# Research topics

Our research focuses on*non-equilibrium statistical mechanics*,

*soft matter*and

*theoretical biological physics*. Key topics include the theory and applications of normal and anomalous stochastic processes, gene regulation, crowding in biological cells, and (bio)polymer physics. Effects of disorder, annealed or quenched, interacting particles, ergodicity and its violation, as well as ageing are studied. Our methods are analytics, numerics, and simulations. We collaborate with a number of international groups.

## Anomalous diffusion

As one of the internationally leading groups we study stochastic processes, in which the mean squared displacement (MSD) deviates from the linear form*<*known from Brownian motion. Mostly we are interested in processes following the power law form

**r**^2(t)>∼t*<*where

**r**^2(t)>∼t^{α}*α≠1*, distinguishing subdiffusion (

*0<α<1*) and superdiffusion (

*α>1*), but we also study ultraslow processes with

*<*log

**r**^2(t)>∼*(t)*such as Sinai diffusion. In particular we are interested in the physical origins of anomalous diffusion, but also in the inference of parameters and mechanisms from measured data. Apart from biological systems our theories find application in solid state physics, physical chemistry, econophysics, and movement ecology.

## Ergodicity and ageing in physical systems

In standard statistical mechanics courses we learn in the spirit of Boltzmann that the long time average of a physical observable, obtained from following a single particle, should provide the identical information as the ensemble average of this observable garnered from following many particles. In many cases this ergodic behaviour is broken. Motivated by a growding body of evidence garnered by superresolution microscopy, we study non-ergodic systems in detail and demonstrate that the understanding of this phenomenon is pivotal for the proper physical interpretation of the dynamics of many complex systems. Concurrently, many of these systems are non-stationary, such that their dynamics changes over time: they are ageing. In our analyses of ageing systems we demonstrate interesting crossover phenomena dependening on the length of the observation time.## Normal diffusion and first passage processes

Even for normal Brownian processes a large number of questions remains open. One of our focus points of study is the effect of geometry and heterogeneity on the dynamics of a test particle. In particular, we are interested in the analytical study of first passage processes, that is the statistics of when a particle reaches a given location in a physical space for the first time. Our current research demonstrates that the typical analysis in terms of the mean first passage time may provide insufficient information on specific processes, in particular, when small concentrations of particles are involved.## Physics of molecular crowding

The cytoplasm of biological cells is*superdense*, crowded by larger biopolymers at volume fractions of some 35%. Sometimes cells can be even more crowded (supercrowded) by large inclusions such as granules or vesicles. We study how such crowding affects the dynamics of both passive diffusion and active motion propelled by molecular motors in such crowded environments. Important clues also come from experimental collaborators. Crowding also occurs in two-dimensional membrane systems, which are studied by partner groups via large scale simulations and experiments. We are interested in a systematic investigation of the physical behaviour of crowded systems.