CPS Research ProjectsTheories of Diffusion-Limited AggregationDiffusion-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.
Physics of Disordered MediaThe 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.
Physical Mechanisms in Liquid WaterWater 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.
Pathological Alterations in Alzheimer's DiseaseEven by inspection of the gross brain, it is evident that Alzheimer disease (AD) neuropathological changes do not occur randomly or uniformly, but instead target the medial temporal lobe and association cortices of the temporal and parietal areas [Fuller, 1911; Brun, 1976; Arnold, 1991; Braak, 1991]. It might be expected, given the complex topography of brain architecture, that on finer examination this distribution would show a pattern of vulnerability definable at the level of neural systems, cytoarchitectural fields, and even specific lamina within cytoarchitectural fields. Our anatomical studies of AD neuropathology have shown that this is indeed the case. Neurofibrillary tangles (NFT), senile plaques (SP) and neuronal loss affect individual cytoarchitectural areas and laminae in an extraordinarily specific and selective manner. Based on application of neuroanatomical principles and extrapolation of connectional data from the nonhuman primate experimental system, we have suggested that these lesions likely destroy major feedforward and feedback projections, leading to disruption of neural systems related to memory and cognition. Our challenge now is to understand the processes that give rise to these amazingly consistent and stereotyped patterns. To do so will require moving beyond traditional anatomic methodology and pattern recognition, to quantitative and topographically precise methods of data collection. In the last 2 years we have been developing such techniques, and we propose to use newly developed stereological counting techniques in conjunction with computer assisted image analysis to generate quantitative maps of human cortical cytoarchitecture, and of AD-related alterations. The utility of these data is to develop models and test models of AD pathophysiology. A major new thrust is a collaboration with Dr. Gene Stanley, head of the Boston University Center for Polymer Physics, and an international leader in applying the mathematical techniques of complex systems analysis to biological systems. With Dr. Stanley and his group, we will bring powerful mathematical analytical techniques to bear on the geometrical problems posed by these quantitative topographic maps of neural architecture. This unique collaboration has allowed us already to generate and test new hypotheses about the normal organization of the cortex and disruption of cytoarchitectural features by neuronal loss, NFT formation, and SP formation in AD.
Long-range Corellations in DNA Base Pair SequencesRecently, it was discovered by researchers in our group that non-coding DNA sequences exhibit scale-invariant long-range correlations quantitatively measured by a power law decay. The exponent characterizing the power law decay of the correlations is well-defined for infinite sequences. However, for DNA sequences the accuracy of the analysis is limited by the length of the available nucleotide chains (i.e., there are only a few samples of published nucleotide sequences with length >100,000 base pairs). The goal of this project is to investigate the possible broad-based implications of this discovery. |