Structure functions in a model of turbulent energy dissipation

A stochastic activity-transfer model, previously proposed to apply to turbulence, is studied and simulated on a 256×256 lattice. Introduction of random self-activation does not allow stable fronts to develop in the limit of small growth probability. By assigning discrete density values equal to the threshold values in a related continuous and deterministic model, the structure functions for distancesr in the lattice are calculated. They have a functional form different from the power behavior which in the case of the deterministic version was interpreted as another sign of self-organized criticality. Future studies of these and other models may be facilitated by the algorithm developed for structure function calculations.     

Diffusion on random lattices

We study the motion of a point particle along the bonds of a two-dimensional random lattice, whose sites are randomly occupied with right and left rotators, which scatter the particle according to deterministic scattering rules. We consider both a Poisson (PRL) and a vectorized random lattice (VRL) and fixed as well as flipping scatterers. On both lattices, for fixed scatterers and equal concentrations of right and left rotators the same anomalous diffusion of the particle is obtained as before for the triangular lattice, where the mean square displacement is t, the diffusion process non-Gaussian, and the particle trajectories exhibit scaling behavior as at a percolation threshold. For unequal concentrations the particle is trapped exponentially rapidly. This system can be considered as an extreme case of the Lorentz lattice gases on regular lattices discussed before or as an example of the motion of a particle along cracks or (grain or cellular) boundaries on a two-dimensional surface.     

对流扩散方程的格点模型

推广流体力学的格点法解一般的数学物理方程,建立了一维对流扩散方程的简单和复杂的格点模型,并利用此模型模拟了几种不同初边值条件下的对流扩散方程     

Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications

In this paper we derive deterministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium     

Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.Dedicated to Res Jost and Arthur WightmanSupported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft     

A high accuracy stochastic estimation of a nonlinear deterministic model

In this Letter, an approach to estimating a nonlinear deterministic model is presented. We introduce a stochastic model with extremely small variances so that the deterministic and stochastic models are essentially indistinguishable from each other. This point is explained in the Letter. The estimation is then carried out using stochastic optimization based on Markov chain Monte Carlo (MCMC) methods.     

Diffusion in the Lorentz Gas

The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann-Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.     

Nonintersecting string model and graphical approach: Equivalence with a Potts model

Using a graphical method we establish the exact equivalence of the partition function of aq-state nonintersecting string (NIS) model on an arbitrary planar, even-valenced, lattice with that of a q²-state Potts model on a related lattice. The NIS model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types, and the resulting Potts model generally has multispin interactions. For the square and Kagomé lattices this leads to the equivalence of a staggered NIS model with Potts models with anisotropic pair interactions, indicating that these NIS models have a first-order transition forq > 2. For the triangular lattice the NIS model turns out to be the five-vertex model of Wu and Lin and it relates to a Potts model with two- and three-site interactions. The most general model we discuss is an oriented NIS model which contains the six-vertex model and the NIS models of Stroganov and Schultz as special cases.     

Structural relaxations,phonons, and Ising models

Lattice models such as Ising or Potts models are very often successfully applied to order-disorder phenomena in solids (e.g., for alloys) or on surfaces (e.g., for physisorption). In this contribution it is shown how to derive such models from a microscopic Hamiltonian in the framework of classical statistical mechanics. Both structural relaxations and thermal fluctuations can be incorporated within the (temperature-dependent) parameters of the lattice model.     

Three-body interactions,rotational invariance and the elastic consistency of lattice dynamical models for face-centred tetragonal indium

The origin of the elastic inconsistency ofdaf, mas andgtf models for non-cubic solids and the failure of their force constants to comply with all the rotational invariance conditions are analysed by resolving the atomic displacements of face-centred tetragonal indium along three mutually perpendicular directions. It is shown that a lattice dynamical model suffers from these deficiencies as a consequence of its neglect of three-body interactions as well as the mixed neighbour interactions associated with the angular forces, while thecgw model which incorporates both these interactions is elastically consistent and its potential energy rotationally invariant. The degree of equivalence that exists among the force constants ofdaf, mas, gtf andcgw models, the distortions introduced by the elastic inconsistency into the phonon dispersion curves of fct indium as well as the consequences of imposing the rotational invariance conditions on the force constants of a lattice dynamical model are discussed.     

Coupled map lattices as computational systems

The coupled map lattice (CML) as a mathematical model for a computer is considered. Using the theory of synchronous concurrent algorithms, it is shown that the CML is a valid new model for a parallel deterministic analog machine, but that, in principle, such a CML computer does not generate computations that cannot be reproduced by the standard mathematical models for computing on real numbers. The analysis is based on new general mathematical definitions of CMLs, and an axiomatic approach to determining which models of computation can be used to simulate CMLs.     

On the theory of surface diffusion: kinetic versus lattice gas approach

The present paper extends the results of a recent analytic kinetic theory of particle-on-substrate diffusion. The approach treats explicitly the molecule–surface interaction and takes into account inter-molecular interaction within the hard particle approximation. The physics influencing the diffusion pre-exponential factor and mechanisms determining the density dependence of collective diffusivity are discussed. The kinetic results are compared with those of the traditional lattice gas hopping models. Analytical expressions for jump rates in the low density limit are derived, and the density dependence of effective jump rates at finite occupancy is discussed. It is shown how the traditional hopping model oversimplifies the picture of diffusion by neglecting the collision part of the hopping process.     

