Ensemble-averaged Density Research Articles

Yielding under compression and the polyamorphic transition in silicon

We investigate the behavior of amorphous silicon under hydrostatic compression using molecular simulations. During compression, amorphous silicon undergoes a discontinuous nonequilibrium transition from a low-density to a high-density structure at a pressure of around $13$-$16$~GPa. Ensemble-averaged density and elastic constants change discontinuously across the transition. Densification of individual glassy samples occurs through a series of discrete plastic events, each of which is accompanied by a vanishing shear modulus. This is the signature of a series of elastic instabilities, similar to shear transformation zones observed during shear yielding of glasses. We compare the structure obtained during compression with a near-equilibrium form of amorphous silicon obtained by quenching a melt at constant pressure. This gives structures identical to nonequilibrium compression at low and high pressure, but the transition between them occurs gradually rather than discontinuously. Our observations indicate that the polyamorphic transition is of a nonequilibrium nature, and it has the characteristics of a yield transition that occurs under compression instead of shear.

Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with k-body interactions.

Embedded random matrix ensembles with k-body interactions are well established to be appropriate for many quantum systems. For these ensembles the two point correlation function is not yet derived, though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues, say E and E^{'}. Fluctuation measures such as the number variance and Dyson-Mehta Δ_{3} statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with k-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called q-normal distribution. With this, the eigenvalue density can be expanded by starting with the q-normal form and using the associated q-Hermite polynomials He_{ζ}(x|q). Covariances S_{ζ}S_{ζ^{'}}[over ¯] (overline representing ensemble average) of the expansion coefficients S_{ζ} with ζ≥1 here determine the two-point function as they are a linear combination of the bivariate moments Σ_{PQ} of the two-point function. Besides describing all these, in this paper formulas are derived for the bivariate moments Σ_{PQ} with P+Q≤8, of the two-point correlation function, for the embedded Gaussian unitary ensembles with k-body interactions [EGUE(k)] as appropriate for systems with m fermions in N single particle states. Used for obtaining the formulas is the SU(N) Wigner-Racah algebra. These formulas with finite N corrections are used to derive formulas for the covariances S_{ζ}S_{ζ^{'}}[over ¯] in the asymptotic limit. These show that the present work extends to all k values, the results known in the past in the two extreme limits with k/m→0 (same as q→1) and k=m (same as q=0).

One-dimensional turbulence modeling of compressible flows: II. Full compressible modification and application to shock–turbulence interaction

One-dimensional turbulence (ODT) is a simulation methodology that represents the essential physics of three-dimensional turbulence through stochastic resolution of the full range of length and time scales on a one-dimensional domain. In the present study, full compressible modifications are incorporated into ODT methodology, based on an Eulerian framework and a conservative form of the governing equations. In the deterministic part of this approach, a shock capturing scheme is introduced for the first time. In the stochastic part, one-dimensional eddy events are modeled and sampled according to standard methods for compressible flow simulation. Time advancement adjustments are made to balance comparable time steps between the deterministic and stochastic parts in compressible flows. Canonical shock–turbulence interaction cases involving Richtmyer–Meshkov instability at Mach numbers 1.24, 1.5, and 1.98 are simulated to validate the extended model. The ODT results are compared with available reference data from large eddy simulations and laboratory experiments. The introduction of a shock capturing scheme significantly improves the performance of the ODT method, and the results for turbulent kinetic energy are qualitatively improved compared with those of a previous compressible Lagrangian ODT method [Jozefik et al., “Simulation of shock–turbulence interaction in non-reactive flow and in turbulent deflagration and detonation regimes using one-dimensional turbulence,” Combust. Flame 164, 53 (2016)]. For the time evolution of profiles of the turbulent mixing zone width, ensemble-averaged density, and specific heat ratio, the new model also yields good to reasonable results. Furthermore, it is found that the viscous penalty parameter Z of the ODT model is insensitive to compressibility effects in turbulent flows without wall effects. A small value of Z is appropriate for turbulent flows with weak wall effects, and the parameter Z serves to suppress extremely small eddy events that would be dissipated instantly by viscosity.

