Random Lattice Model Research Articles

Vibrational spectrum of Granular packings with random matrices

The vibrational spectrum of granular packings can be used as a signature of the jamming transition, with the density of states at zero frequency becoming nonzero at the transition. It has been proposed previously that the vibrational spectrum of granular packings can be approximately obtained from random matrix theory. Here, we show that the autocorrelation function of the density of states shows good agreement between dynamical numerical simulations of frictionless bead packs near the jamming point and the analytic predictions of the Laguerre orthogonal ensemble of random matrices; there is clear disagreement with the Gaussian orthogonal ensemble, establishing that the Laguerre ensemble correctly reproduces the universal statistical properties of jammed granular matter and excluding the Gaussian orthogonal ensemble. We also present a random lattice model which is a physically motivated variant of the random matrix ensemble. Numerical calculations reveal that this model reproduces the known features of the vibrational density of states of frictionless granular matter, while also retaining the correlation structure seen in the Laguerre random matrix theory.Graphic abstract

Comparing Microscopic and Macroscopic Dynamics in a Paradigmatic Model of Glass-Forming Molecular Liquid.

Glass transition is a most intriguing and long-standing open issue in the field of molecular liquids. From a macroscopic perspective, glass-forming systems display a dramatic slowing-down of the dynamics, with the inverse diffusion coefficient and the structural relaxation times increasing by orders of magnitude upon even modest supercooling. At the microscopic level, single-molecule motion becomes strongly intermittent, and can be conveniently described in terms of “cage-jump” events. In this work, we investigate a paradigmatic glass-forming liquid, the Kob–Andersen Lennard–Jones model, by means of Molecular Dynamics simulations, and compare the macroscopic and microscopic descriptions of its dynamics on approaching the glass-transition. We find that clear changes in the relations between macroscopic timescales and cage-jump quantities occur at the crossover temperature where Mode Coupling-like description starts failing. In fact, Continuous Time Random Walk and lattice model predictions based on cage-jump statistics are also violated below the crossover temperature, suggesting the onset of a qualitative change in cage-jump motion. Interestingly, we show that a fully microscopic relation linking cage-jump time- and length-scales instead holds throughout the investigated temperature range.

Random walk discrete element lattice boltzmann model for scalar transport in fluid and particle flows with strict scalar mass conservation

Scalar transport in fluid-particle systems can be found in many natural processes and engineering applications. Modeling these systems can be challenging due to complex moving boundaries involved, such as fluid-particle, solute-particle interfaces. One often overlooked issue is: the scalar mass within the fluid domain is not strictly conserved even the no-flux boundary condition is applied in many grid-based methods. Here we introduce a hybrid model for simulating scalar transport in particulate flows with strict scalar mass conservation. It is achieved by adapting the Random Walk Method (RWM) to simulate the scalar transport and coupled with the Discrete Element Lattice Boltzmann Method (DE-LBM) for fluid-particle interactions. A moving boundary condition is proposed to prevent the scalar from penetrating solid boundaries and special treatments are presented in detail. The proposed model is validated by a series of benchmarks and experiments, then the mass conservation for both RWM and grid-based methods are checked. Finally, the model is applied to simulate scalar (solute) dispersion with particles in an oscillator which demonstrate its capability on simulating fluid-particle-solute systems. Furthermore, it is straightforward to extend the proposed model for non-spherical particles and reactive boundary conditions.

Tracer diffusion on a crowded random Manhattan lattice

We study, by extensive numerical simulations, the dynamics of a hard-core tracer particle (TP) in presence of two competing types of disorder—frozen convection flows on a square random Manhattan lattice and a crowded dynamical environment formed by a lattice gas of mobile hard-core particles. The latter perform lattice random walks, constrained by a single-occupancy condition of each lattice site, and are either insensitive to random flows (model A) or choose the jump directions as dictated by the local directionality of bonds of the random Manhattan lattice (model B). We focus on the TP disorder-averaged mean-squared displacement, (which shows a super-diffusive behaviour ∼t4/3, t being time, in all the cases studied here), on higher moments of the TP displacement, and on the probability distribution of the TP position X along the x-axis, for which we unveil a previously unknown behaviour. Indeed, our analysis evidences that in absence of the lattice gas particles the latter probability distribution has a Gaussian central part , where u = X/t2/3, and exhibits slower-than-Gaussian tails for sufficiently large t and u. Numerical data convincingly demonstrate that in presence of a crowded environment the central Gaussian part and non-Gaussian tails of the distribution persist for both models.

