Faculty Members -- R. Brower, R. Giles, W. Klein, G. Murthy, S. Redner, H.E. Stanley, Z. Wang, C. Willis
Research Faculty & Associates -- S. Buldyrev, E. Cabarcos, L. Cruz, R. Cuerno, P. Falus, L. Frachebourg, P. Garik, S. Havlin, B. Jovanovic, R. Mategna, J. Mendes, E. Taylor, P. Trunfio
Graduate Students -- S. Datta, N. Dokholyan, C. Ferguson, P. Gopikrishnan, I. Grosse, S. Harrington, I. Isploatov, P. Ivanov, B. Jovanovic, Y. Liu, H. Makse, C. Muratov, R. Sadr-Lahijany, F. Starr, A. Tandon, G. Viswanathan, S. Zapperi
Theorists in condensed-matter physics and statistical mechanics are trying to understand the behavior of the complex systems formed by large numbers of particles in solids, liquids, and complex biological molecules. Their interests range from high-temperature superconductivity to the percolation of water through porous materials.
Acceleration Algorithms -- R. Giles, W. Klein, R. Brower
Computer simulations have become a major source of information in all areas of physics. The detailed microscopic mechanisms of systems undergoing phase transitions can, to a large extent, only be obtained through computer simulation. A serious flaw of these simulations is the small system size that can be simulated compared to the ~1023 particles in a real system. This is particularly acute near phase transitions where large numbers of particles act coherently. The group has begun a program to design algorithms which move, not single particles in one step, but large correlated regions, thereby mimicking the physical processes associated with large regions of coherence. These so-called acceleration algorithms have been extremely successful in removing the critical slowing down associated with critical points. They are attempting to adapt these algorithms to first-order transitions and the modeling of non-equilibrium processes.
Kinetics of Phase Transitions -- W. Klein
One of the most important problems in condensed-matter physics is to understand the evolution of systems far from equilibrium. For example, understanding of the process by which a metastable state decays into equilibrium is important in areas as diverse as alloy structure, cloud formation, membrane structure, and decay of the false vacuum in the early universe. The properties of non-equilibrium systems have been studied for over a hundred years, yet there are still fundamental questions that are unanswered. This class of problems is extremely difficult because the proper theoretical tools to describe evolution far from equilibrium do not yet exist and microscopic level experimental information has been hard to obtain.
In the last decade, two advances in seemingly unrelated fields have stimulated research along new and very promising lines in the physics of non-equilibrium systems.The first of these advances, massively parallel computation, has made it possible to formulate realistic computer models of systems far from equilibrium and obtain information on a microscopic level. The second advance was in the understanding of clusters and how to formulate condensed-matter physics problems as problems in cluster growth, percolation, and fractals.
The present research program involves the application of the theoretical tools of cluster dynamics and structure, field theory, and renormalization group, as well as the Connection Machine, a massively parallel computer, to the problem of systems far from equilibrium. Currently, we are investigating the occurrence of fractal structures in the early stage mechanisms of nucleation and spinodal decomposition, the effect of conservation laws on the evolution of systems out of equilibrium and the presence and influence of fractal structures in non-linear dynamics. In addition, extensive contact with experimentalists (R. Bansil and K. Ludwig) is maintained.
Physics of Earthquakes -- W. Klein
Earthquakes are a natural hazard that pose a significant threat to both the safety and the economic well-being of a significant fraction of the earth's population. Consequently, it is essential that a reliable assessment of risk can be made. This assessment requires a thorough understanding of the mechanisms of earthquake faults. To attain this understanding, we have constructed, in collaboration with geologists from the University of Colorado, several mathematical and computer models of both single faults and networks of earthquake faults.
To analyze these models, the group uses field theoretical techniques, that were previously developed to study the kinetics of phase transitions, as well as supercomputer simulations at The Center for Computational Science. These studies indicate the presence of nucleation events, critical phenomena and scaling, as well as a transition from quasi-periodicity to marginal chaos in synthetic earthquakes on single faults. We are presently expanding our studies to models of fault networks, as well as investigating the relation between our models and real faults.