Strongly correlated zero-bias anomaly in double quantum dot measurements

Experiments in doped transition metal oxides often show suppression in the single-particle density of states at the Fermi level, but disorder-induced zero-bias anomalies in strongly correlated systems remain poorly understood. Numerical studies of the Anderson-Hubbard model have identified a zero-bias anomaly that is unique to strongly correlated materials, with a width proportional to the intersite hopping amplitude t [S. Chiesa, P. B. Chakraborty, W. E. Pickett, and R. T. Scalettar, Phys. Rev. Lett. 101, 086401 (2008)]. In ensembles of two-site systems, a zero-bias anomaly with the same parameter dependence also occurs, suggesting a similar physical origin [R. Wortis and W. A. Atkinson, Phys. Rev. B 82, 073107 (2010)]. We describe how this kinetic-energy-driven zero-bias anomaly in ensembles of two-site systems may be seen in a mesoscopic realization based on double quantum dots. Moreover, the double-quantum-dot measurements provide access not only to the ensemble-average density of states but also to the details of the transitions which give rise to the zero-bias anomaly.

Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the basic reproduction numberR0, which is the average number of secondary infections generated from every infected. We show that the R0 formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model. The method can be generically implemented to analyze other reaction-diffusion models.

Average density of states of amorphous Hamiltonians: role of phonon mediated coupling of nano-clusters

Based on a description of an amorphous solid as a collection of coupled nanosize molecular clusters referred as basic blocks, we analyse the statistical properties of its Hamiltonian. The information is then used to derive the ensemble averaged density of the vibrational states (non-phonon) which turns out to be a Gaussian in the bulk of the spectrum and an Airy function in the low frequency regime. A comparison with experimental data for six glasses confirms validity of our theoretical predictions.

An Analytic Model for the Subgalactic Matter Power Spectrum in Fuzzy Dark Matter Halos

Abstract Fuzzy dark matter (FDM), a scalar particle coupled to the gravitational field without self-interaction, whose mass range is m ∼ 10−24–10−20 eV, is one of the promising alternative dark matter candidates to cold dark matter. The quantum interference pattern, which is a unique structure of FDM, can be seen in halos in cosmological FDM simulations. In this paper, we first provide an analytic model of the subgalactic matter power spectrum originating from quantum clumps in FDM halos, in which the density distribution of the FDM is expressed by a superposition of quantum clumps whose size corresponds to the de Broglie wavelength of the FDM. These clumps are assumed to be distributed randomly, such that the ensemble average density follows a halo profile such as the Navarro–Frenk–White profile. We then compare the convergence power spectrum projected along the line of sight around the Einstein radius, which is converted from the subgalactic matter power spectrum, to that measured in the strong lens system SDSS J0252 + 0039. While we find that the current observation provides no useful constraint on the FDM mass, we show that future deep, high spatial resolution observations of strong lens systems can tightly constrain FDM with a mass around 10−22 eV.

Molecule Mechanics from Cryo-EM Images and Multiple Reconstructed Densities

After the resolution revolution, cryo-EM moves from sets of individual structures, to understanding properties of biological samples as a whole, like pathways between resolved structures and binding affinities. Extracting these molecule mechanics as probability distribution of arbitrary observables, like open and close configuation, or small molecule binding probabilities however, is still challenging despite increasing resolution and numbers of resolved structures of a single complex, due to the non-trivial connection three dimensional structures and all the possible molecule conformations they represent. Here, we present how to resolve all molecule conformations that are represented by three dimensional structures and pathways between multiple reconstructed densities. The opening and closing dynamics of adenylate kinase are used to verify our the method, using ensemble-averaged density maps of open and closed state. In the ribosome-selB complex, our method further refines structural transitions for GTPase activation. In our work, the free energy for each molecule conformation under a cryo-EM experiment is derived from a probabilistic formulation of the complete cryo-EM measurement process. It is shown why a set of reconstructed densities is a good approximation to the rigorous but computationally challenging calculation of conformation weights from all cryo-EM images. It is found that morphing densities and refining structures into cryo-EM maps is tightly coupled through the measure of goodness-of-fit. With these insights, we asess how much of all underlying configurations are represented by single densities and sample pathways between densities, that represent physical intermediate states of the ribosome-selB complex upon GTPase activation.

Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

We consider the effects of impurities on topological insulators and superconductors. We start by identifying the general conditions under which the eigenenergies of an arbitrary Hamiltonian H belonging to one of the Altland-Zirnbauer symmetry classes undergo a robust zero energy crossing as a function of an external parameter which can be, for example, the impurity strength. We define a generalized root of \det H, and use it to predict or rule out robust zero-energy crossings in all symmetry classes. We complement this result with an analysis based on almost degenerate perturbation theory, which allows a derivation of the asymptotic low-energy behavior of the ensemble averaged density of states $\rho \sim E^\alpha$ for all symmetry classes, and makes it transparent that the exponent \alpha\ does not depend on the choice of the random matrix ensemble. Finally, we show that a lattice of impurities can drive a topologically trivial system into a nontrivial phase, and in particular we demonstrate that impurity bands carrying extremely large Chern numbers can appear in different symmetry classes of two-dimensional topological insulators and superconductors. We use the generalized root of \det H(k) to reveal a spiderweblike momentum space structure of the energy gap closings that separate the topologically distinct phases in p_x + i p_y superconductors in the presence of an impurity lattice.

Correlation functions and correlation widths in quantum-chaotic scattering for mesoscopic systems and nuclei.

We derive analytical expressions for the correlation functions of the electronic conductance fluctuations of an open quantum dot under several conditions. Both the variation of energy and that of an external parameter, such as an applied perpendicular or parallel magnetic fields, are considered in the general case of partial openness. These expressions are then used to obtain the ensemble-averaged density of maxima, a measure recently suggested to contain invaluable information concerning the correlation widths of chaotic systems. The correlation width is then calculated for the case of energy variation, and a significant deviation from the Weisskopf estimate is found in the case of two terminals. The results are extended to more than two terminals. All of our results are analytical. The use of these results in other fields, such as nuclei, where the system can only be studied through a variation of the energy, is then discussed.

Frozen-density embedding theory with average solvent charge densities from explicit atomistic simulations.

Besides molecular electron densities obtained within the Born-Oppenheimer approximation (ρB(r)) to represent the environment, the ensemble averaged density (〈ρB〉(r)) is also admissible in frozen-density embedding theory (FDET) [Wesolowski, Phys. Rev. A, 2008, 77, 11444]. This makes it possible to introduce an approximation in the evaluation of the solvent effect on quantum mechanical observables consisting of replacing the ensemble averaged observable by the observable evaluated at ensemble averaged ρB(r). This approximation is shown to affect negligibly the solvatochromic shift in the absorption of hydrated acetone. The proposed model provides a continuum type of representation of the solvent, which reflects nevertheless its local structure, and it is to be applied as a post-simulation analysis tool in atomistic level simulations.

A molecular dynamics study of phase transition in strongly coupled pair-ion plasmas

Existence of phase transition in strongly coupled pair-ion plasmas with soft core is investigated. Extensive Molecular Dynamics (MD) simulations are performed in the canonical ensemble, for such plasmas, at different temperatures, to analyze phase stability. Our studies show interesting phase co-existence between liquid-like and vapor-like phases. The different phases are identified by calculating the ensemble averaged density. This and the corresponding critical properties are calculated directly from MD simulation. The critical temperature of vapor-liquid coexistence is obtained, and the corresponding critical value of density is also estimated for different sizes of the soft core. We have used a novel method that allows the location of phase coexistence through a constant density simulation in which the temperature is changed in a single time-step (quenching) in order to place the system in a thermodynamically and mechanically unstable state, resulting in spontaneous separation of two coexisting phases. The results obtained from this temperature quench MD method also show the coexistence of vapor-liquid phase in pair-ion plasmas. The critical exponents obtained directly from MD simulation are found to be in close agreement with the values predicted by a mean-field theory.

