Biomedical subjects

P M Duxbury

Publications and source records attributed to P M Duxbury.

12 recordsLinked to original sources

PubMed2006

Maximum independent set on diluted triangular lattices.

Core percolation and maximum independent set on random graphs have recently been characterized using the methods of statistical physics. Here we present a statistical physics study of these problems on bond diluted triangular lattices. Core percolation critical behavior is found to be consistent with the standard percolation values, though there are strong finite size effects. A transfer matrix method is developed and applied to find accurate values of the density and degeneracy of the maximum independent set on lattices of limited width but large length. An extrapolation of these results to the infinite lattice limit yields high precision results, which are tabulated. These results are compared to results found using both vertex based and edge based local probability recursion algorithms, which have proven useful in the analysis of hard computational problems, such as the satisfiability problem.

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PubMed2006

Ab initio determination of solid-state nanostructure.

Advances in materials science and molecular biology followed rapidly from the ability to characterize atomic structure using single crystals. Structure determination is more difficult if single crystals are not available. Many complex inorganic materials that are of interest in nanotechnology have no periodic long-range order and so their structures cannot be solved using crystallographic methods. Here we demonstrate that ab initio structure solution of these nanostructured materials is feasible using diffraction data in combination with distance geometry methods. Precise, sub-ångström resolution distance data are experimentally available from the atomic pair distribution function (PDF). Current PDF analysis consists of structure refinement from reasonable initial structure guesses and it is not clear, a priori, that sufficient information exists in the PDF to obtain a unique structural solution. Here we present and validate two algorithms for structure reconstruction from precise unassigned interatomic distances for a range of clusters. We then apply the algorithms to find a unique, ab initio, structural solution for C60 from PDF data alone. This opens the door to sub-ångström resolution structure solution of nanomaterials, even when crystallographic methods fail.

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PubMed2005

Culling avalanches in bootstrap percolation.

We study the culling avalanches which occur after the "death" of a single randomly chosen site in a network where sites are unstable, and are culled, if they have coordination less than an integer parameter m. Avalanche distributions are presented for triangular and cubic lattices for values of m where the associated bootstrap transitions are either first or second order. In second order cases, the culling avalanche distribution is found to be exponential, while in first order cases it follows a power law. We present an exact relation between culling avalanches and conventional bootstrap percolation and show that a relation proposed by Manna [Physica A 261, 351 (1998)] can be a good approximation for strongly first order bootstrap transitions but not for continuous bootstrap transitions.

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PubMed2005

Statistical physics of grain-boundary engineering.

Percolation theory is now standard in the analysis of polycrystalline materials where the grain boundaries can be divided into two distinct classes, namely "good" boundaries that have favorable properties and "bad" boundaries that seriously degrade the material performance. Grain-boundary engineering (GBE) strives to improve material behavior by engineering the volume fraction c and arrangement of good grain boundaries. Two key percolative processes in GBE materials are the onset of percolation of a strongly connected aggregate of grains, and the onset of a connected path of weak grain boundaries. Using realistic polycrystalline microstructures, we find that in two dimensions the threshold for strong aggregate percolation c(SAP) and the threshold for weak boundary percolation c(WBP) are equivalent and have the value c(SAP) = c(WBP) =0.38 (1) , which is slightly higher than the threshold found for regular hexagonal grain structures, c(RH) =2 sin (pi/18) =0.347... . In three dimensions strong aggregate percolation and weak boundary percolation occur at different locations and we find c(SAP) =0.12 (3) and c(WBP) =0.77 (3) . The critical current in high T(c) materials and the cohesive energy in structural systems are related to the critical manifold problem in statistical physics. We develop a theory of critical manifolds in GBE materials, which has three distinct regimes: (i) low concentrations, where random manifold theory applies, (ii) critical concentrations where percolative scaling theory applies, and (iii) high concentrations, c> c(SAP) , where the theory of periodic elastic media applies. Regime (iii) is perhaps most important practically and is characterized by a critical length L(c) , which is the size of cleavage regions on the critical manifold. In the limit of high contrast epsilon-->0 , we find that in two dimensions L(c) proportional, gc/ (1-c) , while in three dimensions L(c) proportional, g exp [ b(0) c/ (1-c) ] / [c (1-c) ](1/2) , where g is the average grain size, epsilon is the ratio of the bonding energy of the weak boundaries to that of the strong boundaries, and b(0) is a constant which is of order 1. Many of the properties of GBE materials can be related to L(c) , which diverges algebraically on approach to c=1 in two dimensions, but diverges exponentially in that limit in three dimensions. We emphasize that GBE percolation processes and critical manifold behavior are very different in two dimensions as compared to three dimensions. For this reason, the use of two dimensional models to understand the behavior of bulk GBE materials can be misleading.

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PubMed2003

Scaling laws for critical manifolds in polycrystalline materials.

