Discrete Lattice Model Research Articles

Strong clustering of noninteracting, sliding passive scalars driven by fluctuating surfaces

We study the clustering of passive, noninteracting particles moving under the influence of a fluctuating field and random noise, in one and two dimensions. The fluctuating field in our case is provided by surfaces governed by the Kardar-Parisi-Zhang (KPZ) and the Edwards-Wilkinson (EW) equations, and the sliding particles follow the local surface slope. As the KPZ equation can be mapped to the noisy Burgers equation, the problem translates to that of passive scalars in a Burgers fluid. Monte Carlo simulations on discrete lattice models reveal very strong clustering of the passive particles for all sorts of dynamics under consideration. The resulting strong clustering state is defined using the scaling properties of the two point density-density correlation function. Our simulations show that the state is robust against changing the ratio of update speeds of the surface and particles. We also solve the related equilibrium problem of a stationary surface and finite noise, well known as the Sinai model for random walkers on a random landscape. For this problem, we obtain analytic results which allow closed form expressions to be found for the quantities of interest. Surprisingly, these results for the equilibrium problem show good agreement with the nonequilibrium KPZ problem.

Collective Dynamics of Interacting Molecular Motors

The collective dynamics of N interacting processive molecular motors are considered theoretically when an external force is applied to the leading motor. We show, using a discrete lattice model, that the force-velocity curves strongly depend on the effective dynamic interactions between motors and differ significantly from those of a simple approach where the motors equally share the force. Moreover, they become essentially independent of the number of motors if N is large enough (N> or approximately 5 for conventional kinesin). We show that a two-state ratchet model has a very similar behavior to that of the coarse-grained lattice model with effective interactions. The general picture is unaffected by motor attachment and detachment events.

Effect of constrained molecular layers on resonance heat transport through interface between two media

The surface of water layers on surfaces of crystalline substrates is a subject of theoretical modeling and experimental research. The results of the present work give the additional information about dynamic properties of layers formed on the surface of Si crystals. We investigate the propagation of acoustic waves through interface of two crystals on microscopic level using the discrete lattice model. The essential result is existence of “cut-off” frequency for energy transmission. The cut-off frequency is a result of discrete structure of interface. This principal feature can't be derived in the frame of continual theory point of elasticity. Results are compared with the experiments on ballistic heat transport through point contact. We show that anomalous decreasing of the reduced heat flux in the temperature range 10 K < T < 25 K are induced by the resonance transport of phonons through the layer of impurity atoms.

Discrete-lattice model approach to the tunneling between d-wave superconductors: Interference of tunnel bonds

The Josephson current between d-wave superconductors is investigated in the framework of tight-binding lattice model. The junction is modelled by a small number of connecting bonds. It is obtained that the Josephson current through one bond vanishes when at least one of the superconductors has $(110)$ interface-to-crystal orientation. Interference between the nearest bonds is appeared to be very important. In particular, it is the interference term that leads to the nonzero Josephson current for $(110)$ orientation. Also, interference of two connecting bonds manifests itself in non-monotonic behavior of the critical Josephson current in dependence on the distance between the bonds.

Elastic Energy Analysis of Carbide and Nitride-Type Precipitates in an Fe&amp;ndash;Mn&amp;ndash;Si&amp;ndash;Cr Shape Memory Alloy

The application of the microscopic theory of elasticity in a discrete lattice model is made on the transitional metal carbide and nitride precipitates formed in the Fe–Mn–Si–Cr shape memory alloy in conjunction with the improvement of shape memory effect (strain) and the improvement of strength. Two distinguishable methods of analysis have been established using the microscopic theory of elasticity in a discrete lattice model: One is the establishment of the description of precipitate and misfit dislocations in Fourier space. The second is the rigorous estimation of interaction energies among precipitate and misfit dislocations. The results could successfully describe the shape of the precipitate observed in the experimental investigation. It was also concluded that the elastic strain energy increases with the lattice parameter of the precipitate. Among the transition carbides and nitrides under investigations, VN, which revealed the minimum value of the elastic energy, is manifested to be the most favored one for the precipitation enhanced Fe–Mn–Si–Cr shape memory alloy. Homogeneously precipitated VN containing materials could show large deformability, higher strength by precipitation hardening and the higher shape recovery strain due to the nucleation sites of the precipitates in its reverse phase transformation.