QCWAVE – A Mathematica quantum computer simulation update

This Mathematica 7.0/8.0 package upgrades and extends the quantum computer simulation code called QDENSITY. Use of the density matrix was emphasized in QDENSITY, although that code was also applicable to a quantum state description. In the present version, the quantum state version is stressed and made amenable to future extensions to parallel computer simulations. The add-on QCWAVE extends QDENSITY in several ways. The first way is to describe the action of one, two and three-qubit quantum gates as a set of small (2×2, 4×4 or 8×8) matrices acting on the 2nq amplitudes for a system of nq qubits. This procedure was described in our parallel computer simulation QCMPI and is reviewed here. The advantage is that smaller storage demands are made, without loss of speed, and that the procedure can take advantage of message passing interface (MPI) techniques, which will hopefully be generally available in future Mathematica versions.Another extension of QDENSITY provided here is a multiverse approach, as described in our QCMPI paper. This multiverse approach involves using the present slave–master parallel processing capabilities of Mathematica 7.0/8.0 to simulate errors and error correction. The basic idea is that parallel versions of QCWAVE run simultaneously with random errors introduced on some of the processors, with an ensemble average used to represent the real world situation. Within this approach, error correction steps can be simulated and their efficacy tested. This capability allows one to examine the detrimental effects of errors and the benefits of error correction on particular quantum algorithms.Other upgrades provided in this version include circuit-diagram drawing commands, better Dirac form and amplitude display features. These are included in the add-ons QCWave.m and Circuits.m, and are illustrated in tutorial notebooks.In separate notebooks, QCWAVE is applied to sample algorithms in which the parallel multiverse setup is illustrated and error correction is simulated. These extensions and upgrades will hopefully help in both instruction and in application to QC dynamics and error correction studies. Program summaryProgram title: QCWAVECatalogue identifier: ADXH_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXH_v3_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 94 159No. of bytes in distributed program, including test data, etc.: 734 158Distribution format: tar.gzProgramming language: Mathematica 7.0 and 8.0.Computer: Any supporting Mathematica.Operating system: Any operating system that supports Mathematica; tested under Microsoft Windows XP, Macintosh OSX, and Linux FC4.Has the code been vectorised or parallelized?: Utilises Mathematicaʼs (7.0 and 8.0) parallel computing features.Classification: 4.15.Catalogue identifier of previous version: ADXH_v2_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 474.Does the new version supersede the previous version?: Yes. This version supersedes all prior versions of QDENSITY.Nature of problem: Simulation of quantum circuits, quantum algorithms, noise and quantum error correction.Solution method: A Mathematica package containing commands to create and analyze quantum circuits is upgraded and extended, with emphasis on state amplitudes. Several Mathematica notebooks containing relevant examples are explained in detail. The parallel computing feature of Mathematica is used to develop a multiverse approach for including noise and forming suitable ensemble averaged density matrix evolution. Error correction is simulated.Reasons for new version: The new version updates QDENSITY to run on Mathematica 7.0 and 8.0 and makes it compatible with our extension QCWAVE. QCWAVE emphasizes wavefunctions with efficient gate operations and also extends the code to use the parallel computing features of Mathematica 7.0–8.0. Circuit diagram and amplitude display are new features. Dirac display of states is also provided.Summary of revisions: The revisions include working with state vectors and the implementation of efficient Op1, Op2 and Op3, one, two, three gate operators. Parallel processing is used to form a multiverse approach for simulating noise effects and error corrections in quantum operations. Drawing circuit diagrams and displaying amplitude evolution has been added. A simple Dirac display feature DForm has also been provided.Running time: The notebooks provided in the distribution package take only a matter of minutes to execute.

On the interplay between sedimentation and phase separation phenomena in two-dimensional colloidal fluids

Colloidal particles that are confined to an interface effectively form a two-dimensional fluid. We examine the dynamics of such colloids when they are subject to a constant external force, which drives them in a particular direction over the surface. Such a situation occurs, for example, for colloidal particles that have settled to the bottom of their container, when the container is tilted at an angle, so that they ‘sediment’ to the lower edge of the surface. We focus in particular on the case when there are attractive forces between the colloids which causes them to phase separate into regions of high density and low density and we study the influence of this phase separation on the sedimentation process. We model the colloids as Brownian particles and use both Brownian dynamics computer simulations and dynamical density functional theory (DDFT) to obtain the time evolution of the ensemble average one-body density profiles of the colloids. We consider situations where the external potential varies only in one direction so that the ensemble average density profiles vary only in this direction. We solve the DDFT in one-dimension, by assuming that the density profile only varies in one direction. However, we also solve the DDFT in two dimensions, allowing the fluid density profile to vary in both the x- and y-directions. We find that in certain situations the two-dimensional DDFT is clearly superior to its one-dimensional counterpart when compared with the simulations and we discuss this issue.