Strongly Interacting Electron Systems -- G. Murthy
There are many interesting physical problems where strong interactions between electrons seem to be essential to the physics. These phenomena include high-temperature superconductivity, fullerene superconductivity, heavy fermion systems, persistent currents in mesoscopic rings, and the spectroscopy of quantum dots. A multipronged approach is being followed to understand these systems. The techniques used are mean-field theory, exact diagonalizations of small systems, and numerical renormalization group analysis.
Attention is also being focused on the interplay between electronic and photonic effects in fullerenes. Since the electronic time scale is comparable to the phonon time scale in these materials, the phonons have the time to relax into a distorted configuration on a fullerene, the Jahn-Teller effect. A novel approach to superconductivity in these materials based on the semiclassical approximation is being pursued.
Kinetics of Stochastic Processes -- S. Redner
Stochastic processes underlie a wide variety of non-equilibrium phenomena in physics. One important area where the application of stochastic methods and the theory of random walks has led to major advances is in characterizing the kinetics of chemical reactions. Considerable insight has now been gained in understanding the dynamics of disparate classes of reactions, such as aggregation, catalysis, recombination, and trapping. A crucial feature is that spatial fluctuations in the reactant densities can play a critical role in influencing in the rate of various chemical reactions. For example, in competitive reactions between two equivalent species, an initially homogeneous system can evolve into a continuously coarsening mosaic of single species domains. The rate of reaction is profoundly affected by this spatial organization and the concomitant coarsening processes.
The insights gained from Redner's investigations of idealized models are now being applied to understand the spatio-temporal behavior in more complicated, but phenomenologically richer situations. For example, in exothermic reactions, the coupling between heat production and the local reaction rate may give rise to a turbulent reaction, as well as to spatial and temporal instabilities. In a related vein, the essential role of spatial heterogeneity is being investigated in a variety of population biology models, such as prey-predator systems, as well as in biological morphogenesis. Finally, these approaches are being adapted to account for the rich variety of co-operative and competitive social and economic phenomena which can be described within the rubric of stochastic reaction models.
Transport and Collective Processes in Disordered and Granular Media -- S. Redner
Dynamical process in disordered and granular media pose a range of fascinating questions with wide-ranging practical ramifications. For example, by monitoring the spread of dynamically-neutral Brownian tracer particles within a fluid which flows through a porous medium, one can gain fundamental insights about the structure of the medium itself. A comprehensive theoretical description of this hydrodynamic "dispersion" process has been developed in which the competition between the flow and molecular diffusion leads to new types of stochastic transport laws. This work is being extended to non-linear processes, such as dispersion in the high Reynolds number flow regime, and also the plugging of porous media by the filtering of "dirt" (suspended tracer particles) as a dirty fluid asses through a porous matrix. These removed particles gradually plug the pore space, giving rise to feedback between transport and filtration for which theoretical understanding has yet to be developed. Finally, investigations of the dynamics and clustering phenomena of inelastic granular media are beginning. Such a granular medium exhibits largescale particle clustering as it "cools" due to the inelastic collisions. Both simulations and analytical theories are being developed to help explain this intriguing, and as yet unexplained, behavior.
Physics of Disordered Media -- S. Redner, H.E. Stanley
The physical properties of disordered media pose a wide range of fascinating questions and open problems, with both theoretical and experimental ramifications. Our recent work has focused on theoretical studies of percolation models of disordered materials. Redner and Stanley have developed new tools, such as scaling and the renormalization group, and developed efficient (parallel) large-scale numerical simulation techniques. Recently, it has been found that there is an underlying hierarchical structure which governs many aspects of disordered media. The implications of this hierarchical organization are far-reaching, as many well-established ideas from critical phenomena need to be reformulated. Instead of a single scale, or ``fractal dimension'' describing a disordered system, there can be a multiplicity of scales, or ``multifractality''. The insights gained from these advances have ramifications for problems such as fluid flow and chemical reactions in porous media, and transport processes in disordered materials.
One important application is the electrical conductance of disordered materials. This quantity is of relevance to a range of phenomena, such as the anomalous behavior of the elastic modulus and shear viscosity near the sol-gel transition. The group has exploited the Einstein relation to connect the electrical conductance to properties of random walkers in the same disordered medium. By this equivalence, we can compute the electrical conductance of two-component composite media. Through the isomorphism between electrical conductance and viscoelastic properties of disordered networks, the mechanical behavior near the sol-get transition has been investigated.
