Seminars of the Focus Area Complex Systems

Prof. Dr. C. Beta, Prof. Dr. K. Dethloff, Prof. Dr. R. Engbert, Prof. Dr. M. Holschneider, Prof. Dr. W. Huisinga, Prof. Dr. Ralf Metzler, Prof. Dr. A. Pikovsky, Prof. Dr. S. Reich, Prof. Dr. M. Rosenblum, Prof. Dr. G. Rüdiger, Prof. Dr. T. Scheffer, Prof. Dr. F. Scherbaum, Prof. Dr. J. Selbig, Prof. Dr. F. Spahn


Speaker: Igor Goychuk, Inst. f. Theoretische Physik, Uni Potsdam

Title: Nonlinear anomalous diffusion and transport: fractional dynamics in and out of thermal equilibrium * Colloquium on Complex and Biological Systems

Time: Fri, Feb 8, 2013, 10:15am

Place: bldg 28, room 0.108

I will discuss an approach to anomalous diffusion and transport based on the Generalized Langevin Equation (GLE). It can be derived from a Hamiltonian dynamics of one particle coupled to a thermal bath of harmonic oscillators within a molecular dynamics like setup (dynamical approach by Bogolyubov, Ford, Kac, Zwanzig, and others). Randomness comes about from the initial canonical distribution of bath oscillators (a quasi-infinite set). Profoundly non-Markovian stochastic dynamics with long-lasting memory emerges as a projection of underlying hyper-dimensional Markovian dynamics onto the subspace of relevant variables. GLE memory kernel and random noise are related by the fluctuation-dissipation theorem which guarantees thermal detailed balance at thermal equilibrium. Algebraically decaying memory kernels (fractional GLE) yield anomalous diffusion and transport. The corresponding linear dynamics can be both anomalously slow (sub-Ohmic thermal bath) and anomalously fast (super-Ohmic thermal bath). It can be treated analytically. A nonlinear dynamics, e.g. in a periodic potential, is however generally highly nontrivial and very difficult for both analytical and numerical treatment.

I will discuss a number of highly nontrivial results obtained recently using a finite-dimensional Markovian embedding of underlying GLE dynamics. These include:

(i) universality class of GLE subdiffusion and transport in tilted washboard potentials;
(ii) emergence of a transient hyperdiffusion for superdiffusive GLE dynamics in tilted washboard potentials;
(iii) subdiffusive ratchet transport. The approach is based on approximation of the memory kernel by a sum of exponentials. It is numerically highly efficient and very reliable. Moreover, it agrees with the exact analytical results whenever available. In the case of subdiffusive GLE dynamics the approach allows for a vivid physical interpretation within a generalized Maxwell-Langevin model of viscoelasticity.

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Students' seminar: Theoretical Physics, PIK, Modeling & TSA Berlin-Potsdam-Colloquia: PhysGesellschaft Berlin, TU Berlin, Pro Physik, AIP, AEI, MPI-KGF, GFZ, HMI, PIK, AWI, Max Planck Institute for the History of Science, Mathematik, DPG Disputationen, & Vorschau UP

Udo Schwarz, Zentrum für Dynamik komplexer Systeme,
Universität Potsdam, Campus Golm Karl-Liebknecht-Str. 24, 14476 Potsdam, building 28, room 2.107
Phone: (+49-331) 977-1658, Fax : (+49-331) 977-1045

Email: Udo.Schwarz AT

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