Karin A. Dahmen
Associate Professor of Physics
Professor Dahmen received her Vordiplom in physics from the Universität Bonn, Germany, in 1989, and her Ph.D in physics from Cornell in 1995. Before joining the faculty at Illinois in 1999, she was a Junior Fellow at Harvard University. She has wide-ranging interests in "soft" condensed matter physics, including nonequilibrium dynamical systems, hysteresis, avalanches, earthquakes, population biology, and disorder-induced critical behavior.
Research Areas: condensed Matter physics, nonequilibrium dynamical systems, including pattern formation in homogeneous systems and inhomogeneous systems with quenched disorder
Description of Current Research
Generally, I have wide ranging interests in non-equilibrium dynamical systems, including pattern formations in homogeneous systems and inhomogeneous systems with quenched disorder. There are new experiments showing interesting non-equilibrium phenomena, that are only being studied recently. I am also interested in other aspects of condensed matter physics and mathematical physics, and in areas of biophysics and geophysics, where methods of condensed matter physics can be fruitfully applied.
In physical systems consisting of a large number of atoms or molecules statistical fluctuations usually become small. The signals we perceive are mostly averages over the complex microscopic behavior of the system. Equilibrium properties, for example, are given by ensemble averages that obey simple thermodynamical, hydrodynamical and statistical mechanical laws. Systems driven far from equilibrium, can also be usefully described by simple laws on long length scales, where local fluctuations in the amount of disorder are averaged out. There has been much progress made recently in the study of collective behavior in systems ranging from advancing domain walls in magnets to event size distributions in earthquakes. The methods applied include ideas from dynamical systems and chaos, critical phenomena, hydrodynamics and disordered systems theory.
Hysteresis and Hierarchies: Dynamics of Disorder Driven First Order Phase Transformations
We have discovered a critical point in the behavior of hysteretic systems. Adding disorder to the system, we find a second order transition from hysteresis loops with a macroscopic jump or burst (roughly as seen in the supercooling of liquids) to smoothly varying hysteresis loops (as seen in most magnets). We study the critical point in the nonequilibrium zero temperature random field Ising model using mean field theory, renormalization group techniques, and numerical simulations in 2, 3, 4, and 5 dimensions.
Hysteresis, Avalanches, and the Breakdown of Hyperscaling: Critical Exponents in 6-e Dimensions,
We have performed a history dependent renormalization group calculation for the critical point described above, using the Martin Siggia Rose formalism in a dynamical description of the system. We extracted the critical exponents in a 6-e-expansion. Using a mapping to the pure Ising model, we performed a Borel resummation for the correlation length exponents to Oe5. Close to the critical point discussed above, the avalanche size distribution on long length scales also has a universal scaling form.
Statistics of Earthquakes in Simple Models of Heterogeneous Faults
Observations have shown that earthquakes exhibit apparently universal scaling of the rupture size distributions and related quantities. We have studied simple models for ruptures along a heterogeneous earthquake fault zone, focusing on the interplay between the roles of disorder and dynamical effects. A class of models were found to operate naturally at a critical point whose properties yield power law scaling of earthquake statistics.
Phase Diagram for the Statistics of Earthquakes in a Mean-Field Model of Heterogeneous Faults: Switching from Gutenberg–Richter to Characteristic Earthquake Distribution and Back
The results from the earthquake project, above, for weak dynamical effects are extended to strong dynamical effects in mean-field theory, which can be treated analytically. A two-parameter phase diagram is found, spanned by the amplitude of dynamical weakening effects e and the normal distance L of the driving forces from the fault. For small e and small L, the fault produces Gutenberg–Richter type power law statistics with an exponential cutoff, while large e and large L lead a to distribution of small events combined with characteristic system size events. In a certain parameter regime, nucleation from one phase to the other is possible on time scales determined by the fault size and other model parameters.
Effects of Quenched Disorder in Population Biology
We study a reaction-diffusion model for population growth of bacteria in a spacially inhomogeneous environment (simulating for example spatial fluctuations in the density of nutrients or toxins, or an inhomogeneous illumination pattern projected onto, e.g., photosynthetic bacteria). Specifically, we have used the Fisher equation, which describes diffusive spreading, growth and decay of a scalar density (of species).
Results in the Presence of a Drift Term
Recent work by David Nelson and Nadav Schnerb (preprint 1997) shows that if convection is added to the problem above (without discreteness), a delocalization transition can be seen as a function of drift velocity: If at zero velocity the bacteria are localized in separated oases of nutrients, the addition of a drift term with high enough velocity can delocalize the bacteria by enabling them to move through the desert from oasis to oasis.
Fisher Waves in Random Media
In the of limit positive, uniform, growth rate, in the absence of a drift term, the Fisher equation has wave (soliton) solutions, that describe the invasion of a front of high bacteria concentration into a region of high nutrient concentration. Interesting questions arise concerning the effect of spatial fluctuations in the nutrient concentration on the shape of the wave front and its mode of propagation.
Depinning of a domain wall in the 2d random field Ising model, We study the behavior of a driven domain wall in the two--dimensional random--field Ising model closely above the depinning threshold, at zero temperature. (This could correspond for example to a fluid invading a 2d porous medium). It is found that not only for large, but even for very weak disorder, the domain wall propagates through the system in a percolative fashion. A scaling theory in terms of the disorder strength and the magnetic field strength is being worked out, which gives exact values for most exponents and would suggest that two is the lower critical dimension of the interface depinning transition with a correlation length scaling exponentially as R ® 0.
Nonequilibrium Transport
In a recently started collaboration with Nadav Shnerb, we are studying interesting experimental results on nonequilibrium transport and slow relaxation in hopping conductivity. The hope is that a very simple model may explain some of the observed features.
Selected Publications
Alex Travesset, Robert A. White, and Karin A. Dahmen, "Crackling Noise, Power Spectra, and Disorder-Induced Critical Scaling," Phys. Rev. B 66, 024430/1-11 (2002).
Robert A. White and Karin A. Dahmen, "Driving Rate Effects on Crackling Noise," Phys. Rev. Lett. 91, 085702/1-4 (2003).
John H. Carpenter and Karin A. Dahmen, "Barkhausen Noise and Critical Scaling in the Demagnetization Curve," Phys. Rev. B 67, 020412/1-4 (2003) Rapid Communication.
John H. Carpenter, Karin A. Dahmen, Andrea Mills, Michael Weissman, Andreas Berger, Olav Hellwig, "Disorder, Noise, Subloops, and History-Induced Critical Behavior," Phys. Rev. B 72, 052410/1-4 (2005).
Honors and Awards
- Sloan Foundation Research Fellowship, 2001
- Beckman Fellow, University of Illinois Center for Advanced Study, 2001