Random beam lattice modeling method for catastrophic crack growth simulation of brittle-like materials

A random and irregular beam lattice modeling method is developed to predict the crack growth process of brittle-like materials. The model representing the random degree of beam lattice model is defined, which is used to study the influence of lattice mesh on the crack path prediction results. In the simulation, the coupling interactionsamong cracks initiation and growth can be considered. Crack growth process of two double-edge notched specimens, respectively, under shear and tension and direct-tension were simulated and compared with the corresponding experimental results. By comparison of simulation and experimental results, effectiveness of the developed beam lattice modeling method is verified, and can be used to simulate crack growth for studying the fracture mechanisms of brittle-like materials. And the regular beam lattice model can affect the crack paths prediction, and random and irregular beam lattice model is more suitable for simulating crack growth.

Dual random lattice modeling of backward erosion piping

This manuscript provides a novel random lattice model to simulate the progressive degradation of soil embankments induced by the backward erosion piping (BEP) phenomenon. The progressive evolution of the piping process is described by expressing apparent diffusivity of the soil as a function of the local hydraulic gradient. In order to accurately compute local field gradients, a dual lattice approach that evaluates the response on both Delaunay and Voronoi nodes is formulated. The response computed at the Delaunay nodes is then employed to augment field gradient computation on the Voronoi grid. The proposed dual lattice model is numerically verified and validated by comparing numerical results with BEP experiments available in the literature.

Percolation and the electron–electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 μm in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage V g , the conductivity of the array varies by five orders of magnitude in the temperature range T from 1.4 to 77 K in good agreement with the formula σ(V g ) = (V g −V * (T))β with β = 4. The saturation of σ(T) at low temperatures is absent because of the electron–electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence σ(T, V g ) and makes it possible to determine the fraction of microcontacts x(V g , T) with conductances higher than σ. It is found that the dependence x(V g ) is nonlinear and the critical exponent in the formula σ ∝ − (x - 1/2) t in the range 1.3 < t(T, V g ) < β.

Application of the quasi-random lattice model to rare-earth halide solutions for the computation of their osmotic and mean activity coefficients

This work dealt with the computation of the mean activity coefficients of rare-earth halide aqueous solutions at 25°C, by means of the Quasi Random Lattice (QRL) model. The osmotic coefficients were then calculated consistently, through the integration of the Gibbs-Duhem equation. Using of QRL was mainly motivated by its dependence on one parameter, given in the form of an electrolyte-dependent concentration, which was also the highest concentration at which the model could be applied. For all the electrolyte solutions here considered, this parameter was experimentally known and ranged from 1.5 to 2.2 mol/kg, at 25 °C. Accordingly, rare-earth halide concentrations from strong dilution up to 2 mol/kg about could be considered without need for best-fit treatment in order to compute their osmotic and mean activity coefficients. The experimental knowledge about the parameter was an advantageous feature of QRL compared to existing literature models. Following a trend already observed with low charge electrolytes, a satisfactory agreement was obtained with the experimental values for all the investigated rare-earth chlorides and bromides. For the sake of compactness, in this work the considered rare-earth halides were all belonging to the P63/m space group in their crystalline (anhydrous) form.

Failure simulation of RC structures under highly dynamic conditions using random lattice models

The failure behavior of concrete structures is simulated using a random lattice model, in which the rate dependent features are described by a visco-plastic damage model. Steel reinforcement is explicitly incorporated within the model without introducing additional degrees of freedom to the system. As part of a calibration process, direct tensile tests are simulated at various strain rates. Validation efforts are made through simulations of four-point bending tests of RC beams under different loading rates. Finally, the enhancement of dynamic failure resistance by various reinforcement conditions is investigated through the simulations of four-point bending tests under a higher loading rate.

A crossover random lattice fluid model for hydrocarbons and carbon dioxide

A random lattice fluid model with finite coordination number is a versatile molecular-based lattice fluid equation of state, but this model fails to reproduce the non-analytical, singular behavior of fluids in the critical region. In this work, a method of obtaining the classical critical properties is presented in the random lattice fluid model. This model is combined with the crossover theory to obtain the crossover random lattice fluid model and to calculate the thermodynamic properties of hydrocarbons and carbon dioxide. This crossover random lattice fluid model presents much better agreement with experimental data near to and far from the critical region than the classical random lattice fluid model.

