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Martensitic transformations, viewed as continuous transformations between triply periodic minimal surfaces (TPMS), as originally proposed by Hyde and Andersson [Z. Kristallogr. 174, 225 (1986)], is extended to include paths between the initial and final phases. Bravais lattices correspond to particular TPMS whose lattice points are flat points, where the Gaussian curvature vanishes. Reversible transformations, which correspond to shape memory materials, occur only if lattice points remain at flat points on a TPMS throughout a continuous deformation. For the shape memory material NiTi, density-functional theory (DFT) yields irreversible and reversible paths with and without energy barriers, respectively. Although there are TPMS for face-centered gamma-Fe) and body-centered (alpha-Fe) cubic lattices, gamma to alpha deformation paths are not reversible, in agreement with non-vanishing energy barriers obtained from DFT.

The limitations of the classical Black-Scholes model are examined by comparing calculated and actual historical prices of European call options on stocks from several sectors of the S&P 500. Persistent differences between the two prices point to an expanded model proposed by Segal and Segal (1998) in which information not simultaneously observable or actionable with public information can be represented by an additional pseudo-Wiener process. A real linear combination of the original and added processes leads to a commutation relation analogous to that between a boson field and its canonical momentum in quantum field theory. The resulting pricing formula for a European call option replaces the classical volatility with the norm of a complex quantity, whose imaginary part is shown to compensate for the disparity between prices obtained from the classical Black-Scholes model and actual prices of the test call options. This provides market evidence for the influence of a non-classical process on the price of a security based on non-commuting operators., Comment: 11 pages, 3 figures (Supplement - 26 pages, 20 figures)

The path integral for Darcy's law with a stochastic conductivity, which characterizes flow through random porous media, is used as a basis for Wilson renormalization-group (RG) calculations in momentum space. A coarse graining procedure is implemented by integrating over infinitesimal shells of large momenta corresponding to the elimination of the small scale modes of the theory. The resulting one-loop $\beta$-functions are solved exactly to obtain an effective conductivity in a coarse grained theory over successively larger length scales. We first carry out a calculation with uncorrelated Gaussian conductivity fluctuations to illustrate the RG procedure before considering the effect of a finite correlation length of conductivity fluctuations. We conclude by discussing applications and extensions of our calculations, including comparisons with the numerical evaluation of path integrals, non-Gaussian fluctuations, and multiphase flow, for which the path integral formulation should prove particularly useful., Comment: 17 pages, 2 figures

The pressure and flow statistics of Darcy flow through a random permeable medium are expressed in a form suitable for evaluation by the method of simulated annealing. There are several attractive aspects to using simulated annealing: (i) any probability distribution can be used for the permeability, (ii) there is no need to invert the transmissibility matrix which, while not a factor for single-phase flow, offers distinct advantages for the case of multiphase flow, and (iii) the action used for simulated annealing is eminently suitable for coarse graining by integrating over the short-wavelength degrees of freedom. In this paper, we show that the pressure and flow statistics obtained by simulated annealing are in excellent agreement with the more conventional finite-volume calculations.

The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes both a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice and calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schr\"odinger's equation., Comment: Published in European Journal of Physics

The path integral for classical statistical dynamics is used to determine the properties of one-dimensional Darcy flow through a porous medium with a correlated stochastic permeability for several spatial correlation lengths. Pressure statistics are obtained from the numerical evaluation of the path integral by using the Markov chain Monte Carlo method. Comparisons between these pressure distributions and those calculated from the classic finite-volume method for the corresponding stochastic differential equation show excellent agreement for Dirichlet and Neumann boundary conditions. The evaluation of the variance of the pressure based on a continuum description of the medium provides an estimate of the effects of discretization. Log-normal and Gaussian fits to the pressure distributions as a function of position within the porous medium are discussed in relation to the spatial extent of the correlations of the permeability fluctuations.

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multi-variate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

The formation of nanostructures during metalorganic vapor-phase epitaxy on patterned (001)/(111)B GaAs substrates is reviewed. The focus of this review is on the seminal experiments that revealed the key kinetic processes during nanostructure formation and the theory and modelling that explained the phenomenology in successively greater detail. Experiments have demonstrated that V-groove quantum wires and pyramidal quantum dots result from self-limiting concentration profiles that develop at the bottom of V-grooves and inverted pyramids, respectively. In the 1950s, long before the practical importance of patterned substrates became evident, the mechanisms of capillarity during the equilibration of non-planar surfaces were identified and characterized. This was followed, from the late 1980s by the identification of growth rate anisotropies (i.e. differential growth rates of crystallographic facets) and precursor decomposition anisotropies, with parallel developments in the fabrication of V-groove quantum wires and pyramidal quantum dots. The modelling of these growth processes began at the scale of facets and culminated in systems of coupled reaction-diffusion equations, one for each crystallographic facet that defines the pattern, which takes account of the decomposition and surface diffusion kinetics of the group-III precursors and the subsequent surface diffusion and incorporation of the group-III atoms released by these precursors. Solutions of the equations with optimized parameters produced concentration profiles that provided a quantitative interpretation of the time-, temperature-, and alloy-concentration dependence of the self-ordering process seen in experiments., Comment: 26 pages, 13 images

We give an introduction to the calculation of path integrals on a lattice, with the quantum harmonic oscillator as an example. In addition to providing an explicit computational setup and corresponding pseudocode, we pay particular attention to the existence of autocorrelations and the calculation of reliable errors. The over-relaxation technique is presented as a way to counter strong autocorrelations. The simulation methods can be extended to compute observables for path integrals in other settings.