多孔介质中流体流动及扩散的耦合格子Boltzmann模型

多孔介质中高Péclet数和大黏性比下混溶流体的流动和扩散广泛存在于二氧化碳驱油、化工生产等工业过程中.用数值方法对该问题进行研究时,关键在于如何正确描述高Péclet数和大黏性比下多孔介质内流体的行为.为此,提出了一种基于多松弛模型和格子动理模型的耦合格子Boltzmann模型.通过Chapman-Enskog分析,证明该模型能有效求解不可压Navier-Stokes方程和对流扩散方程.数值结果表明,该模型不仅具有二阶精度和良好的稳健性,而且对于高Péclet数和大黏性比的问题具有良好的数值稳定性,为模拟此类问题提供了有效工具.     

Lattice Models Solvable Through the Full Interval Method on Links

A two state model on a one dimensional lattice is considered, where the evolution of the state of each site is determined by the states of that site and its neighboring sites. Corresponding to this original lattice, a derived lattice is introduced the sites of which are the links of the original lattice. It is shown that there is only one reaction on the original lattice, which results in the derived lattice being solvable through the full interval method. And that reaction corresponds to the one dimensional stochastic non-consensus opinion model. A one dimensional non-consensus opinion model with deterministic evolution has already been introduced. Here this is extended to be a model which has a stochastic evolution. Discrete time evolution of such a model is investigated, including the two limiting cases of small probabilities for evolution (resulting to an effectively continuous-time evolution), and deterministic evolution. The formal solution to the evolution equation is obtained and the large time behavior of the system is investigated. Some special cases are studied in more detail.     

A deterministic sandpile automaton revisited

The Bak-Tang-Wiesenfeld (BTW) sandpile model is a cellular automaton which has been intensively studied during the last years as a paradigm for self-organized criticality. In this paper, we reconsider a deterministic version of the BTW model introduced by Wiesenfeld, Theiler and McNamara, where sand grains are added always to one fixed site on the square lattice. Using the Abelian sandpile formalism we discuss the static properties of the system. We present numerical evidence that the deterministic model is only in the BTW universality class if the initial conditions and the geometric form of the boundaries do not respect the full symmetry of the square lattice. Received 19 August 1999     

Diffusion and propagation in triangular Lorentz lattice gas cellular automata

The diffusion process of point particles moving on regular triangular and random lattices, randomly occupied with stationary scatterers (a Lorentz lattice gas cellular automaton), is studied, for strictly deterministic scattering rules, as a function of the concentration of the scatterers. In addition to the normal and various kinds of retarded diffusion found before on the regular square lattice, straight-line propagation through the scatterers is observed.     

On the Ising model for the simple cubic lattice

The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.     

Lorentz lattice gases: Basic theory

We present several ballistic models of the Lorentz gas in two-dimensional lattices with deterministic and stochastic deflection rules, and their corresponding Liouville equations. Boltzmann-level-equation results are obtained for the diffusion coefficient and velocity autocorrelation function for models with stochastic deflection rules. The long-time behavior of the mean square displacement is briefly discussed and the possibility of abnormal diffusion indicated. Even if the diffusion coefficient exists, its low-density limit may not be given correctly by the Boltzmann equation.     

Non-Equilibrium Behaviour in Unidirectionally Coupled Map Lattices

A spatially one dimensional coupled map lattice with a local and unidirectional coupling is introduced. This model is studied analytically by a perturbation theory that is valid for small coupling strength. In parameter space three phases with different ergodic behaviour are observed. Via coarse graining the deterministic model is mapped to a stochastic spin model that can be described by a master equation. Because of the anisotropic coupling non-equilibrium behaviour is found on the coarse grained level. However, the stationary statistical properties of the spin dynamics can still be described with a nearest neighbour Ising model whereby the ordering is predominantly antiferromagnetic.     

Discrete, Amorphous Physical Models

Discrete models of physical phenomena are an attractive alternative to continuous models such as partial differential equations. In discrete models, such as cellular automata, space is treated as having finitely many locations per unit volume, and physical processes are modelled by rules that depend on a small number of nearby locations. Such models depend critically on a regular (crystalline) lattice, as well as the global synchronization of all sites. We should ask, on the grounds of minimalism, whether the global synchronization and crystalline lattice are inherent in any discrete formulation. Is it possible to do without these conditions and still have a useful physical model? Or are they somehow fundamental? We will answer this question by presenting a class of models that are “extremely local” in the sense that the update rule does not depend on synchronization with the other sites, or on knowledge of the lattice geometry. All interactions involve only a single pair of sites. The models have the further advantage that they exactly conserved the analog of quantities such as momentum and energy which are conserved in physics. An example model of waves is given, and evidence is given that it agrees well qualitatively and quantitatively with continuous differential equations.