Three-dimensional Lattice Models Research Articles

Application of the fourth-order three-stage symplectic integrators in chaos determination

Global symplectic integrators mean that both the equations of motion and their variational equations of Hamiltonian systems do simultaneously use symplectic integrations. The numerical approaches are very helpful to obtain reliable chaos indicators like Lyapunov exponents and fast Lyapunov indicators, so that correct chaos determination can be ensured. The two new fourth-order three-stage symplectic integrators proposed recently by us are particularly suitable for integrating variational equations of any Hamiltonian in which the kinetic energy is a quadratic form of momenta and the potential energy depends on position coordinates, and also are efficient to solve a class of Hamiltonian systems including the lattice φ 4 models. As a result, both one of the two integrators and the method of fast Lyapunov indicators are recommended to provide some insight into the dynamics of three-dimensional lattice φ 4 models so that a great deal of the computational cost can be saved. A transition to chaos on the variation of single parameter and the structure of associated initial-condition sets for chaotic and regular motions is described.

Modeling of phase interfaces during pre-critical crack growth in concrete

Abstract Recent developments in the Rigid-Body-Spring Network (RBSN) modeling of concrete are presented, with candid assessments of research needs for the modeling of pre-critical crack growth under multi-axial stress conditions. Fracture of micro-concrete specimens is investigated through the combined use of such discrete (lattice) models, X-ray tomography, and imaging techniques. The specimens contain small amounts (10–50 wt.% fraction) of spherical glass aggregates. Acid-etching of the glass aggregate surfaces is done to vary the bond properties of the matrix–aggregate interface. Tomographic images of the unloaded specimens provide the initial configurations of the three-dimensional lattice models, which are based on a three-phase representation of the material meso-structure: fine-grained mortar matrix, glass aggregate inclusions, and matrix–aggregate interfaces. The numerical and physical test results agree well with respect to peak loads and the associated crack patterns. Material structure and interface properties affect pre-critical cracking, in accordance with expectations. Dependence of composite strength on aggregate content and arrangement is studied through simulations of large sets of nominally identical models, which differ only in random positioning of the aggregates.

Dissection of the Zipping-and-Assembly Mechanism for Folding of Model Proteins

Zipping-and-assembly mechanism (ZAM) is a new mechanism describing the kinetics of protein folding. To dissect the validity of this mechanism for various protein-like systems, a prediction test based on three-dimensional HP lattice models is carried out. It is found that only the native structures of a part of protein-like models could be predicted with a ZAM-based method. The detailed comparisons between the model proteins which are predicted or failed with the ZAM-based method suggest that the ZAM is likely to be applicable for the model proteins with the weak hydrophobicity, the low contact order for native conformations, and the large separation between the energies of native state and denatured states. These observations bring us more information about the protein-like systems for which the ZAM could be applied.

One- and three-dimensional lattice models with two repulsive ranges: simple systems with complex phase behaviour

The properties of a simple one-dimensional lattice model with two repulsive ranges (a hard core repulsion that covers two sites and a next-to-nearest neighbour repulsive interaction) and a long-ranged staggered mean field, are studied in terms of its analytic solution. A related three-dimensional lattice model is analysed using computer simulation. Both systems present a phase diagram that displays a disorder–order–disorder transition (or a fluid–solid–fluid transition, in lattice gas language). When the usual mean field attraction is incorporated in the one-dimensional model, the system also exhibits a vapour–liquid transition. The overall appearance of the phase diagram when dispersion forces are included bears some resemblance with the phase diagram of water and related systems.

Influence of particle density on 3D size effects in the fracture of (numerical) concrete

In this paper, three-dimensional beam lattice models are extended and used for simulating size effects on strength of 3-point bending fracture experiments on concrete. At the meso-level concrete is schematized as a three-phase material, consisting of aggregate particles, cement matrix and the bond zone, which separates these two phases. Displacement-controlled 3-point bending experiments are simulated varying the particle density P k ( P k = 0%, 15%, 35% and 55%) and the specimen size, which is scaled in all three dimensions in a range of 1:8 (volume range 1:512), containing between 15,703 and 7,448,373 lattice elements. The numerical analyses show particle density dependent scaling behaviour of strength. For very low (0%) and high (55%) particle density, scaling comes close to classical Weibull theory; for intermediate densities a significantly different power emerges caused by stable pre-critical crack growth leading to hardening.

