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     

The zero-point energy in the molecular hydrogen crystal

We propose a modified Einstein approximation to describe zero-point energy vibrations in a quantum crystal. Our aim was to develop a computationally cheap tool suitable for lattice structure optimisation. As in the classical Einstein model the representative atom vibrates in an effective potential due to the surrounding atoms of the crystal; the atoms however are not strictly placed at the positions corresponding to the crystal potential energy minima but their positions are described by the quantum mechanical density distributions. The effective potential computed that way is suitable for the application in solid para-hydrogen in contrast to the normal (unmodified) Einstein approximation. We compute the cohesive energy of the para-hydrogen crystal and perform lattice structure optimisation. The hexagonal closed packed is more stable than the fcc closed packed lattice and the lattice constants obtained are in very good agreement with the experimental values.     

Logics in realization and modeling from data

Systemrealization is the construction of a state-space model given input-output data of a system. One approach, briefly summarized here, is the subspace method. In the deterministic realization problem, the data are used in alinear fashion, whereas the stochastic realization problem usesquadratic forms in the data. This dichotomy is related to the basic assumptions of repeatability or nonrepeatability of the input-output experiments performed on the system. In particular, the logic of the system is constructed, closely following the axiomatic foundations of physics. It is shown that this logic is Boolean in the deterministic and quantal in the stochastic case. The system dynamics is obtained from the data-induced measures one can define on the lattice.     

Multi-grid Monte Carlo (III). Two-dimensional O(4)-symmetric non-linear σ-model

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation applied to the two-dimensional O(4)-symmetric non-linear σ-model [ = SU(2) principal chirral model]], on lattices up to 256×256. We find a dynamic critical exponent for the W-cycle and for the V-cycle, compared to for the single-site heat-bath algorithm (subjective 68% confidence intervals). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. For a 256×256 lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.     

One-Dimensional Lorentz Gas with Rotating Scatterers: Exact Solutions

The simplest solutions (orbits) to the recently introduced Lorentz gas with rotating scatterers are found by considering its one-dimensional one-particle reduction. This model has only one parameter which can be viewed as the amount of energy transfer between the scatterers and the particle during a collision. Exact solutions of the system are found for several values of this parameter. For some of these values, the dynamics is shown to be in many respects similar to the dynamics of the deterministic Lorentz lattice gases.     

Dynamical Localization for Unitary Anderson Models

This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle. We implement the method of Aizenman-Molchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes.     

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.     

Lattice dynamical study of body-centred tetragonal indium

The phonon dispersion curves, phonon frequency distribution function as well as the lattice specific heat of body-centred tetragonal indium have been deduced using a lattice dynamical model which includes central, angular and volume forces. Six elastic constants, four zone boundary frequencies and an equilibrium condition were used in the evaluation of the force constants. It is shown that this model is elastically consistent and satisfies the symmetry requirements of the lattice, the phonon frequencies of indium deduced from it are in very good agreement with the experimental values of Reichardt and Smith and the theoretical values of Garrett and Swihart, and theθ _D values compare well with the experimental values over a wide temperature range. The apparent discrepancies in the phonon dispersion curves and theθ _D-T curves obtained from deficient models, importance of umklapp processes and the significance of angular forces in the lattice dynamical models are discussed.     

Lorentz lattice gases,abnormal diffusion,and polymer statistics

Diffusive behavior in various Lorentz lattice gases, especially wind-tree-like models, is discussed. Comparisons between lattice and continuum models as well as deterministic and probabilistic models are made. In one deterministic model, where the scatterers behave like double-sided mirrors, a new kind of abnormal diffusion is found, viz., the mean square displacement is proportional to the time, but the probability density distribution function is non-Gaussian. The connections of this mirror model with the percolation problem and the statistics of polymer chains on a lattice are also discussed.     

Griffiths' singularities in diluted ising models on the Cayley tree

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.     

