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Tetratic order in the phase behavior of a hard-rectangle system

Physical Review B 2006年第5期73卷 054109-054109页

作者： Aleksandar Donev Joshua Burton Frank H. Stillinger Salvatore Torquato Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA PRISM Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA

Previous Monte Carlo investigations by Wojciechowski et al. have found two unusual phases in two-dimensional systems of anisotropic hard particles: a tetratic phase of fourfold symmetry for hard squares [Comput. Methods Sci. Tech. 10, 235 (2004)], and a nonperiodic degenerate solid phase for hard-disk dimers [Phys. Rev. Lett. 66, 3168 (1991)]. In this work, we study a system of hard rectangles of aspect ratio two, i.e., hard-square dimers (or dominos), and demonstrate that it exhibits phases with both of these unusual properties. The liquid shows quasi-long-range tetratic order, with no nematic order. The solid phase we observe is a nonperiodic tetratic phase having the structure of a random tiling of the square lattice with dominos with the well-known degeneracy entropy 1.79kB per particle. Our simulations do not conclusively establish the thermodynamic stability of this orientationally disordered solid; however, there are strong indications that this phase is glassy. Our observations are consistent with a two-stage phase transition scenario developed by Kosterlitz and co-workers with two continuous phase transitions, the first from isotropic to tetratic liquid, and the second from tetratic liquid to solid. We obtain similar results with both a classical Monte Carlo method using true rectangles and a novel molecular dynamics algorithm employing rectangles with rounded corners.

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Mechanism of enhanced broadening of the ionization level of Li outside transition metal surfaces

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Physical Review B 2006年第11期74卷 115109-115109页

作者： Keith Niedfeldt Peter Nordlander Emily A. Carter Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544-5263 USA Department of Physics and Astronomy M.S. 61 Rice University P.O. Box 1892 6100 South Main Street Houston Texas 77251-1892 USA

We show that the d-band of a transition metal surface induces intra-atomic hybridization of an atom in its vicinity. It is demonstrated that such hybridization can have a profound influence on the resonance width and hence lifetime of the atomic ionization level. The degree of Li s−p hybridization is found to be directly correlated to the overlap of the d-band density of states with the Li 2s and 2p levels, and is shown to be not due solely to long-range electrostatic image potential effects.

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Packing hyperspheres in high-dimensional Euclidean spaces

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Physical Review E 2006年第4期74卷 041127-041127页

作者： Monica Skoge Aleksandar Donev Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA PRISM Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

We present a study of disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d=4, 5, and 6 to be ϕMRJ≈0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form ϕMRJ=c1∕2d+(c2d)∕2d, where c1=−2.72 and c2=2.56, which appears to be consistent with the high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g2(r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed “decorrelation principle,” which, among other things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d→∞. As in three dimensions (where ϕMRJ≈0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact with power-law exponent ≈0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2(r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, ϕF≈0.32 and ϕM≈0.39, respectively, for d=4, and ϕF≈0.20 and ϕM≈0.25, respectively, for d=5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension.

关键词： RANDOM CLOSE PACKING 4TH VIRIAL-COEFFICIENTS HARD-PARTICLE PACKINGS ARBITRARY DIMENSIONALITY BOUNDARY-CONDITIONS MONTE-CARLO SPHERES EQUATION STATE SIMULATION

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Self-assembly of the simple cubic lattice with an isotropic potential

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Physical Review E 2006年第2期74卷 021404-021404页

作者： Mikael C. Rechtsman Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology Materials Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

Conventional wisdom presumes that low-coordinated crystal ground states require directional interactions. Using our recently introduced optimization procedure to achieve self-assembly of targeted structures [M. C. Rechtsman et al., Phys. Rev. Lett. 95, 228301 (2005); Phys. Rev. E 73, 011406 (2006)], we present an isotropic pair potential V(r) for a three-dimensional many-particle system whose classical ground state is the low-coordinated simple cubic lattice. This result is part of an ongoing pursuit by the authors to develop analytical and computational tools to solve statistical-mechanical inverse problems for the purpose of achieving targeted self-assembly. The purpose of these methods is to design interparticle interactions that cause self-assembly of technologically important target structures for applications in photonics, catalysis, separation, sensors, and electronics. We also show that standard approximate integral-equation theories of the liquid state that utilize pair correlation function information cannot be used in the reverse mode to predict the correct simple cubic potential. We report in passing optimized isotropic potentials that yield the body-centered-cubic and simple hexagonal lattices, which provide other examples of non-close-packed structures that can be assembled using isotropic pair interactions.

关键词： CRYSTALS CLUSTERS DENSE

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Collective coordinate control of density distributions

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Physical Review E 2006年第3期74卷 031104-031104页

作者： Obioma U. Uche Salvatore Torquato Frank H. Stillinger Department of Chemical Engineering Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied & Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

Real collective density variables C(k) [cf. Eq. (1.3)] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor S(k), where k denotes the wave vector. Our objective is to prescribe C(k) and then to find many-particle configurations that correspond to such a target C(k) using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control S(k) in the neighborhood of ∣k∣=0. The optimization method employed generates multiparticle configurations for which S(k)∝∣k∣α, ∣k∣⩽K, and α=1, 2, 4, 6, 8, and 10. The case α=1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid He4, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground states are configurationally degenerate and disordered.