Grating-based holographic X-ray diffraction: theory and application to confined fluids

A grating-based holographic X-ray diffraction technique has been developed for reconstructing density profiles of nano-scale fluids confined in channel arrays. Within this approach, the reference wave is due to diffraction from the fabricated channel array, whereas the object wave is generated by the confinement-induced ordering of the fluid. The ensemble-averaged density profile of the fluid across the confining channel, which constitutes a weak phase object, is then determined in a model-independent manner from the interference between the reference and object waves by direct Fourier inversion. The validity of the linear holographic approach and its connection to the autocorrelation function, the inclusion of channel tapering, and volume-diffraction effects are discussed in detail.

Random walks on random Koch curves

Diffusion processes in porous materials are often modeled as random walks on fractals. In order to capture the randomness of the materials random fractals are employed, which no longer show the deterministic self-similarity of regular fractals. Finding a continuum differential equation describing the diffusion on such fractals has been a long-standing goal, and we address the question of whether the concepts developed for regular fractals are still applicable. We use the random Koch curve as a convenient example as it provides certain technical advantages by its separation of time and space features. While some of the concepts developed for regular fractals can be used unaltered, others have to be modified. Based on the concept of fibers, we introduce ensemble-averaged density functions which produce a differentiable estimate of probability explicitly and compare it to random walk data.

Theory of charged impurity scattering in two-dimensional graphene

We review the physics of charged impurities in the vicinity of graphene. The long-range nature of Coulomb impurities affects both the nature of the ground state density profile and graphene’s transport properties. We discuss the screening of a single Coulomb impurity and the ensemble averaged density profile of graphene in the presence of many randomly distributed impurities. Finally, we discuss graphene’s transport properties due to scattering off charged impurities both at low and high carrier density.

Solid-liquid coexistence of polydisperse fluids via simulation

We describe a simulation method for the accurate study of the equilibrium freezing properties of polydisperse fluids under the experimentally relevant condition of fixed polydispersity. The approach is based on the phase switch Monte Carlo method of Wilding and Bruce [Phys. Rev. Lett. 85, 5138 (2000)]. This we have generalized to deal with particle size polydispersity by incorporating updates which alter the diameter sigma of a particle, under the control of a distribution of chemical potential differences mu(sigma). Within the resulting isobaric semi-grand-canonical ensemble, we detail how to adapt mu(sigma) and the applied pressure such as to study coexistence, while ensuring that the ensemble averaged density distribution rho(sigma) matches a fixed functional form. Results are presented for the effects of small degrees of polydispersity on the solid-liquid transition of soft spheres.

Structure of confined fluids by x-ray interferometry using diffraction gratings

We develop a novel method for structure determination of confined fluids using diffraction-grating-based x-ray interferometry.Within this approach, diffraction from a microfluidic array, which acts both as confinement and transmission diffraction grating, provides the reference wave, whereas the density modulations of the confined fluid, acting as a weak phase object, generate the object wave. The ensemble-averaged density profile of the fluid perpendicular to the confining channel is then unambiguously obtained from the interference between the reference and object waves by direct Fourier inversion.

Confinement-induced liquid ordering investigated by x-ray phase retrieval

Using synchrotron x-ray diffraction, we have determined the ensemble-averaged density profile of colloidal fluids within confining channels of different widths. We observe an oscillatory ordering-disordering behavior of the colloidal particles as a function of the channel width, while the colloidal solution remains in the liquid state. This phenomenon has been suggested by surface force studies of hard-sphere fluids and also theoretically predicted, but here we see it by direct measurements of the structure for comparable systems.