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. We have 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. We are attempting to adapt these algorithms to first-order transitions and the modeling of non-equilibrium processes.
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 in the physics of non-equilibrium systems along new and very promising lines. 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.
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 of the earth. 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, we use field theoretical techniques, that we previously developed to study the kinetics of phase transitions, as well as CM5 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.
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 phononic 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.
Stochastic processes underlie a wide variety of non-equilibrium phenomena in many-particle systems and have been of enduring interest to physicists, mathematicians, and chemists. One important example is the random walk process. We have undertaken a variety of fundamental studies to elucidate the effects of trapping and adsorption on the statistical properties of random walks. Perhaps the most useful applications of these results is to the understanding of the kinetics of chemical reactions. Applications of ideas from the theory of random walks have provided considerable insight into understanding the dynamics of disparate classes of reactions, such as aggregation, catalysis, recombination, and trapping. In many situations, the temporal evolution of the reaction can be described in terms of a system of interacting random walks. We have found that spatial fluctuations in the densities of the reactants can play a profound role in influencing the rates of 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. The insights gained from these studies may be helpful for understanding more complicated, but phenomenologically richer problems, such as reversible and oscillatory reacting systems, prey-predator population biology models, and biological morphogenesis.
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. We have developed new tools, such as scaling and the renormalization group, and developed efficient (parallel) large-scale numerical simulation techniques. Recently, we have 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. We have 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.
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.
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.
Research is being done to understand the mechanism and critical properties on the phase transitions between different quantized plateau phases in the Quantum Hall Effect. Our theories have predicted universal values for the critical conductivities at the plateau transitions in agreement with experiments on macroscopic samples. The understanding of mesoscopic conductance fluctuations in simple disordered metals has been well established. Our research concentrates on the poorly-understood quantum Hall systems when a strong magnetic field is present, with particular emphasis on the critical regime of the metal-insulator transitions. This work is expected to provide, for the first time, a microscopic theory of conductance fluctuations in the critical regime, which will then be contrasted to the theory of universal conductance fluctuations in simple disordered metals.
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 4 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 derivation transition in DNA.