Discrete lattice surface models and their quantum spin counterparts

Discrete lattice surface models and their quantum spin counterparts

Some Properties of Annulus SLE

An annulus $\mathrm{SLE}_{\kappa}$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus $\mathrm{SLE}_4$, $\mathrm{SLE}_{8}$ and $\mathrm{SLE}_{8/3}$. From the local martingale for annulus $\mathrm{SLE}_4$ we find a candidate of discrete lattice model that may have annulus $\mathrm{SLE}_4$ as its scaling limit. The local martingale for annulus $\mathrm{SLE}_{8/3}$ is similar to those for chordal and radial $\mathrm{SLE}_{8/3}$. But it seems that annulus $\mathrm{SLE}_{8/3}$ does not satisfy the restriction property

Statistical physics models for nacre fracture simulation

Natural biological materials such as nacre (or mother-of-pearl), exhibit phenomenal fracture strength and toughness properties despite the brittle nature of their constituents. For example, nacre's work of fracture is three orders of magnitude greater than that of a single crystal of its constituent mineral. This study investigates the fracture properties of nacre using a simple discrete lattice model based on continuous damage random thresholds fuse network. The discrete lattice topology of the proposed model is based on nacre's unique brick and mortar microarchitecture, and the mechanical behavior of each of the bonds in the discrete lattice model is governed by the characteristic modular damage evolution of the organic matrix that includes the mineral bridges between the aragonite platelets. The analysis indicates that the excellent fracture properties of nacre are a result of their unique microarchitecture, repeated unfolding of protein molecules (modular damage evolution) in the organic polymer, and the presence of fiber bundle of mineral bridges between the aragonite platelets. The numerical results obtained using this simple discrete lattice model are in excellent agreement with the previously obtained experimental results, such as nacre's stiffness, tensile strength, and work of fracture.

Enhanced continua and discrete lattices for modelling granular assemblies

This article discusses the derivation of continuum models that can be used for modelling the inhomogeneous mechanical behaviour of granular assemblies. These so-called kinematically enhanced models are of the strain-gradient type and of the strain-gradient micro-polar type, and are derived by means of homogenizing the micro-structural interactions between discrete particles. By analysis of the body wave dispersion curves, the enhanced continuum models are compared to corresponding discrete lattice models. Accordingly, it can be examined up to which deformation level the continuum models are able to accurately describe the discrete particle behaviour. Further, the boundary conditions for the enhanced continuum models are formulated, and their stability is considered. It is demonstrated how to use the body wave dispersion relations for the assessment of stability.

Diffusion in the presence of a pole: From the continuous Gaussian to a discrete lattice model

The continuous Gaussian model with delta function potentials which has been used to describe the effects that interacting centers (poles) bring to the diffusion of a random walker, provides an exact solution only in dimension d = 1 . Its application in dimension d ⩾ 2 is in need of introduction of cutoffs coming from the finite sizes of poles. We therefore formulate an analogous discrete model in the d -dimensional lattice space where no cutoffs or their regularization are necessary. The discrete model is amenable to numerical treatment and the Green's function, its first moments and properties derived from them can be determined numerically to all dimensionalities. In the one-dimensional (1-D) case, in the presence of a single center, the outcomes of the Gaussian model are compared, with the corresponding results of the discrete lattice model. The possibility of extension of the lattice model to higher dimension permits the study of the two-dimensional (2-D) case too. The intensities of interactions considered before to obtain positive values are extended to negative values in order to describe attractions also.

