Stochastic processes and applications to complex systems

The temporal evolution of nonequilibrium systems is governed by the interplay of nonlinear dynamics and dynamic fluctuations generated by the fast and irregular motion of the microscopic constituents. The investigation of such systems requires
the application of methods from the theory of stochastic processes. The following issues are currently under investigation:

• Motion of diffusive particles in random media
• Continuous representations of L'evy processes and superdiffusion
• Continuous time random walks
• Subdiffusive nonlinear reaction diffusion systems

The work on diffusion in random media aims at determining the currents in disordered media influenced by external stationary and time periodic electric fields.  It is investigated how the nonlinear conductivity of ionic conductors is influenced by randomness. This work is performed in collaboration with Prof. Roling (Marburg) and Prof. Heuer (Münster) within the SFB 458: Ionic Motion in Materials with Disordered Structure-From Elementary Steps to Macroscopic Transport.

In collaboration with Prof. I. Lyubashewski (Moscow) and Prof. Heuer (Münster) we have derived a microscopic description of L'evy processes, which are used
to model superdiffusion in a variety of systems ranging from fluid turbulence to the diffusion of cells.

Continous time random walks are extensions of the usual random walk processes, which take into account trapping events of particles in localized structures. The work on  CTRW's is  motivated by the investigation of anomalous transport in plasma turbulence. In detail we investigate the CTRW of particles with inertia. Furthermore, numerical tools for the calculations of realizations of CTRW's are developed.

Our work on continuous time random walks has enabled us to formulate nonlinear reaction diffusion equations for chemical species moving subdiffusively due to trapping events. Our investigations show that trapping of chemical species and continous reaction leads to nonlinear diffusion with temporal memory.