Nonlinear Dynamics and Chaos

Participating group members: David Speer, Peter Reimann
Main cooperation partners: Ralf Eichhorn, NORDITA (Stockholm);
Dieter Koelle and Reinhold Kleiner (Solid State Physics, Tübingen)

NAR A first main issue are unusual transport properties of an underdamped Brownian particle in a periodic potential, e.g. particle current v and static force F of opposite signs. Their origin is a subtle interplay between the stability of coexisting attractors, noise-induced metastability, and transient chaos.

The theoretically predicted unusual transport against force has been verified experimentally by measuring a a negative resistance of about -1 Ohm in a Josephson junction. The correspondence between the electrical properties of a Josephson junction and the transport properties of a Brownian particle in a periodic potential is established by the Stewart-McCumber model for Josephson junctions.


Left: Generalized potential, describing the exponentially leading behavior of the invariant probability density for weak noise at the onset of chaos along the Feigenbaum route. Right: Pre-exponential part of the invariant density.

A second main topic is the influence of weak noise on systems with chaotic deterministic dynamics in discrete time (deterministic diffusion, Feigenbaum-scenario, crises, fully developed chaos) and on anomalous transport properties of Hamiltonian chaotic systems and their possible regularization through weak random fluctuations by means of renormalization methods (stochastic Markov-chain- and Markov-tree-models).

One cornerstone of this research is the general mathematical formalism for describing the decay of complex metastable states, in particular the concept of generalized (non-equilibrium) potentials (see figure).

Main publications:

P. Reimann and C. Van den Broeck
Intermittent Diffusion in the Presence of Noise
Physica D 75, 509 (1994)

P. Reimann
Suppression of Deterministic Diffusion by Noise
Phys. Rev. E 50, 727 (1994)

P. Reimann
Noisy One-Dimensional Maps near a Crisis
J. Stat. Phys. 82, 1467 (1996); 85, 403 (1996)

D. Speer, R. Eichhorn, and P. Reimann
Brownian motion: Anomalous response due to noisy chaos
Europhys. Lett. 79, 10005 (2007)

D. Speer, R. Eichhorn, and P. Reimann
Transient chaos induces anomalous transport properties of an underdamped Brownian particle
Phys. Rev. E 76, 051110 (2007)

J. Nagel, D. Speer, T. Gaber, A. Sterck, R. Eichhorn, P. Reimann, K. Ilin, M. Siegel, D. Koelle, and R. Kleiner
Observation of Negative Absolute Resistance in a Josephson Junction
Phys. Rev. Lett. 100, 217001 (2008)

D. Speer, R. Eichhorn, and P. Reimann
Directing Brownian Motion on a Periodic Surface
Phys. Rev. Lett. 102, 124101 (2009)

D. Speer, R. Eichhorn, and P. Reimann
Exploiting Lattice Potentials for Sorting Chiral Particles
Phys. Rev. Lett. 105, 090602 (2010)

D. Speer, R. Eichhorn, and P. Reimann
Anisotropic diffusion in square lattice potentials: giant enhancement and control
EPL 97, 60004 (2012)

Last modified on 2012-03-14