关键词： STATISTICAL MECHANICS REALIZABILITY SYSTEM

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Rank-width and Well-quasi-ordering of Skew-symmetric Matrices. (extended abstract)

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Electronic Notes in Discrete mathematics 2005年 22卷 281-285页

作者： Oum, Sang-il Program in Applied and Computational Mathematics Princeton University Princeton NJ 08540 United States

Robertson and Seymour prove that a set of graphs of bounded tree-width is well-quasi-ordered by the graph minor relation. By extending their methods to matroids, Geelen, Gerards, and Whittle prove that a set of matroids representable over a fixed finite field are well-quasi-ordered if it has bounded branch-width. More recently, it is shown that a set of graphs of bounded rank-width (or clique-width) is well-quasi-ordered by the graph vertex-minor relation. The proof of the last one uses isotropic systems defined by A. Bouchet. We obtain a common generalization of the above three theorems in terms of skew-symmetric matrices over a fixed finite field. © 2005 Elsevier B.V. All rights reserved.

关键词： branch-width isotropic system principal pivot rank-width skew-symmetric matrix tree-width well-quasi-ordering

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Multiple time scale numerical methods for the inverted pendulum problem

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Lecture Notes in computational Science and Engineering 2005年 44卷 241-261页

作者： Sharp, Richard Tsai, Yen-Hsi Engquist, Björn Program in Applied and Computational Mathematics Princeton University NJ 08544 United States Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin TX 78712 United States

ISBN: (纸本)9783540253358

In this article, we study a class of numerical ODE schemes that use a time filtering strategy and operate in two time scales. The algorithms follow the framework of the heterogeneous multiscale methods (HMM) [1]. We apply the methods to compute the averaged path of the inverted pendulum under a highly oscillatory vertical forcing on the pivot. The averaged equation for related problems has been studied analytically in [9]. We prove and show numerically that the proposed methods approximate the averaged equation and thus compute the average path of the inverted pendulum.

关键词： Inverted pendulum

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Core Collapse via Coarse Dynamic Renormalization

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Physical Review Letters 2005年第8期95卷 081102-081102页

作者： Andras Szell David Merritt Ioannis G. Kevrekidis Department of Chemical Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In the context of the recently developed “equation-free” approach to computer-assisted analysis of complex systems, we extract the self-similar solution describing core collapse of a stellar system from numerical experiments. The technique allows us to sidestep the core “bounce” that occurs in direct N-body simulations due to the small-N correlations that develop in the late stages of collapse, and hence to follow the evolution well into the self-similar regime.

关键词： Large scale systems

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Multiscale simulations in simple metals: A density-functional-based methodology

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Physical Review B 2005年第9期71卷 094101-094101页

作者： Nicholas Choly Gang Lu Weinan E Efthimios Kaxiras Division of Engineering and Applied Sciences Harvard University Cambridge Massachusetts 02138 USA Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We present a formalism for coupling a density-functional-theory-based quantum simulation to a classical simulation for the treatment of simple metallic systems. The formalism is applicable to multiscale simulations in which the part of the system requiring quantum-mechanical treatment is spatially confined to a small region. Such situations often arise in physical systems where chemical interactions in a small region can affect the macroscopic mechanical properties of a metal. We describe how this coupled treatment can be accomplished efficiently, and we present a coupled simulation for a bulk aluminum system.

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Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings

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Physical Review E 2005年第1期71卷 011105-011105页

作者： Aleksandar Donev Salvatore Torquato Frank H. Stillinger Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA PRISM Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA

We study the approach to jamming in hard-sphere packings and, in particular, the pair correlation function g2(r) around contact, both theoretically and computationally. Our computational data unambiguously separate the narrowing δ-function contribution to g2 due to emerging interparticle contacts from the background contribution due to near contacts. The data also show with unprecedented accuracy that disordered hard-sphere packings are strictly isostatic: i.e., the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom, once rattlers are removed. For such isostatic packings, we derive a theoretical connection between the probability distribution of interparticle forces Pf(f), which we measure computationally, and the contact contribution to g2. We verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state. We clearly observe a maximum in Pf and a nonzero probability of zero force, shedding light on long-standing questions in the granular-media literature. We computationally observe an unusual power-law divergence in the near-contact contribution to g2, persistent even in the jamming limit, with exponent −0.4 clearly distinguishable from previously proposed inverse-square-root divergence. Additionally, we present high-quality numerical data on the two discontinuities in the split-second peak of g2 and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features. Finally, we present the computational data on the contact contribution to g2 for vacancy-diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings. We find that the contact network remains isostatic even when ordering is present. Unlike previous studies, we find that ordering has a significant impact on the sh

关键词： MOLECULAR-DYNAMICS SIMULATION DISK PACKINGS CRYSTALLIZATION DISTRIBUTIONS GENERATION PARTICLES PRESSURE

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