Discrete breathers in two-dimensional nonlinear lattices

Properties of discrete breathers in two- and three-dimensional nonlinear lattice models are discussed with special emphasis on possible structures of localization in two-dimensional case. In contrast to the extensive studies addressing the various features of discrete breathers in one-dimensional lattices, less studies have been done in regard to the higher dimensional breathers. To begin with a brief survey of existing results for discrete breathers in higher dimensional lattices, we attempt to look into the possible structures of higher dimensional breathers. Based on the survey, we address the following topics concerning the discrete breathers in two-dimensional lattices: types of localized structures; chaotic breathers; quasi-continuum approximation; discrete breathers versus solitons in continuum.

Macroscopic isotropy of two- and three-dimensional elastic lattice models

Particle methods such as the meso-particle method or the method of movable cellular automata provide new possibilities for the simulation of wear, mass-transfer, plastic deformation and melting processes. One crucial point, however, is to ensure that in the continuum limit macroscopic isotropy is guaranteed. We show how this can be done in two- and three-dimensional lattices.

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra $U_q(\hat{sl}(n))$, where the rank $n$ coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the $sl(n)$-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.

Importance of chirality and reduced flexibility of protein side chains: a study with square and tetrahedral lattice models.

Side chains of amino acid residues are the determining factor that distinguishes proteins from other unstable chain polymers. In simple models they are often represented implicitly (e.g., by spin states) or simplified as one atom. Here we study side chain effects using two-dimensional square lattice and three-dimensional tetrahedral lattice models, with explicitly constructed side chains formed by two atoms of different chirality and flexibility. We distinguish effects due to chirality and effects due to side chain flexibilities, since residues in proteins are L residues, and their side chains adopt different rotameric states. For short chains, we enumerate exhaustively all possible conformations. For long chains, we sample effectively rare events such as compact conformations and obtain complete pictures of ensemble properties of conformations of these models at all compactness region. This is made possible by using sequential Monte Carlo techniques based on chain growth method. Our results show that both chirality and reduced side chain flexibility lower the folding entropy significantly for globally compact conformations, suggesting that they are important properties of residues to ensure fast folding and stable native structure. This corresponds well with our finding that natural amino acid residues have reduced effective flexibility, as evidenced by statistical analysis of rotamer libraries and side chain rotatable bonds. We further develop a method calculating the exact side chain entropy for a given backbone structure. We show that simple rotamer counting underestimates side chain entropy significantly for both extended and near maximally compact conformations. We find that side chain entropy does not always correlate well with main chain packing. With explicit side chains, extended backbones do not have the largest side chain entropy. Among compact backbones with maximum side chain entropy, helical structures emerge as the dominating configurations. Our results suggest that side chain entropy may be an important factor contributing to the formation of alpha helices for compact conformations.

Evidence for a Phase Transition in Three-Dimensional Lattice Models

It was recently discovered that an eigenvector structure of commutative families of layer-to-layer matrices in three-dimensional lattice models is described by a two-dimensional spin lattice generalizing the notion of one-dimensional spin chains. We conjecture the relations between the two-dimensional spin lattice in the thermodynamic limit and the phase structure of three-dimensional lattice models. We consider two simplest cases: the homogeneous spin lattice related to the Zamolodchikov–Bazhanov–Baxter model and a “chess spin lattice” related to the Stroganov–Mangazeev elliptic solution of the modified tetrahedron equation. Evidence for the phase transition is obtained in the second case.