Slow relaxation and equilibrium dynamics in a two-dimensional Coulomb glass: Demonstration of stretched exponential energy correlations

We have simulated energy relaxation and equilibrium dynamics in Coulomb glasses using the random energy lattice model. We show that in a temperature range where the Coulomb gap is already well developed $(T=0.03--0.1)$, the system still relaxes to an equilibrium behavior within the simulation time scale. For all temperatures $T$, the relaxation is slower than exponential. Analyzing the energy correlations of the system at equilibrium $C(\ensuremath{\tau})$, we find a stretched exponential behavior, $C(\ensuremath{\tau})={e}^{\ensuremath{-}{(\ensuremath{\tau}/{\ensuremath{\tau}}_{0})}^{\ensuremath{\gamma}}}$. We study the temperature dependence of ${\ensuremath{\tau}}_{0}$ and $\ensuremath{\gamma}$. ${\ensuremath{\tau}}_{0}$ is shown to increase faster than exponentially with decreasing $T$. $\ensuremath{\gamma}$ is proportional to $T$ at low temperature and approaches unity for high temperature. We define a time ${\ensuremath{\tau}}_{\ensuremath{\gamma}}$ from these stretched exponential correlations and show that this time corresponds well with the time required to reach equilibrium. From our data it is not possible to determine whether ${\ensuremath{\tau}}_{\ensuremath{\gamma}}$ diverges at any finite temperature, indicating a glass transition, or whether this divergence happens at zero temperature. While the time dependence of the system energy can be well fitted by a random walker in a harmonic potential for high temperatures $(T=10)$, this simple model fails to describe the long time scales observed at lower temperatures. Instead we present an interpretation of the configuration space as a structure with fractal properties and the time evolution as a random walk on this fractal-like structure.

Dipolar interactions and constant dielectric loss spectra

Frequency-independent dielectric loss spectra constitute an ubiquitous phenomenon in strongly disordered materials. The degree of universality of this effect calls for a microscopic model that takes into account interactions and structural disorder, but otherwise requires only minimal assumptions. Under this point of view we review and further investigate the cooperative relaxation behaviour in an assembly of defect centres at frozen random positions. These centres carry an internal dipolar degree of freedom and interact by dipole-dipole forces. Such a description essentially amounts to studying the cooperative dynamics of interacting “asymmetric double well potential” (ADWP)-like centres, without invoking an ad hoc barrier distribution. Kinetic Monte Carlo simulations of an associated random dipolar lattice model show a transition from Debye to constant dielectric loss behaviour upon lowering the temperature. These results are interpreted with the help of analytic theories that focus on slow relaxation within strongly interacting nearest neighbour pairs of defects. Finally, a collective mode representation of the dielectric response derived from stochastic field theory is presented.

Emergence of conduction channels in lithium silicate

The existence of conduction channels in lithium silicate $({\mathrm{Li}}_{2}\mathrm{O})$ $(\mathrm{Si}{\mathrm{O}}_{2})$ is investigated. Regions of the system, where many different ions pass by, form channels and are thus spatially correlated. For a closer analysis the properties of the individual ionic sites are elucidated. The mobility of ions in single sites is found to depend strongly on the number of bridging oxygens in the coordination shell. The channels are not reflected in the network structure as obtained from the distribution of the bridging oxygens. Spatial correlations similar to those found in the silicate also emerge from studying the dynamics of particles in a simple random lattice model. This supports the suggestion that the observed spatial correlations can be viewed in analogy to the emergence of percolation paths.

Electron hopping between localized states: A simulation of the finite-temperature Anderson problem using density functional methods

It is of fundamental interest to develop a deeper understanding of transport in disordered materials. For metals, a simple qualitative relation between the conductivity s and the diffusion coefficient D of a mobile entity is given by Einstein’s relation D}s. Only diffusive states will contribute to the metallic transport. For amorphous semiconductors, transport theory requires a different foundation. In celebrated work, Anderson, 1 using a random lattice model, proved that without thermal activation no diffusion takes place for a sufficiently disordered one-band model. Sophisticated hopping theory 2‐ 4 proposes that the quantum jump of electrons is the essential transport mechanism through the localized states, in which thermal-induced delocalization is the driving mechanism. This picture is essentially correct, but highly incomplete, since it does not incorporate known information about the topological disorder in materials and to date has not used highly accurate descriptions of electronic states. In this paper, we explore electron dynamics with the time

Toward a Micromechanics-Based Procedure to Characterize Fatigue Performance of Asphalt Concrete

A lattice-based micromechanics approach is proposed to characterize the cracking performance of asphalt concrete. A random truss lattice model was introduced and investigated for simulating the following: ( a) linear elastic and viscoelastic deformation of homogeneous materials in axial compression and shear loading experiments, ( b) linear elastic deformation and the stress field in heterogeneous materials in an axial compression loading experiment, and ( c) damage evolution in elastic solids under an indirect tensile test. The simulation results match well with the theoretical solutions and show excellent promise in predicting cracking patterns in the indirect tensile test. A brief discussion about ongoing work is also presented.