Viscous coupling based lattice Boltzmann model for binary mixtures

A new lattice Boltzmann model for binary mixtures, which can naturally include both the two-fluid approach and the single-fluid approach, is developed. The model is derived from the continuous kinetic model proposed by Hamel, which independently takes into account self-collisions and cross collisions. The original kinetic model is discussed in order to appreciate that cross collisions realize an internal coupling force, proportional to the diffusion velocity, and an additional coupling effect in the effective stress tensor, proportional to the deformation of the barycentric velocity field. For this reason, Hamel’s model is the natural forerunner of all linearized models based on the two-fluid approach and allows us to describe binary mixtures at different limiting regimes consistently. A discrete lattice Boltzmann model, which recovers the original Hamel’s model with second-order accuracy in both time and space, is proposed. This discrete model can analyze ordinary diffusion, pressure diffusion, and forced diffusion.

A continuous damage random thresholds model for simulating the fracture behavior of nacre

This study investigates the fracture properties of nacre using a discrete lattice model based on continuous damage random threshold fuse network. The discrete lattice topology of the model is based on nacre's unique brick and mortar microarchitecture. The mechanical behavior of each of the bonds in the discrete lattice model is governed by the characteristic modular damage evolution of the organic matrix and the mineral bridges between the aragonite platelets. The numerical results obtained using this simple discrete lattice model are in very good agreement with the previously obtained experimental results, such as nacre's stiffness, tensile strength, and work of fracture. The analysis indicates that nacre's superior toughness is a direct consequence of ductility (maximum shear strain) of the organic matrix in terms of repeated unfolding of protein molecules, and its fracture strength is a result of its ordered brick and mortar architecture with significant overlap of the platelets, and shear strength of the organic matrix.

On the extension of goal-oriented error estimation and hierarchical modeling to discrete lattice models

The Goals algorithm for adaptive modeling is extended to the case of discrete models such as those characterized by lattice structures (or the static behavior molecular systems) by using surrogate models obtained from continuum approximations of the lattice. The result is a technique that could provide for scale-bridging between continuum models and atomistic models, although the present development concerns only simple algebraic systems. An example is provided in which quantities of interest in system with a large number of degrees of freedom are computed to a preset tolerance in relatively few low-order approximations.

Dual geometric worm algorithm for two-dimensional discrete classical lattice models.

We present a dual geometrical worm algorithm for two-dimensional Ising models. The existence of such dual algorithms was first pointed out by Prokof'ev and Svistunov [Phys. Rev. Lett. 87, 160601 (2001)]]. The algorithm is defined on the dual lattice and is formulated in terms of bond variables and can therefore be generalized to other two-dimensional models that can be formulated in terms of bond variables. We also discuss two related algorithms formulated on the direct lattice, applicable in any dimension. These latter algorithms turn out to be less efficient but of considerable intrinsic interest. We show how such algorithms quite generally can be "directed" by minimizing the probability for the worms to erase themselves. Explicit proofs of detailed balance are given for all the algorithms. In terms of computational efficiency the dual geometrical worm algorithm is comparable to well known cluster algorithms such as the Swendsen-Wang and Wolff algorithms, however, it is quite different in structure and allows for a very simple and efficient implementation. The dual algorithm also allows for a very elegant way of calculating the domain wall free energy.

Algorithmic Generation of Freely Jointed Hard Sphere Chains and Properties of Inertial Tensors

A statistical algorithm, capable of generating a large number of freely jointed hard sphere chains, is presented. This is the first of a series of algorithms being developed to model unfolded proteins by different modes of hard sphere chains. The aim of these studies is to systematically investigate the effects of different factors, such as atomic radii, bond angles, torsion angles, chain length, etc., on the conformation of unfolded proteins and other random polymers. As continuous models, various types of hard sphere chains enable one to isolate the aforementioned factors one at a time for investigation and thus are advantageous over discrete lattice models. In particular, the freely jointed hard sphere chain model allows one to evaluate the excluded volume effect. As a first step in this endeavor, the average determinant D(N,r) and the average trace T(N, r) of the inertial tensor A of the random chains were calculated at various sphere radii r and chain lengths N. It is found that both the average determinant D(N,r) and the average trace T(N,r) scale linearly with chain length N. after logarithmic transformation. However, the critical exponent of D(N,r) increases with r faster than that of T(N,r) as a result of the non-commutativity between the det operator and the average operator < >. The significance of the algorithm and the results obtained on understanding random polypeptide chains are discussed.

