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**Lattice Lorentz Gas:**

The random walk of a single tracer particle on a lattice with randomly distributed obstacles is known as lattice Lorentz gas. It can be treated analytically for all times in the limit of small obstacle densities. 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. Such long-time anomalies constitute a generic feature of much more complicated systems like Brownian motion of a mesoscopic particle suspended in a solvent where hydrodynamic memory is considered.

Fig. 1: Random walk of tracer on a lattice in the presence of obstacles.

Since the velocity autocorrelation function in the lattice Lorentz gas is known for all times, 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.

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. We explore this regime of confinement by simulating hard-spheres confined in a slit or wedge geometry. Our study suggests 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 very small tilt angle can promote a liquid-glass coexistence. Since there is a growing interest for smaller system in modern applications, these new findings can be utilized for future technological advancements.

Fig. 2: Hard-sphere liquids confined in a wedge geometry.

**Dynamics of self-propelled particles:**

Locomotion by swimming is a crucial aspect to optimize survival strategies of microorganisms such as bacteria, unicellular protozoa, or spermatozoa. Recently, physicists have been able to build artificial self-propelled particles that mimic the motion of the microorganisms by converting some energy supply into directed motion, thereby producing entropy at a steady rate. These self-propelled particles are driven out of equilibrium and show interesting statistical properties, that differ significantly from those of their passive counterparts. We explore theoretically the statistical properties of a single self-propelled particle in terms of Langevin equations. The intermediate scattering function, that contains the full information about the stochastic process, for these microswimmers is analytically derived and tested against computer simulations. These findings can help to further understand phenomena arising at the collective level.

**Liquid of Infinitely Thin Needles:**

Liquids of infinitely thin needles 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. With increasing density of needles transport slows down drastically and algebraic decay of the self-diffusion coefficients is obtained in the dense regime.

The main quantity of interest for the dynamics of a needle is the intermediate scattering function which encodes all moments of the movement of the needle. It can be calculated analytically for a single needle in the absence of other needles and it can be used to describe the dynamics a needle in the presence of other needles in the dense regime. This effective description is obtained by inserting the long-time diffusion coefficients obtained from simulations into the analytic theory of a single needle.