonstrate that the TAMSD scaling is determined by the stationary part of the VACF. This resolves the contrast to the EAMSD scaling, which is mostly affected by the non-stationary part. This thesis and the explored dichotomy between the EAMSD and TAMSD scaling highlights the difficulty in describing cell motility and other forms of stochastic motion both qualitatively and quantitatively, but also hints on how the behavior of the TAMSD can be utilized to attain a deeper understanding of the underlying mechanics itself.