On the Orthocomplementation of State–Property–Systems of Contextual Systems

We adopt an operational approach to quantum mechanics in which a physical system is defined by the mathematical structure of its set of states and properties. We present a model in which the maximal change of state of the system due to interaction with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the case N=2 the system reduces to a model for the spin measurements on a quantum spin-1/2 particle. In the limit N→∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate situations the change of state due to measurement is neither ‘maximal’ (i.e. quantum) nor ‘zero’ (i.e. classical). We show that two of the axioms used in Piron’s representation theorem for quantum mechanics are violated, namely the covering law and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability of a state transition in general not only depends on the two states involved, but also on the measurement context which induces the state transition.     

Growth kinetics in multicomponent fluids

The hydrodynamic effects on the late-stage kinetics in spinodal decomposition of multicomponent fluids are examined using a lattice Boltzmann scheme with stochastic fluctuations in the fluid and at the interface. In two dimensions, the three- and four-component immiscible fluid mixture (with a 1024² lattice) behaves like an off-critical binary fluid with an estimated domain growth oft ^0.4±0.03 rather thant ^1/3 as previously estimated, showing the significant influence of hydrodynamics. In three dimensions (with a 256³ lattice), we estimate the growth ast ^0.96±0.05 for both critical and off-critical quenches, in agreement with phenomenological theory.     

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     

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.Received: 12 March 2004, Published online: 23 July 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 64.60.-i General studies of phase transitions - 72.80.Tm Composite materials     

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in ℝ ^k , called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in ℝ ^k . We also discuss a simple signaling pathway related to cancer research, called p53 module.     

Lattices of Quantum Automata

We defined and studied three different types of lattice-valued finite state quantum automata (LQA) and four different kinds of LQA operations, discussed their advantages, disadvantages, and various properties. There are four major results obtained in this paper. First, no one of the above mentioned LQA follows the law of lattice value conservation. Second, the theorem of classical automata theory, that each nondeterministic finite state automaton has an equivalent deterministic one, is not necessarily valid for finite state quantum automata. Third, we proved the existence of semilattices and also lattices formed by different types of LQA. Fourth, there are tight relations between properties of the original lattice l and those of the l-valued lattice formed by LQA.     

Existence of enantiomeric phase separation in a three-dimensional lattice gas model

A three-dimensional lattice gas model for enantiomeric phase separation is introduced. The enantiomeric molecules (d andl) are the two nonsuperimposable mirror images having the molecular structure C(AB)₂, where C is a tetrahedrally bonded carbon atom with one bond to each end of two AB groups. The lattice gas model consists of a body-centered cubic lattice, each site of which can be either vacant or occupied by a molecule oriented so that the A and B groups point toward neighboring lattice sites. Pairs of molecules interact with short-range, orientationally-dependent interactions. For a domain of interaction parameters, the Pirogov-Sinai extension of the Peierls argument is used to prove thatd-rich andl-rich phases exist in the model at sufficiently low temperature. For another domain of interaction parameters, at sufficiently high chemical potential there is an infinite number of ground states, each containing a racemic mixture ofd andl molecules.     

YM2:Continuum expectations,lattice convergence,and lassos

The two dimensional Yang-Mills theory (YM₂) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM₂-measure space.     

Distribution of dipole fields in disordered crystalline and amorphous ferromagnetic alloys

By means of computer simulation we have calculated the distribution functions of dipole fields in disordered crystalline and amorphous ferromagnetic alloys A_1-xB_x. It is shown that for all cubic lattice sites in simple cubic, body-centred cubic and face-centred cubic materials as well as for amorphous materials the envelopes of the distribution functions may be obtained in a satisfactory approximation by considering only the nearest contributing atoms. Whereas in crystalline materials we have a complicated structure of the distribution functions for arbitrary values ofx, we obtain simple Gaussian distributions for the case of amorphous materials, using Heimendahl's model of the amorphous structure. The influence of isotropic and anisotropic short-range order is discussed in detail.