Discrete and continuum models of heat conduction problems in periodic lattice structures

AbstractA new modelling approach to the hyperbolic heat conduction problems in periodic lattice structures of an arbitrary form is proposed. To this end we introduce a special description of the periodic lattice geometry which leads to the discrete lattice model governed by the system of finite difference equations. The continuum models are derived from the discrete model, by using the principle of stationary action.

Solute segregation transition and drag force on grain boundaries

We investigate solute segregation and transition at grain boundaries and the corresponding drag effect on grain boundary migration. A continuum model of grain boundary segregation based on gradient thermodynamics and its discrete counterpart (discrete lattice model) are formulated. The model differs from much previous work because it takes into account several physically distinctive terms, including concentration gradient, spatial variation of gradient-energy coefficient and concentration dependence of solute–grain boundary interactions. Their effects on the equilibrium and steady-state solute concentration profiles across the grain boundary, the segregation transition temperature and the corresponding drag forces are characterized for a prototype planar grain boundary in a regular solution. It is found that omission of these terms could result in a significant overestimate or underestimate (depending on the boundary velocity) of the enhancement of solute segregation and drag force for systems of a positive mixing energy. Without considering these terms, much higher transition temperatures are predicted and the critical point is displaced towards much higher bulk solute concentration and temperature. The model predicts a sharp transition of grain boundary mobility as a function of temperature, which is related to the sharp transition of solute concentration of grain boundary as a function of temperature. The transition temperatures obtained during heating and cooling are different from each other, leading to a hysteresis loop in both the concentration–temperature plot and the mobility–temperature plot. These predictions agree well with experimental observations.

ATOMIC DISPLACEMENTS DUE TO POINT DEFECTS IN Ni DILUTE ALLOYS

The Embedded atom method has been used to investigate the strain field due to substitutional transition metal impurities in Ni. The calculations are carried out in the discrete lattice model of the metal using Kanzaki lattice static method. The results for atomic displacements due to 3d, 4d and 5d impurities (Cu, Pd, Pt and Au) in Ni are given up to 20 NN's of impurity and are compared with the earlier calculations and with the available experimental data. The maximum displacements of 3.6% of 1NN distance are found for NiAu, while the minimum displacements of 0.78% of 1NN distance are found for NiCu alloy respectively. The relaxation energy for Cu are found less than those for Pd, Au and Pt impurities in the Ni host.

The algorithmics of folding proteins on lattices

It should be possible to predict the fold of a protein into its native conformation, once we are given the sequence of the constituent amino acids. This is known as the protein structure prediction problem and is sometimes referred to as the problem of deciphering the second half of the genetic code. While large proteins fold in nature in seconds, computational chemists and biologists have found that folding proteins to their minimum energy conformations is a challenging unsolved optimization problem. Computational complexity theory has been useful in explaining, at least partially, this (Levinthal's) paradox. The pedagogic cross-disciplinary survey by Ngo, Marks and Karplus (Computational Complexity, Protein Structure Prediction and the Levinthal Paradox, Birkhauser, Basel, 1994) provides an excellent starting point for non-biologists to take a plunge into the problem of folding proteins. Since then, there has been remarkable progress in the algorithmics of folding proteins on discrete lattice models, an account of which is presented herein.

Lattice Model for the Drape and Bending Properties of Nonwoven Fabrics

A discrete lattice model is presented to study the factors controlling the bending stiffness of nonwoven fabrics. The approach lends itself very easily to a detailed study of the importance of bonding pattern, laydown nonuniformities, fiber length, and orientation distribution. We find that nonwovens with a three-dimensional fiber orientation distribu tion have a much lower bending stiffness that those with a planar distribution. Variations in fiber density across the fabric thickness are also of great importance. Thus, random variations due to nonuniformities in the laydown process typically increase the bending stiffness, whereas the latter can be considerably decreased by concentrating most of the fiber weight within the neutral (or mid-) plane of a fabric. Our results in the limit of full bonding are in good quantitative agreement with those obtained from Freeston and Platt's classic model.