Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Anomalous Stochastic Processes

If the time-evolution of the mean-squared displacement of some quantity is non-linear,
the system is said to exhibit anomalous diffusion. The underlying mechanisms leading to such anomalous diffusion can be multifold. Our group focuses on processes whose
anomalous behavior is due to heavy-tailed distributions of either the waiting time
distribution between the displacements or of the displacements themselves.
 
 One topic of our group is the path integral description of Continuous Time Random Walks (CTRWs) which are random walks where the time between successive displacements is random and governed by some probability distribution. While a path integral formulation of CTRWs shows not only an interesting mathematical structure, it furthermore enables the application of many sophisticated approximation schemes for CTRWs in non-linear potentials or with multiplicative noise, which were originally invented in the context of quantum field theory.
 
Furthermore we are interested in the properties of two-dimensional anomalous diffusion processes. Here we use the concept of the convex hull which is the minimal convex
polygon enclosing a sample trajectory. We apply our results to model the home-range
of certain foraging animals.
 
Non-exponential waiting times cannot only significantly alter the transport properties
of complex systems but also change the birth and death dynamics of constituents within the system itself. These considerations lead to anomalous reaction kinetics. We focus on the mathematical description of reactions which have one reaction channel being governed by anomalous, non-Poissonian kinetics while all other channels exhibit regular, Poissonian kinetics. The developed models are relevant in many biological systems such as gene transcription and ion channel dynamics. 


Contact:  Stephan Eule 

Members working within this Project:

 Stephan Eule 

Former Members:

 Mirko Lukovic 

Selected Publications:

S. Eule (2016).
Response Behavior of Aging Systems with Temporal Disorder
J. Stat. Mech.:043210. download file

S. Eule, and J.J. Metzger (2016).
Non-equilibrium steady states of stochastic processes with intermittent resetting
New J. Phys. 18:033066. download file

S. Eule, and R. Friedrich (2014).
Path probabilities of continuous time random walks
J. Stat. Mech(P12005). download file

M. Lukovic, T. Geisel, and S. Eule (2013).
Area and perimeter covered by anomalous diffusion processes
New J. Phys. 15:063034. download file

S. Eule, V. Zaburdaev, R. Friedrich, and T. Geisel (2012).
Langevin description of superdiffusive Levy processes
Phys. Rev. E 86(041134).

H. Affan, R. Friedrich, and S. Eule (2009).
Anomalous diffusion in a field of randomly distributed scatterers
Phys. Rev. E 80:011137.

S. Eule, and R. Friedrich (2009).
Subordinated Langevin equations for anomalous diffusion in external potentials —Biasing and decoupled external forces
EPL 86(3):30008.

S. Eule, R. Friedrich, and F. Jenko (2008).
Anomalous transport in a one-dimensional Lorentz gas model
J. Chem. Phys 129(174308).

S. Eule, R. Friedrich, F. Jenko, and I. Sokolov (2008).
Continuous-time random walks with internal dynamics and subdiffusive reaction-diffusion equations
Phys. Rev. E (R)(78):060102.

S. Eule, R. Friedrich, and F. Jenko (2007).
Anomalous diffusion of particles with inertia in external Potentials
J. Phys. Chem. B 111:13041--13046.

S. Eule, R. Friedrich, F. Jenko, and D. Kleinhans (2007).
Langevin approach to fractional diffusion equations including inertial effects
J. Phys. Chem. B 111:11474--11477.

T. Hauff, F. Jenko, and S. Eule (2007).
Intermediate non-Gaussian transport in plasma core turbulence
Physics of Plasmas 14(102316).

R. Friedrich, F. Jenko, A. Baule, and S. Eule (2006).
Exact solution of a generalized Kramers-Fokker-Planck equation retaining retardation effects
Phys. Rev. E 74(041103).

R. Friedrich, F. Jenko, A. Baule, and S. Eule (2006).
Anomalous diffusion of inertial, weakly damped particles
Phys. Rev. Lett. 96(230601).

S. Eule, and R. Friedrich (2006).
A note on the forced Burgers equation
Phys. Lett. A 351:238--241.