## Research Statement

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.

**Unifying theory of universal “quake” statistics in plasticity: from nanocrystals to earthquakes**

(PRL 2009, 2010, 2011, Nature Physics 2011, Nature Communications 2011, Advanced Functional Materials 2012, and submitted to Nature, 2012).
Since my Sabbatical in 2007 we have pursued a new direction of research, studying slip avalanches and plasticity in sheared solids, ranging from sheared nano-crystals and micro-crystals, and dislocation dynamics, to sheared granular materials, to amorphous materials, to earthquakes. We have been developing simple analytical models to describe the universal (i.e. detail-independent) aspects of the slip avalanche statistics, acoustic emission, stress strain curves, and other properties of these systems. These theoretical predictions are currently leading ahead of experiments, motivating the design of new experiments and data analysis methods.
Our results are receiving much attention from experimentalists, theorists, materials scientists, mechanical engineers, geophysicsists, applied mathematicians, and biophysicsists, as evidenced by recent invited lectures at the annual meetings of various professional societies, such as the American Physical Society, the Americal Geophysical Union, The Minerals, Metals & Materials Society (TMS), the Materials Research Society, the Aspen Center of Physics, the Kavli Institute of Theoretical Physics, and the Mathematical Geophysics conference. About 12 different experimental groups (nationally and internationally) are currently setting up and running experiments to test our results. Initial experimental results confirm our predictions. In collaboration with groups at Purdue and Harvard we are also applying related ideas to scaling behavior of stripes in high Tc Superconductors. We have also contributed to experiments in biophysics through collaborations with the CPLC, the IGB, and with outside groups, focusing on avalanches in neural networks, stochastic gene-expression, and the statistics of bacterial motion in inhomogeneous environments.
Some of our most recent studies are relevant for deformation studies of bulk metallic glasses, magnesium and twin boundary dynamics, high entropy alloys, colloids, granular materials, cement, and bone. Each of these materials has many applications and it is important to gain an understanding of their deformation properties. We expect that there will be ample funding opportunities for future research in these directions. Below we also list brief summmaries of additional research projects on other topics.

**Hysteresis and Hierarchie***s***: 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¨CRichter 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**¨C**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 ¡ú 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.