COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE MODEL BASED ON THE NEHRING–SAUPE INTERACTION POTENTIAL

1999 ◽  
Vol 13 (32) ◽  
pp. 3879-3902 ◽  
Author(s):  
R. HASHIM ◽  
S. ROMANO
Keyword(s):  
Lattice Model ◽  
Lattice Models ◽  
Interaction Potentials ◽  
Cylindrically Symmetric ◽  
Positive Quantity ◽  
Nearest Neighbours ◽  
Anisotropic Lattice ◽  
Scalar Products ◽  
Scalar Invariants ◽  
Unit Vectors

By now, nematogenic lattice models have been extensively studied in the literature; they usually involve cylindrically symmetric (uniaxial) particles and isotropic interaction potentials defined by even functions of the scalar products between unit vectors defining their orientations; anisotropic interaction potentials involving other scalar invariants, i.e. also depending on the orientations of the two particles with respect to the intermolecular vector, have been considered far less often. A model of the latter kind was proposed by Nehring and Saupe over 25 years ago; we have considered here its restriction to nearest neighbours, having the form [Formula: see text] Here the three-component vectors xj∈ Z3define centre-of-mass coordinates of the particles, and ukare three-component unit vectors defining their orientations; ∊ is a positive quantity setting energy and temperature scales (i.e. T*=kBT/∊); this model is seen to be the anisotropic counterpart to the generic Lebwohl–Lasher lattice model.The model has been addressed by simulation; comparisons are reported with other anisotropic lattice models recently studied in the literature.

COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE-GAS MODEL BASED ON THE NEHRING–SAUPE INTERACTION POTENTIAL

2001 ◽  
Vol 15 (04n05) ◽  
pp. 137-155 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Lattice Model ◽  
Lattice Gas ◽  
Chemical Potential ◽  
Mean Field Theory ◽  
Mean Field ◽  
Centre Of Mass ◽  
Positive Quantity ◽  
Occupation Numbers ◽  
Nearest Neighbours ◽  
Unit Vectors

Over 25 years ago, Nehring and Saupe proposed an anisotropic nematogenic lattice model, whose restriction to nearest neighbours has the form [Formula: see text] Here the three-component vectors xj∈Z3define centre-of-mass coordinates of the particles, and ukare three-component unit vectors defining their orientations; ∊ is a positive quantity setting energy and temperature scales (i.e. T*=kBT/∊); this model is seen to be the anisotropic counterpart to the generic Lebwohl–Lasher lattice model. It has often been used for approximate calculations of elastic properties, and has recently been studied by simulation [Hashim and Romano, Int. J. Mod. Phys.B13, 3879 (1999)]. We study here its lattice-gas extension, whose Hamiltonian is defined by [Formula: see text] where νk=0,1 denote occupation numbers, ∑{j<k}denotes sum over all distinct nearest-neighbouring pairs of lattice sites, and μ is the chemical potential. The model has been addressed by Monte Carlo simulation; comparisons are reported with Mean Field theory as well as with the Lebwohl–Lasher counterpart.

scholarly journals Convergent perturbation theory for lattice models with fermions

2016 ◽  
Vol 31 (13) ◽  
pp. 1650072 ◽  
Author(s):  
V. K. Sazonov
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Expansions ◽  
Lattice Model ◽  
Lattice Models ◽  
Convergent Series ◽  
Standard Perturbation Theory

The standard perturbation theory in QFT and lattice models leads to the asymptotic expansions. However, an appropriate regularization of the path or lattice integrals allows one to construct convergent series with an infinite radius of the convergence. In the earlier studies, this approach was applied to the purely bosonic systems. Here, using bosonization, we develop the convergent perturbation theory for a toy lattice model with interacting fermionic and bosonic fields.

