Coupled map lattices as cellular automata

We approach the problem of the complex dynamics of coupled map lattices (CML) by proposing a reduction to deterministic cellular automata (CA) with more than two states per site. The reduction scheme replaces the local map by an approximation in terms of a step function based on a straightforward analysis of the local dynamics. The variation of the spatial coupling in the CML then translates itself as a path in the spaces of rules for the equivalent deterministic CA. The transition to turbulence via spatiotemporal intermittency in the CML is then interpreted as a transition in the space of rules. The observed nonuniversality of this transition can be traced back to the nature of the rules involved on both sides of the transition region and to the character of the escape process from the turbulent state, either strongly deterministic or quasiprobabilistic. The relation between CML, deterministic, and probabilistic CA and the possibility of a mean-field treatment of the dynamics of CML are discussed at a more formal level.     

Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for “pure” frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.     

Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for pure frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln, FRG     

Algebraic properties of cellular automata

Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.Address from January 1983     

An effective-field study of Ising metamagnet in an external field

The phase diagrams and magnetization curves of a two-sublattice Ising metamagnet at finite temperature with longitudinal crystal field H are investigated by the use of an effective-field theory (EFT) with correlations. In addition to the second-order transition lines, the first-order transition lines are also presented, since a method to calculate the Gibbs free energy numerically at finite temperature within EFT is found in this work. The results show that there is no fourth-order critical point or reentrant phenomenon in the phase diagrams given by using EFT as found by using mean-field theory (MFT).     

Quantum fluctuations, temperature, and detuning effects in solid-light systems

The superfluid to Mott insulator transition in cavity polariton arrays is analyzed using the variational cluster approach, taking into account quantum fluctuations exactly on finite length scales. Phase diagrams in one and two dimensions exhibit important non-mean-field features. Single-particle excitation spectra in the Mott phase are dominated by particle and hole bands separated by a Mott gap. In contrast to Bose-Hubbard models, detuning allows for changing the nature of the bosonic particles from quasilocalized excitons to polaritons to weakly interacting photons. The Mott state with density one exists up to temperatures T/g > or = 0.03, implying experimentally accessible temperatures for realistic cavity couplings g.     

On vacuum instability in quantum field theory

The imaginary part of the phase of the vacuum to vacuum transition amplitude is computed for a class of diagrams involving renormalons. It is found that for these particular diagrams the imaginary part of the phase is finite. We suggest that in general the divergences are at most logarithmic and that they are therefore hopefully renormalizable. In asymptotically free theories this implies that the vacuum is unstable.     

Traffic states and fundamental diagram in cellular automaton model of vehicular traffic controlled by signals

We present a cellular automaton (CA) model for vehicular traffic controlled by traffic lights. The CA model is not described by a set of rules, but is given by a simple difference equation. The vehicular motion varies highly with both signals’ characteristics and vehicular density. The dependence of tour time on both cycle time and vehicular density is clarified. In the dilute limit of vehicles, the vehicular motion is compared with that by the nonlinear-map model. The fundamental diagrams are derived numerically. It is shown that the fundamental diagram depends highly on the signals’ characteristics. The traffic states are shown for various values of cycle time in the fundamental diagram. We also study the effect of a slow vehicle on the traffic flow.     

Magnetic fluids in an external field

Within mean field approximation we investigate the phase diagrams of magnetic fluids in presence of a magnetic field. In a finite field the magnetic phase transition is absent, but instead a line of first order liquid-liquid transitions ending in a critical point occurs for a magnetic interaction, which is sufficiently strong. Varying the magnetic field these critical points extend from the tricritical point at H=0 to a critical endpoint. For a fluid with Ising spins we calculate the critical lines and several tricritical exponents analytically. For Heisenberg fluids we obtain the phase diagrams from a numerical solution of the mean field equations of state. Received 20 March 1998     

Zero momentum calculations for finite temperature field theory

Problems with calculations of Feynaman diagrams with no external leg carrying non-zero momentum for a finite temperature scalar field theory using a real time path integral approach are considered. The necessary extra rules for all such calculations are derived, covering all the usual methods of calculating the effective potential, and several problems with such calculations from the path integral viewpoint are resolved.     

热场动力学中的切割图方法及热格林函数的虚部

在热场动力学的框架下,依据有限温度下的切割定理提出了一套直观的费曼图切割方法及其相应的费曼规则,从而给出了一条计算热格林函数应部的方便途径.作为应用的例子我们分别讨论了二点、三点和四点格林函数的虚部.     