Superconducting fluctuation effect on the de Haas–van Alphen oscillations in the vortex liquid state of quasi-2D superconductors

The damping of the de Haas–van Alphen (dHvA) oscillations in the mixed state of a 2D superconductor is studied within a new analytical approach to the problem of vortex-lattice fluctuations at high magnetic fields. It is shown that in the ‘liquid’ state around mean field H c2( T), averaging over phase fluctuations leads to damping of the dHvA osillations, which can be well described by the random vortex lattice model. The calculated damping rate is in a good quantitative agreement with recent experiments carried out on a quasi-2D organic superconductor. It is predicted that deep in the mixed state, the freezing into an ordered vortex lattice is accompanied by a sharp upward turn in the slope of the ‘Dingle plot’.

Vortex states and quantum magnetic oscillations in conventional type-II superconductors

The theory of pure type-II superconductors at high magnetic fields and low temperatures has recently attracted much attention due to the discovery of de Haas--van Alphen oscillations deep in the vortex state. In this article the authors review the state of the art in this rapidly growing new field of research. The very existence of quantum magnetic oscillations deep in the vortex state poses challenging questions to the theorists working in this field. For a conventional extreme type-II superconductor in magnetic fields just below the upper critical field ${H}_{c2},$ the quasiparticle spectrum is gapless and the de Haas--van Alphen effect is suppressed with respect to the corresponding normal-state signal due to superconducting induced currents near the vortex cores, which are of paramagnetic nature. Numerical simulations of the quasiparticle band structure in the Abrikosov vortex lattice show the existence of well-separated Landau bands below ${H}_{c2}.$ An analytical perturbative approach, which emphasizes the importance of phase coherence in quasiparticle scattering by the pair potential in the Abrikosov lattice, predicts a relatively weak magnetic breakdown of the corresponding cyclotron orbits. In contrast to the situation in the Abrikosov lattice state, a theory based on a random vortex lattice model yields large exponential decay of the de Haas--van Alphen oscillations with the superconducting order parameter below ${H}_{c2}.$ The disordered nature of the vortex state near ${H}_{c2}$ in real superconductors, where long-range phase coherence in the superconducting order parameter is destroyed, could explain the success of this model in interpreting experimental data below ${H}_{c2}.$ In the Abrikosov vortex lattice state, which usually stabilizes well below ${H}_{c2},$ the residual damping of the de Haas--van Alphen amplitude is significantly reduced. In quasi-two-dimensional superconductors, phase fluctuations associated with sliding Bragg chains along principal axes in the vortex lattice lead to a weak first-order melting transition far below the mean-field ${H}_{c2}.$ Superconducting fluctuations dominate the additional damping of the de Haas--van Alphen oscillations in this vortex liquid state. Below the first-order freezing point, this damping is predicted to weaken signifiFantly.

ON THE COUNTING OF COLORED TANGLES

The connection between matrix integrals and links is used to define matrix models which count alternating tangles n which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n=2, the corresponding matrix integral is that recently solved in the study of the random lattice six-vertex model. The generating function of alternating 2-color tangle is provided in terms of elliptic functions, expanded to 16-th order (16 crossings) and its asymptotic behaviors is given.

GOR'KOV'S EXPANSION AND THE dHvA EFFECT IN THE VORTEX STATE

The validity of Gor'kov's expansion in the small superconducting order parameter near the upper critical field H c2 of type II superconductors at low temperatures is examined in the light of the accumulated experimental data of the dHvA effect in the vortex state. Using a 2D electron gas model, it is shown that a small amount of thermal smearing is sufficient to ensure the validity of the expansion within a significantly broad field range below H c2 . The lowest order (i.e. quadratic) term in this expansion is insensitive to the inhomogeneity of the pair potential in the vortex state and tends to cancel the zeroth order (i.e. normal electrons) term. Higher order terms are found, in the asymptotic limit, E F/ℏω c ≫1 , to be much smaller than the corresponding terms in the random vortex lattice model, thus reflecting weak `scattering' of quasi-particles by the vortex lattice. This picture seems to account qualitatively for a common feature observed in almost all experiments so far, namely the crossover of the Dingle plot from a relatively large slope just below H c2 to a smaller slope at lower fields.

ON THE SCHILD ACTION FOR D=0 AND D=1 STRINGS

It is shown that the integration measure over the matrix Y in the matrix representation of the Schild action can be fixed by comparing the Schild matrix model with the random lattice string model for D=0. It is further checked that the given measure is consistent with the case D= 1 as well.