Entropic sampling of simple polymer models within Wang–Landau algorithm

In this paper we apply a new simulation technique proposed in Wang and Landau (WL) (2001 Phys. Rev. Lett. 86 2050) to sampling of three-dimensional lattice and continuous models of polymer chains. Distributions obtained by homogeneous (unconditional) random walks are compared with results of entropic sampling (ES) within the WL algorithm. While homogeneous sampling gives reliable results typically in the range of 4?5 orders of magnitude, the WL entropic sampling yields them in the range of 20?30 orders and even larger with comparable computer effort. A combination of homogeneous and WL sampling provides reliable data for events with probabilities down to 10?35.For the lattice model we consider both the athermal case (self-avoiding walks, SAWs) and the thermal case when an energy is attributed to each contact between nonbonded monomers in a self-avoiding walk. For short chains the simulation results are checked by comparison with the exact data. In WL calculations for chain lengths up to N = 300 scaling relations for SAWs are well reproduced. In the thermal case distribution over the number of contacts is obtained in the N-range up to N = 100 and the canonical averages?internal energy, heat capacity, excess canonical entropy, mean square end-to-end distance?are calculated as a result in a wide temperature range.The continuous model is studied in the athermal case. By sorting conformations of a continuous phantom freely joined N-bonded chain with a unit bond length over a stochastic variable, the minimum distance between nonbonded beads, we determine the probability distribution for the N-bonded chain with hard sphere monomer units over its diameter a in the complete diameter range, 0 ? a ? 2, within a single ES run. This distribution provides us with excess specific entropy for a set of diameters a in this range. Calculations were made for chain lengths up to N = 100 and results were extrapolated to N ? ? for a in the range 0 ? a ? 1.25.

Stable Optimization of a Tensor Product Variational State

We consider a variational problem for three-dimensional (3D) classical lattice models. We construct the trial state as a two-dimensional product of local variational weights that contain auxiliary variables. We propose a stable numerical algorithm for the maximization of the variational partition function per layer. The numerical stability and efficiency of the new method are examined through its application to the 3D Ising model.

Simultaneous analysis of several models in the three-dimensional Ising universality class.

We investigate several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling. These models include spin-1 / 2 models with nearest-neighbor interactions on the simple-cubic and on the diamond lattice. For the simple cubic lattice, we also include models with third-neighbor interactions of varying strength, and some "equivalent-neighbor" models. Also included are a spin-1 model and a hard-core lattice gas. Separate analyses of the numerical data confirm the Ising-like critical behavior of these systems. On this basis, we analyze all these data simultaneously such that the universal parameters occur only once. This leads to an improved accuracy. The thermal, magnetic, and irrelevant exponents are determined as y(t)=1.5868(3), y(h)=2.4816(1), and y(i)=-0.821(5), respectively. The Binder ratio is estimated as Q=<m(2)>(2)/<m(4)>=0.62 341(4).

Explicit free parametrization of the modified tetrahedron equation

The modified tetrahedron equation (MTE) with affine Weyl quantum variables at the Nth root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parametrized in terms of eight free parameters and 16 discrete phase choices, thus providing a broad starting point for the construction of three-dimensional integrable lattice models. The Fermat-curve points parametrizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N = 2 we write the MTE in full detail.

Dependence of Folding Rates on Protein Length

Using three-dimensional Go lattice models with side chains for proteins, we investigate the dependence of folding times on protein length. In agreement with previous theoretical predictions, we find that the folding time grows as a power law with the chain length N with exponent $\lambda \approx 3.6$ for the Go model, in which all native interactions (i.e., between all side chains and backbone atoms) are uniform. If the interactions between side chains are given by pairwise statistical potentials, which introduce heterogeneity in the contact energies, then the power law fits yield large $\lambda$ values that typically signifies a crossover to an underlying activated process. Accordingly, the dependence of folding time is best described by the stretched exponential \exp(\sqrt{N}). The study also shows that the incorporation of side chains considerably slows down folding by introducing energetic and topological frustration.