At a more geometric level, ``exact'' closed-form solutions to descriptions of the structure of the incipient infinite cluster in percolation, and the infinite network that form just above the percolation threshold have been achieved. A hybrid model for this structure was proposed that incorporates both singly-connected and multiply-connected bonds. Fundamental studies of this ``links and blobs'' model, using both exact calculations and computer simulations, yielded a rigorous result for the geometry of the singly-connected bonds.
Theories of Diffusion-Limited Aggregation
-- H.E. Stanley
Diffusion-limited growth processes have attracted considerable recent attention because of the myriad of applications to the morphology of growing interfaces and because of fundamental issues associated with disorderly growth processes. The diffusion-limited aggregation (DLA) model, is a particularly attractive realization of such growth processes. DLA growth is completely characterized by assigning to each perimeter site i the number Pi, the probability that site i is the next to grow. Evidence suggests that the numbers Pi, form a multifractal set: that which cannot be characterized by a single exponent, but rather require an infinite hierarchy of exponents. Since the hottest tips of a DLA aggregate grow much faster than the deep fjords, the scaling of Pi, with respect to the aggregate size depends on the particular site. Analogies to percolation systems are made by assigning different sites in the DLA model to particular kinds of percolation bonds. Additionally, conditions for viscous-fingering experiments yield patterns that obey the same scale symmetry as DLA have been found, leading to new questions about the physics underlying the viscous-fingering phenomenon.
Physical Mechanisms in Liquid Water
-- H.E. Stanley
Water is a unique substance. It plays a major role in all living systems, and even small perturbations---such as the substitution of deuterium for hydrogen---are sufficient to destroy biological function. In living systems, essential water-related phenomena occur in restricted geometries in cells and organelles, and at active sites on membranes. While significant progress has been made in the understanding of water in terms of hydrogen-bonded networks, most treatments of liquids fail to take account of their transient nature, an important feature responsible for many of water's unusual properties. In an effort to understand the mechanisms at the molecular level, one must be aware that in confined geometries the physical scale may approach the scale of locally structured regions of the network. Our key idea is to relate structural differences and the dynamics to the behavior of the transient gel network, which is formed by hydrogen bonds.
Quantum Phase Transitions -- Z. Wang
Wang's research centers on the properties of electric systems in which the many-body correlations and/or the breaking of translational invariance due to disorder introduce fundamentally new physics. His recent research has been devoted to the theoretical studies of the interplay between disorder-induced localization and electronic interactions, and its effects on the low-temperature electronic transport properties of materials in strong magnetic fields. The most dramatic manifestation of the new physics under such settings is through the remarkable quantum Hall effect in a two dimensional electron gas trapped between semiconductor interfaces. Wang is developing new field theories and numerical approaches to understand the fundamental new physics associated with the transitions between the adjacent quantum Hall plateau phases. He is particularly interested in the effects of electronic interactions and quasiparticle statistics on the transitions. The new theoretical framework is used to study the transport properties in multi-layer quantum wells, semiconductor superlattices, as well as various aspects of the quantum Hall effect between two and three dimensions. Also underway is the study of the transport properties in bulk-homogeneous semiconductors in which the effects of many-body correlations are important for observing the quantum Hall effect.
Non-Linear Excitations on Surfaces -- C. Willis
Several investigations involving solitons and their relationship to surface physics are being carried out. Theoretical developments include the derivations of Hamiltonian descriptions of collective variables for the Sine-Gordon, double Sine-Gordon, and phi4 systems. The Hamiltonian approach has been applied to double sine soliton collisions, discrete soliton lattices including phonon radiation, and soliton Brownian motion. Present problems include soliton impurity scattering, kink pinning and depinning, exact equilibrium statistical mechanics of the double Sine-Gordon system, and kink transport phenomena. Studies in surface physics also include applications of dynamical and statistical mechanics models to the problem of structural and vibrational properties of atomic layers absorbed on a solid surface. Studies on biological molecules include energy transport on DNA and the phase transitions in DNA.