Effects of Disorder on Chaotic Diffusion: Complex Particle Dynamics in Magnetized Plasmas
Final Report Abstract
On the plasma physics side of the project, very significant progress could be achieved via a stepladder approach, going in a systematic fashion from simpler (idealized) to more complex (realistic) models. This allowed us to investigate in great detail some fundamental phenomena while always making sure that there is direct connection to real-life systems. Initially, superpositions of (typically thousands of) plane waves were used to create “pseudoturbulent” fields whose properties could be very well controlled. Aspects like the influence of the Kubo number and the role of fluctuation anisotropies or zonal flows were studied both numerically and analytically in the framework of this setup. Later, realistic turbulence fields created by simulations with the state-of-the-art gyrokinetic plasma turbulence code GENE (developed in the group of Frank Jenko since 1999) were used instead. Here, many of the phenomena studied in isolation before, occurred simulataneously. Building on the qualitative and quantitative insights from the first project phase, these situations could also be interpreted quite nicely. By means of 3D gyrokinetic data for tokamak geometries, a longstanding paradigm in plasma physics regarding the dynamics of energetic particles in turbulent fields (the applicability of orbit averaging) was replaced by a new one (the fast perpendicular decorrelation of particles w.r.t. background fields). Consequently, the key publication describing this effect has received a lot of attention. It is also noteworthy that this project has been characterized by a large degree of interdisciplinarity. It involved researchers from theoretical and experimental plasma physics, complex systems research, turbulence research, particle astrophysics, and applied mathematics. This allowed for a fruitful exchange of ideas across disciplinary boundaries and led to innovative approaches. On the fundamental side of this project the most important achievements were obtained for Hamiltonian systems in great generality. These are on one hand the formulation and verification of a statistical theory of the breakup of critical Kolmogorov-Arnold-Moser tori in Hamiltonian systems with frozen disorder potentials, and on the other hand the fundamental discovery of ageing and weak ergodicity breaking in Hamiltonian systems showing momentum diffusion. Kolmogorov-Arnold-Moser (KAM) theory and the breakup of critical KAM-tori at the transition to global stochasticity are one of the most celebrated achievements of Hamiltonian dynamics of the last century. Hamiltonian systems with frozen disorder potentials, however, are defined only in an ensemble sense. Therefore a statistical description of the breakup of critical KAM-tori in such systems was asked for. This important step was achieved for the first time by verifying a renormalized extreme value theory. This theory allows to predict the distribution of critical disorder amplitudes of large systems from that of small ones. As a consequence we were also able to derive a universal scaling law for the vanishing of critical KAM-tori in the limit of large systems. These results are relevant for a broad spectrum of applications ranging from plasma and accelerator physics to disordered semiconductor heterostructures and friction problems in solid state physics. Weak ergodicity breaking and ageing are phenomena, which were first detected and investigated in detail in disordered systems, especially in spin glasses. Our results obtained in this project, show for the first time that these are also common phenomena in all Hamiltonian systems where momentum behaves diffusively due to chaoticity or other reasons such as disorder. Weak ergodicity breaking in these systems means that ensemble and time averages for the anomalous increase of the mean-squared displacement in space differ fundamentally. Specifically, the powers of the increase in time differ, and in addition the generalized diffusion constant of the time-averaged mean-squared displacement is a random quantity in the sense that it varies from trajectory to trajectory. We demonstrated these results numerically for the prototypical Standard Map and in addition we are able to calculate all quantities of interest analytically by reducing the problem to that of a randomly accelerated particle, or more generally speaking to integrated Brownian motion. Since the latter is a very general type of motion, our results are not only valid for Hamiltonian systems, but for an even broader class of systems ranging from cold atoms in optical lattices to tracer diffusion in turbulent flows.
Publications
- Turbulent ExB Advection of Charged Test Particles with Large Gyroradii, Physics of Plasmas 13, 102309 (2006)
T. Hau¤ and F. Jenko
- ExB Advection of Trace Ions in Tokamak Microturbulence, Physics of Plasmas 14, 092301 (2007)
T. Hauff and F. Jenko
- Intermediate Non-Gaussian Transport in Plasma Core Turbulence, Physics of Plasmas 14, 102316 (2007)
T. Hauff, F. Jenko, and S. Eule
- Lagrangian Structures and Transport in Turbulent Magnetized Plasmas, New Journal of Physics 9, 400 (2007)
K. Padberg, T. Hau¤, F. Jenko, and O. Junge
- Mechanisms and Scalings of Energetic Ion Transport via Tokamak Microturbulence, Physics of Plasmas 15, 112307 (2008)
T. Hauff and F. Jenko
- Turbulent Transport of Beam Ions, Physics of Plasmas 15, 062508 (2008)
T. Dannert, S. Günter, T. Hauff, F. Jenko, X. Lapillonne, and P. Lauber
- Electrostatic and Magnetic Transport of Energetic Ions in Turbulent Plasmas, Physical Review Letters 102, 075004 (2009)
T. Hauff, M. J. Pueschel, T. Dannert, and F. Jenko
- Runaway Electron Transport via Tokamak Microturbulence, Physics of Plasmas 16, 102308 (2009)
T. Hau¤ and F. Jenko
- Streamer-Induced Transport in Electron Temperature Gradient Turbulence, Physics of Plasmas 16, 102306 (2009)
T. Hauff and F. Jenko
- Scaling Theory for Cross-Field Transport of Cosmic Rays in Turbulent Fields, The Astrophysical Journal 711, 997 (2010)
T. Hauff, F. Jenko, A. Shalchi, and R. Schlickeiser
- Fast Ion Power Loads on ITER First Wall Structures in the Presence of NTMs and Microturbulence, Nuclear Fusion 51, 083041 (2011)
T. Kurki-Suonio, O. Asunta, E. Hirvijoki, T. Koskela, A. Snicker, T. Hauff, F. Jenko, E. Poli, and S. Sipila
- Analyzing anomalous di¤usion processes: the distribution of generalized di¤usivities. In: R. Metzler, W.H. Deng (eds.); Proceedings of Sino-German Bilateral Symposium on Fractional Dynamics: Recent advances (CD-ROM), 2012
T. Albers and G. Radons
- Anomalous Diffusion of Energetic Particles: Connecting Experiment and Simulations, Nuclear Fusion 52, 10301 (2012)
M. J. Pueschel, F. Jenko, M. Schneller, T. Hauff, S. Günter, and G. Tardini
(See online at https://doi.org/10.1088/0029-5515/52/10/103018) - Subdiffusive continuous time random walks: ergodicity breaking analyzed with the distribution of diffusivities. EPL 102, 4006 (2013)
T. Albers and G. Radons
- Weak ergodicity breaking and ageing of chaotic transport in Hamiltonian systems, Physical Review Letters Phys. Rev. Lett. 113, 184101 – Published 31 October 2014
T. Albers and G. Radons
(See online at https://doi.org/10.1103/PhysRevLett.113.184101)