Computer simulation studies of anisotropic systems IV. The effect of translational freedom

1980 ◽  
Vol 373 (1752) ◽  
pp. 111-130 ◽  
Keyword(s):  
Monte Carlo ◽  
Lattice Model ◽  
Short Range ◽  
Orientational Order ◽  
Pair Potential ◽  
Field Approximation ◽  
Isotropic Phase ◽  
Spatial Order ◽  
Cylindrically Symmetric ◽  
Angular Correlation Function

We have simulated the properties of 256 cylindrically symmetric particles interacting via a simple anisotropic potential of the form u 2 (r 12 ) P 2 (cos B 12 ) and with a scalar Lennard-Jones 12:6 potential, using the Monte Carlo technique. The simulations were performed for two forms of u2(r12) in the isothermal-isobaric ensemble and yielded values for volume, enthalpy, second-rank orientational order parameter, radial distribution function and second-rank angular correlation function. The specific heat at constant pressure, isothermal compressibility and isobaric expansivity were also obtained but they are subject to considerable error because they were evaluated from fluctuations. The system is found to exhibit a weak, firstorder transition from a nematic to an isotropic phase on increasing the temperature. The isotropic phase possesses short-range spatial and orientational order; it differs from the nematic phase, which has longrange orientational order but only short-range spatial order. The results of these simulations are used to discuss the influence of the range of the anisotropic potential on the behaviour of the nematogen. Previous Monte Carlo simulations of nematic liquid crystals had employed a lattice model with the anisotropic interactions restricted to nearest neighbours. Our results are used to study the effect of these convenient but unrealistic restrictions on the properties of the nematic. The results of our simulations are in reasonable accord with the properties of the nematogen, 4,4'- dimethoxyazoxybenzene, although no attem pt was made to select a pair potential to mimic the behaviour of any substance. Finally, we use the results of our simulations to test the validity of the molecular field approximation, as applied to nematics. This approximation is one of the foundations of the Maier-Saupe theory and its predictions are compared with the behaviour of the simulated nematics. It would appear that this theory provides a better description of our system than the lattice model, with its enforced spatial order and truncated anisotropic pair potential.

GENERALIZED SKLYANIN ALGEBRA AND INTEGRABLE LATTICE MODELS

1994 ◽  
Vol 09 (13) ◽  
pp. 2245-2281 ◽  
Author(s):  
YAS-HIRO QUANO
Keyword(s):  
Inverse Problem ◽  
Lattice Model ◽  
Lattice Models ◽  
Baxter Equation ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
Integrable Lattice ◽  
Crossing Symmetry ◽  
Sklyanin Algebra ◽  
Integrable Lattice Models

We study three properties of the ℤn⊗ℤn-symmetric lattice model; i.e. the initial condition, the unitarity and the crossing symmetry. The scalar factors appearing in the unitarity and the crossing symmetry are explicitly obtained. The [Formula: see text]-Sklyanin algebra is introduced in the natural framework of the inverse problem for this model. We build both finite- and infinite-dimensional representations of the [Formula: see text]-Sklyanin algebra, and construct an [Formula: see text] generalization of the broken ℤN model. Furthermore, the Yang-Baxter equation for this new model is proved.

scholarly journals Rock-Paper-Scissors Lattice Model

2020 ◽  
Vol 16 (4) ◽  
pp. 400-402
Author(s):  
Nasir Ganikhodjaev ◽  
Pah Chin Hee
Keyword(s):  
Lattice Model ◽  
Statistical Physics ◽  
Lattice Models ◽  
Evolutionary Games ◽  
Payoff Matrix ◽  
Ergodic Transformation ◽  
Translation Invariant ◽  
Periodic Gibbs Measure ◽  
Quadratic Stochastic Operators ◽  
Player Game

In this work, we introduce Rock-Paper-Scissors lattice model on Cayley tree of second order generated by Rock-Paper-Scissors game. In this strategic 2-player game, the rule is simple: rock beats scissors, scissors beat paper, and paper beats rock. A payoff matrix  of this game is a skew-symmetric. It is known that quadratic stochastic operator generated by this matrix is non-ergodic transformation. The Hamiltonian of Rock-Paper-Scissors Lattice Model is defined by this skew-symmetric payoff matrix . In this paper, we discuss a connection between three fields of research: evolutionary games, quadratic stochastic operators, and lattice models of statistical physics. We prove that a phase diagram of the Rock-Paper-Scissors model consists of translation-invariant and periodic Gibbs measure with period 3.