We study the surfaces of lowest energy through model polycrystalline materials in two and three dimensions. When the grain boundaries are sufficiently weak, these critical manifolds (CM's) lie entirely on grain boundaries, while when the grain boundaries are strong, cleavage occurs. A scaling theory for the intergranular to transgranular transition of CM's is developed. The key parameters are the average grain size g, the ratio of grain boundary to the grain interior energy, epsilon, and the sample size L. The key result is that a critical length scale exists, L(c)(g,epsilon), so that on short length scales l L(c), the critical manifold is rough. We develop a scaling theory for L(c) and find that in two dimensions L(c) approximately gx(y(2)), while in three dimensions L(c) approximately g exp(bx(y(3))), where x=epsilon/(1-epsilon) and b is a constant. Data from realistic polycrystalline grain structures are used to test the scaling theory. The exact lowest energy surface through model grain structures is found using a mapping to the minimum-cut/maximum-flow problem in computer science. As a function of grain-boundary energy, we observe the crossover from grain-boundary rupture to mixed mode failure (a mixture of transgramular and intergranular modes) and finally cleavage and that the two-dimensional data are consistent with y(2) approximately 3.0+/-0.3, while the three-dimensional data are more difficult to analyze, but are consistent with y(3) approximately 3.5+/-1.0.

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PubMed2002

Permeability and conductivity of platelet-reinforced membranes and composites.

We present large-scale simulations of the diffusion constant D of a random composite consisting of aligned platelets with aspect ratio a/b>>1 in a matrix (with diffusion constant D0) and find that D/D(0)=1/(1+c(1)x+c(2)x(2)), where x=av(f)/b and v(f) is the platelet volume fraction. We demonstrate that for large aspect ratio platelets the pair term (x(2)) dominates suggesting large property enhancements for these materials. However, a small amount of face-to-face ordering of the platelets markedly degrades the efficiency of platelet reinforcement.

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PubMed2001

Ground state nonuniversality in the random-field Ising model.

Two attractive and often used ideas, namely, universality and the concept of a zero-temperature fixed point, are violated in the infinite-range random-field Ising model. In the ground state we show that the exponents can depend continuously on the disorder and so are nonuniversal. However, we also show that at finite temperature the thermal order-parameter exponent 1/2 is restored so that temperature is a relevant variable. Broader implications of these results are discussed.

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PubMed2001

Minimum spanning trees on random networks.

We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Because of this geometric universality, we are able to characterize the energy of MST using a scaling distribution [P(epsilon)] found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to P(epsilon). We discuss the relationship to invasion percolation, to the directed polymer in a random media, to uniform spanning trees, and also the implications for the broader issue of universality in disordered systems.

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PubMed2001

Extremal statistics in the energetics of domain walls.

We study at T=0 the minimum energy of a domain wall and its gap to the first excited state, concentrating on two-dimensional random-bond Ising magnets. The average gap scales as deltaE1 approximately L(straight theta)f(N(z)), where f(y) approximately [ln y](-1/2), straight theta is the energy fluctuation exponent, L is the length scale, and N(z) is the number of energy valleys. The logarithmic scaling is due to extremal statistics, which is illustrated by mapping the problem into the Kardar-Parisi-Zhang roughening process. It follows that the susceptibility of domain walls also has a logarithmic dependence on the system size.

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PubMed2001

Intermittence and roughening of periodic elastic media.

We analyze intermittence and roughening of an elastic interface or domain wall pinned in a periodic potential, in the presence of random-bond disorder in 1+1 and 2+1 dimensions. Though the ensemble average behavior is smooth, the typical behavior of a large sample is intermittent, and does not self-average to a smooth behavior. Instead, large fluctuations occur in the mean location of the interface and the onset of interface roughening is via an extensive fluctuation which leads to a jump in the roughness of order lambda, the period of the potential. Analytical arguments based on extreme statistics are given for the number of the minima of the periodicity visited by the interface and for the roughening crossover, which is confirmed by extensive exact ground state calculations.

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PubMed1999

Atomic diffusion, step relaxation, and step fluctuations.

We show that the dynamics of the pair correlation function in a step train can pinpoint the dominant relaxation mechanism occurring at a crystal surface. Evaporation-condensation and step-edge diffusion do not produce dynamical correlations between neighboring steps, while terrace diffusion may lead to correlations which fall off like a power law with distance and which are peaked at a characteristic time. We derive these results within a "real space" Langevin formalism which is based on diffusion kernels which are different for each mass transport process. We validate this formalism by reproducing the step fluctuation autocorrelation function. We then derive results on the pair correlation between different steps. Results for solvable limiting cases are summarized in Tables I and II of the paper. As an intermediate step in the analysis we also find expressions for the relaxation time tau(pq) of a mode of wave number q along the steps and wave number p perpendicular to the steps, which we also discuss and compare with prior work.

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PubMed1999

Active clusters in disordered systems.

We introduce an exact algorithm to calculate the distribution of large low energy clusters (droplets) in disordered manifolds and disordered magnets, and we analyze the extent to which these clusters can be treated as independent two-level systems. We show that interfaces in randomly diluted networks always have broad droplet distributions, while diluted antiferromagnets in a field can have either power law or exponential droplet distributions.

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