Interplay of surface preroughening, roughening, and melting in three-dimensional lattice models

We elaborate upon a recently proposed surface model consisting of a domain wall induced by a twisted boundary condition in a three-dimensional N-state Potts model. The model is capable of describing simultaneously surface melting and roughening degrees of freedom and their interplay. The N different spin directions effectively describe the positional entropy, the location of the wall describes the height fluctuations, and both aspects coexist in the same system. Their interplay, known to give rise to an additional preroughening transition of the free-standing surface, is shown here to produce an adsorption phase diagram on an attractive substrate exhibiting the possibility of reentrant layering. The results provide a consistent scenario closely reminiscent of experimental results for multilayers of rare gases.

Self-consistent tensor product variational approximation for 3D classical models

We propose a numerical variational method for three-dimensional (3D) classical lattice models. We construct the variational state as a product of local tensors, and improve it by use of the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG) applied to 2D classical systems. Numerical efficiency of this approximation is investigated through trial applications to the 3D Ising model and the 3D 3-state Potts model.

General recurrence theory of ligand binding on a three-dimensional lattice

Several general conclusions are obtained for the finite linear three-dimensional lattice models which are built by replicating two-dimensional layers in the third dimension. It is shown that the linear lattice, although less symmetrical, has a simpler structure than the circular lattice where the periodical boundary condition is imposed. If the ligands bound in one layer interact with ligands on its d neighbor layers, the size of the transfer matrix M for the linear lattice is equal to the number of unique binding configurations in d consecutive layers, which is usually smaller than the size of the original transfer matrix M′ that determines the recurrence relation of the circular lattice. In certain situations a significant reduction of the matrix size can be achieved. Matrix M′ contains all the eigenvalues of matrix M in addition to other eigenvalues if the binding configurations are degenerate. The global partition functions as well as the contracted partition functions at either end of the linear lattice obey the same unique and minimum recurrence relation determined by the secular equation of M. The two ends of the linear lattice, which break the symmetry of the circular lattice, actually make the linear lattice simpler than the circular lattice. The reduced size of the transfer matrix (or the order of the recurrence relation) for the linear lattice not only makes the model more accessible, but also allows the model to describe linear systems more accurately by making the model closer to the system under study. The general theory is applied to several lattices with simple geometries that are of interest in biology and statistical mechanics.

Interaction between small colloidal particles and polymer chains in a semidilute solution: Monte Carlo simulation

Interaction between flexible-chain polymers and small (nanometric) colloidal particles is studied by Monte Carlo simulation using two-dimensional and three-dimensional lattice models. Spatial distribution of colloidal particles and conformational characteristics of chains in a semidilute solution are considered as a function of the segment adsorption energy, ɛ. When adsorption is sufficiently strong, it induces effective attraction of polymer segments, which results in contraction of macromolecular coils. The strongly adsorbing polymer chains affect the equilibrium spatial distribution of the colloidal particles. The average size of colloidal aggregates exhibits a nontrivial behavior: with ɛ increasing, the value of first decreases and then begins to grow. The adsorption polycomplex formed at strong adsorption exhibits a mesoscopic scale of structural heterogeneity. The results of computer simulations are in a good agreement with predictions of the analytic theory [P.G. Khalatur, L.V. Zherenkova and A.R. Khokhlov (1997) J Phys II (France) 7:543] based on the integral RISM equation technique.

Coarse grained description of protein folding

We consider two- and three-dimensional lattice models of proteins which were characterized previously. We coarse grain their folding dynamics by reducing it to transitions between effective states. We consider two methods of selection of the effective states. The first method is based on the steepest descent mapping of states to underlying local energy minima and the other involves an additional projection to maximally compact conformations. Both methods generate connectivity patterns that allow to distinguish between the good and bad folders. Connectivity graphs corresponding to the folding funnel have few loops and are thus tree-like. The Arrhenius law for the median folding time of a 16-monomer sequence is established and the corresponding barrier is related to easily identifiable kinetic trap states.