- Exact solutions in statistical physics
- Brownian motion and random walks
- Computer simulation of stochastic processes
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 small obstacle density limit. A remarkable feature of this simple model is the existence of persistence 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.
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.
Fig. 2: Snapshot of needles.
- S. Leitmann and T. Franosch, “Nonlinear Response in the Driven Lattice Lorentz Gas“, Physical review letters 111 (19), 190603 (2013).
- S. Leitmann, F. Höfling, and T. Franosch, “Tube Concept for Entangled Stiff Fibers Predicts Their Dynamics in Space and Time” Physical Review Letters (117) 097801 (2016)
- S. Leitmann and T. Franosch, “Time-Dependent Fluctuations and Superdiffusivity in the Driven Lattice Lorentz Gas” Physical Review Letters (118) 018001 (2017)
- C. Kurzthaler, S. Leitmann, and T. Franosch, “Intermediate scattering function of an anisotropic active Brownian particle” Scientific Reports 6, Article number: 36702 (2016)