We report a model for metalorganic vapor-phase epitaxy on non-planar substrates, specifically V-grooves and pyramidal recesses, which we apply to the growth of InGaAs nanostructures. This model, based on a set of coupled reaction-diffusion equations, one for each facet in the system, accounts for the facet-dependence of all kinetic processes (e.g., precursor decomposition, adatom diffusion, and adatom lifetimes) and has been previously applied to account for the temperature, concentration, and temporal-dependence of AlGaAs nanostructures on GaAs (111)B surfaces with V-grooves and pyramidal recesses. In the present study, the growth of In$_{0.12}$Ga$_{0.88}$As quantum wires at the bottom of V-grooves is used to determine a set of optimized kinetic parameters. Based on these parameters, we have modeled the growth of In$_{0.25}$Ga$_{0.75}$As nanostructures formed in pyramidal site-controlled quantum-dot systems, successfully producing a qualitative explanation for the temperature-dependence of their optical properties, which have been reported in previous studies. Finally, we present scanning electron and cross-sectional atomic force microscopy images which show previously unreported facetting at the bottom of the pyramidal recesses that allow quantum dot formation., Comment: 9 pages, 8 figures

As one of the first proposed topologically protected states, the quantum spin Hall effect in graphene relies critically on the existence of a spin-dependent gap at the K/K' points of the Brillouin zone. Using a tight-binding formulation based on the method of invariants, we identify the origin of such an intrinsic gap as the three-center interaction between the pi-orbitals caused by spin-orbit interactions. This methodology incorporates all symmetry compliant interactions previously neglected and has wider applications for comparisons between first-principle calculations and the tight-binding method. It also identifies a correction to the Haldane model and its generalization, which incorporates the spin degrees of freedom and reproduces all the salient features required for the quantum spin Hall effect in graphene., Comment: 5 pages 2 figures plus supplementary material

Weight at delivery is a standard cumulative measure of placental growth. But weight is a crude summary of other placental characteristics, such as the size and shape of the chorionic plate and the location of the umbilical cord insertion. Distributions of such measures across a cohort reveal information about the developmental history of the chorionic plate that is unavailable from an analysis based solely on the mean and standard deviation. Various measures were determined from digitized images of chorionic plates obtained from the Pregnancy, Infection, and Nutrition Study, a prospective cohort study of preterm birth in central North Carolina between 2002 and 2004. The centroids (the geometric centers) and umbilical cord insertions were taken directly from the images. The chorionic plate outlines were obtained from an interpolation based on a Fourier series, while eccentricity (of the best-fit ellipse), skewness, and kurtosis were determined from a shape analysis using the method of moments. The distribution of each variable was compared against the normal, lognormal, and Levy distributions. We found only a single measure (eccentricity) with a normal distribution. All other placental measures required lognormal or "heavy-tailed" distributions to account for moderate to extreme deviations from the mean, where relative likelihoods in the cohort far exceeded those of a normal distribution. Normal and lognormal distributions result from the accumulated effects of a large number of independent additive (normal) or multiplicative (lognormal) events. Thus, while most placentas appear to develop by a series of small, regular, and independent steps, the presence of heavy-tailed distributions suggests that many show shape features which are more consistent with a large number of correlated steps or fewer, but substantially larger, independent steps., Comment: 26 pages, 7 figures, 2 tables

Graphene, a hexagonal sheet of $sp^2$-bonded carbon atoms, has extraordinary properties which hold immense promise for future nanoelectronic applications. Unfortunately, the popular preparation methods of micromechanical cleavage and chemical exfoliation of graphite do not easily scale up for application purposes. Epitaxial graphene provides an attractive alternative, though there are many challenges, not least of which is the absence of an understanding of the complex atomistic assembly kinetics of graphene. Here, we present a simple rate theory of epitaxial graphene growth on close-packed metal surfaces. Based on recent low-energy electron-diffraction microscopy experiments (LEEM) \cite{loginova09}, our theory supposes that graphene islands grow predominantly by the addition of five-atom clusters, rather than solely by the capture of diffusing carbon atoms. With suitably chosen kinetic parameters, our theory produces a time-dependent carbon adatom density that is in quantitative agreement with measured data. The temperature-dependence of this adatom density at the onset of nucleation leads us to predict that the smallest stable precursor to graphene growth is an immobile island composed of six five-atom clusters. Our findings provide a starting point for more detailed simulations which will yield important input to developing strategies for the large-scale production of epitaxial graphene., Comment: 6 pages, 5 figures, 1 table

We propose a method of characterizing radial networks based on a partition function associated with the structural triangulation of the network. The internal energy, Helmholtz free energy, and entropy derived from the partition function are used to group similar networks together and to interrogate the history of their development. We illustrate our methodology for a model for optimal transport in tree leaves., Comment: 4 pages, 4 figures

We investigate the growth kinetics on vicinal GaAs(001) surfaces by making detailed comparisons between reflection high--energy electron--diffraction specular intensity measured near in--phase diffraction conditions and the surface step density obtained from simulations of a solid--on--solid model. Only by including a barrier to interlayer transport and a short--range incorporation process of freshly--deposited atoms can the simulations be brought into agreement with the measurements both during growth and during post--growth equilibration of the surface., Comment: 11 pages, REVTeX 3.0, IC-DDV-93-005