The random walk of a single tracer particle on a lattice with randomly distributed obstacles is known as a lattice Lorentz gas. A remarkable feature of this simple model is the existence of persistent memory in the form of long-time tails in the velocity autocorrelation function of the tracer. The fluctuation-dissipation-theorem can be used to monitor the response of the system under the influence of a small force pulling on the particle. This linear response result can be directly compared to an exact nonlinear response solution for any strength of the force, showing key features of nonlinear driving such as breakdown of persistent memory as well as nonanalytic behavior of the stationary state velocity. 

• S. Leitmann and T. Franosch, Phys. Rev. Lett. 118 (2017)  

In a simple extension to continuous space, known as the Lorentz model, a single particle explores the space between randomly distributed fixed obstacles. As the density of obstacles is increased, this model exhibits a localization transition, above which the particle is confined. Strikingly, from this simple model, subdiffusive motion is predicted at the transition point, in line with what is observed in the crowded environment of biological cells. The nature of this divergence from diffusive behavior is governed by the percolation transition of the geometry of the obstacles, and the specifics of the interaction dynamics between our moving particle and the obstacles.

• C. F. Petersen, T. Franosch, Soft Matter, 15, 3906 (2019) 
• M. Spanner et. al. Phys. Rev. Lett. 116 (2016)

 Single-particle behavior. We investigate the spatiotemporal dynamics of various types of self-propelled agents in terms of exact solutions for the intermediate scattering function, which constitutes the Fourier transform of the probability density of the particle's displacements. In particular, we are interested in the dynamical behavior of catalytic Janus colloids, which can be completely characterized by the paradigmatic active Brownian particle model, and the run-and-tumble behavior of E. coli bacteria using renewal processes. We compare our theoretical predictions to observations of these synthetic and biological micro-swimmers measured by our experimental collaborators. 

• C. Kurzthaler et al., Sci. Rep. 6 (2016), Soft Matter 13 (2017), Phys. Rev. Lett. 121 (2018) 

Active transport in heterogeneous environments. The studies of active particles are usually performed in homogeneous clean spaces, yet in nature their motion is often obstructed by different obstacles. We focus on the different aspects of the interactions of particles with the environment and study their transport properties, which are very important in the context of biological and technological applications. Our goal is to develop a detailed understanding of the dynamics of micro-swimmers in different heterogeneous environments. We perform theoretical investigations of these intricate transport phenomena on the level of computer simulations and analytical analyses.

• W. Schirmacher et al., Phys. Rev. Lett. 115 (2015)
• O. Chepizhko, T. Franosch, Soft Matter, 15, 452-461 (2019), New J. Phys. 22 (2020)

Transport of liquids in nano-sized pores or narrow channels is crucial for most natural and industrial processes. It can exhibit a series of fascinating behaviors when the confinement length scale is comparable to the molecular size.

•  C. F. Petersen et al., J. Stat. Mech. 083216 (2019) 

We explore this regime of confinement by investigating hard spheres confined in a slit, wedge, or torus geometry in terms of a mode coupling theory and extensive computer simulations. Our studies suggest that the confinement length scale appears as another control parameter to the non-equilibrium phase diagram. In particular, we find that a wedge-like confinement with very small tilt angle can promote a liquid-glass coexistence. Since there is a growing interest for smaller systems in modern applications, these findings can be utilized for future technological advancements.

• S. Mandal et al., Nature. Comun. 5 (2014), Phys. Rev. Lett. 118 (2017), Soft Matter 13 (2017) 
• G. Jung, C. F. Petersen, Phys. Rev. Res. 2 (2020)

Thermophoresis is referred to the response of particles to temperature gradients. This phenomenon can be exploited for microfluidic applications and has also been argued to be one of the key mechanisms for the origin of life.

We are interested in the description of a single charged colloid immersed in an electrolyte. In the presence of a temperature gradient, electrical body forces are no longer counter balanced by pressure gradients, thereby, inducing solvent flow and the directed motion of the colloidal particle. We solve numerically the associated field equations including the non-linear Poisson Boltzman equation for the electrical potential and the steady Stokes equation governing the fluid flow. Ultimately, we aim to extract transport properties, such as the Soret coefficient, of experimental systems.

We aim to understand the response of single semiflexible polymers to compression forces in terms of the celebrated wormlike chain model. We derive analytic solutions for the experimentally accessible force-extension relations and elucidate the elastic behavior with respect to the classical Euler buckling instability of a rigid rod. Moreover, we characterize the buckling behavior by computing the associated probability densities.

• C. Kurzthaler, T. Franosch, Phys. Rev. E 95 (2017), Soft Matter 14 (2018) 

We study dynamically crowded solutions of stiff fibers, which have the remarkable property that their static properties are that of an ideal gas, yet their dynamics is governed by a complex behavior since any two needles are not allowed to cross each other. We elaborate analytic expressions for the intermediate scattering function to characterize the spatiotemporal dynamics of a single needle. We use these predictions to elucidate the intricate transport properties of solutions of stiff fibers in the semidilute regime, where the motion of a single needle becomes confined to a narrow tube. This effective description is obtained by inserting the long-time diffusion coefficients obtained from simulations into the analytic theory of a single needle.

• S. Leitmann et al., Phys. Rev. Lett. 114 (2016), Phys. Rev. E 96 (2017)