LATTICE GAS SIMULATION OF OXYGEN ORDERING IN YBa2Cu3O6+x SHOWING DYNAMICAL SCALING

1991 ◽  
Vol 05 (12) ◽  
pp. 827-832
Author(s):  
H. F. POULSEN ◽  
N. H. ANDERSEN ◽  
J. V. ANDERSEN ◽  
H. BOHR ◽  
O. G. MOURITSEN
Keyword(s):  
Charge Transfer ◽  
Transition Temperature ◽  
Superconducting State ◽  
Lattice Model ◽  
Lattice Gas ◽  
Phase Mixing ◽  
Dynamical Scaling ◽  
Oxygen Ordering ◽  
Anisotropic Lattice ◽  
Growth Laws

A 2-dimensional anisotropic lattice model for the oxygen ordering in the high Tc superconductor of the YBa 2 Cu 3 O 7+x type is shown to exhibit an ordering dynamics that obey algebraic growth laws which depend on whether it is an Ortho-I or Ortho-II phase. It is possible to relate this dynamical scaling behavior to a similar scaling in the experimentally observed temporal variation of the superconductivity transition temperature and hence suggesting a specific coupling between the coherence of oxygen order in the basal Cu-O planes and the superconducting state. Furthermore it is possible to explain the variation in the transition temperature with the oxygen density x by a phase mixing model of Ortho-II/Ortho-I domains and an assumption about the charge transfer between the basal and superconducting plane.

MEAN FIELD, TWO-SITE CLUSTER, AND COMPUTER SIMULATION STUDY OF A NEMATOGENIC LATTICE-GAS MODEL

2001 ◽  
Vol 15 (03) ◽  
pp. 259-280 ◽  
Author(s):  
S. ROMANO
Keyword(s):  
Lattice Gas ◽  
Mean Field ◽  
Three Dimensional ◽  
Lattice Gas Model ◽  
Rigorous Results ◽  
Positive Quantity ◽  
Occupation Numbers ◽  
Site Cluster ◽  
Ordering Transition ◽  
Unit Vectors

We have considered a classical lattice-gas model, consisting of a three-dimensional simple-cubic lattice, whose sites host three-component unit vectors; pairs of nearest-neighbouring sites interact via the nematogenic potential [Formula: see text] here P2(τ) denotes the second Legendre polynomial, νj = 0, 1 are occupation numbers, uj are unit vectors (classical spins), and ∊ is a positive quantity setting energy and temperature scales (i.e. T* = k B T/∊); the total Hamiltonian is given by [Formula: see text] where ∑{j<k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model defines the extensively studied Lebwohl–Lasher model, possessing a transition to an orientationally ordered phase at low temperature; according to available rigorous results, there exists a μ0 < 0, such that, for all μ > μ0, the system supports an ordering transition at a finite, μ-dependent, temperature. Continuing along the lines of our previous communication [S. Romano, Int. J. Mod. Phys.B14, 1195 (2000)], we present here a detailed study of the case μ = 0, using Monte Carlo simulation, Mean Field and Two Site Cluster treatments; the latter significantly improves the agreement with simulation results.

Latent symmetry and small branch in anisotropic lattice models

1991 ◽  
Vol 43 (11) ◽  
pp. 1401-1408
Author(s):  
M. S. Gonchar ◽  
V. G. Kozirs'kii
Keyword(s):  
Lattice Models ◽  
Small Branch ◽  
Anisotropic Lattice
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