The phase diagrams of a spin-1 transverse Ising model

The phase diagrams of the spin-1 transverse Ising model with the presence of a crystal field is investigated by using an effective-field theory (EFT). We give a method to calculate the Gibbs free energy numerically at finite temperature within the EFT. The first-order transition lines are obtained by comparing the Gibbs free energy. The phase diagrams and the Gibbs free energy are also compared with those given using the mean-field theory (MFT).     

A numerical simulation of the transition to turbulence in a two-dimensional flow

A numerical simulation of the transition to turbulence is performed using a finite element method. The unsteady Navier-Stokes equations are discretized using a standard Galerkin approximation and a loading strategy for increasing the Reynolds number. The numerical results are then analysed at different Reynolds numbers showing a transition from a steady-state solution to a weakly chaotic one. Phase space diagrams are presented showing the presence of strange attractors. The dimension and Lyapunov's exponents of these attractors are computed and compared with existing results in the literature.     

Freezing transition in the mean-field approximation model of pedestrian counter flow

We study the freezing transition in the counter flow of pedestrians within the channel numerically and analytically. We present the mean-field approximation (MFA) model for the pedestrian counter flow. The model is described in terms of a couple of nonlinear difference equations. The excluded-volume effect and bi-directionality are taken into account. The fundamental diagrams (current-density diagrams) are derived. When pedestrian density is higher than a critical value, the dynamical phase transition occurs from the free flow to the freezing (stopping) state. The critical density is derived by using the linear stability analysis. Also, the velocity and current (flow) at the steady state are derived analytically. The analytical result is consistent with that obtained by the numerical simulation.     

Cellular automaton simulations for mixed traffic with erratic motorcycles’ behaviours

Modeling mixed traffic composed of motorcycles can be a challenging issue because many erratic motorcyclists may not follow the lane disciplines, particularly when traffic is congested. Based upon the refined cellular automaton (CA) model recently developed by the authors [L.W. Lan, Y.C. Chiou, Z.S. Lin, C.C. Hsu, Physica A 388 (2009) 3917-3930], this paper further proposed a sophisticated CA model to elucidate the erratic motorcycle behaviours in mixed traffic contexts. In addition to the conventional moving forward and lane-change rules, the sophisticated CA model also explicated the lateral drift behaviour for cars moving in the same lane, the lateral drift behaviour for motorcycles breaking into two moving cars, and the transverse crossing behaviour for motorcycles through the gap between two stationary cars in the same lane. Fundamental diagrams and space-time trajectories for vehicles with various car-motorcycle mixed ratios are demonstrated.     

Network analysis of the state space of discrete dynamical systems

We study networks representing the dynamics of elementary 1D cellular automata (CA) on finite lattices. We analyze scaling behaviors of both local and global network properties as a function of system size. The scaling of the largest node in-degree is obtained analytically for a variety of CA including rules 22, 54, and 110. We further define the path diversity as a global network measure. The coappearance of nontrivial scaling in both the hub size and the path diversity separates simple dynamics from the more complex behaviors typically found in Wolfram's class IV and some class III CA.     

Phase transition in the Blume-Capel model with second neighbour interaction

A two dimensional antiferromagnetic spin-1 Ising model with negative next- nearest neighbour interaction (J ₂ <0) and under an external magnetic field is investigated by two methods: The mean-field theory and Finite-Size-Scaling based on transfer matrix (TMFSS) calculations. The ground state diagrams exhibit several new phases including frustrated ones. At finite temperature we obtain by these two methods quite rich phase diagrams, with several multicritical points. While Mean field approximation yields phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from transfer matrix finite size scaling calculations. For a certain range of interaction parameters, the model is shown to violate the ordinary universality hypothesis. Received: 3 November 1997 / Revised: 31 March 1998 / Accepted: 7 April 1998     

Two-lane traffic simulations with a blockage induced by an accident car

Based on the two-lane traffic model proposed by Chowdhury et al., a highway traffic model with a blockage induced by an accident car is proposed, in which both symmetric lane changing rules and asymmetric lane changing rules are adopted. The fundamental diagrams and spatial-temporal profiles are presented after the numerical simulation and the jam transition is studied. It is shown that the accident car not only causes a local jam behind the accident car, but also causes vehicles to cluster in the bypass lane. The asymmetric lane changing rules are more advantageous in reducing the local jam than the symmetric lane changing rules when the accident car is in the right lane, and the symmetric lane changing rules are superior when the accident car is in the left lane. Furthermore the curves of lane-changing frequency against the total density are given. It is found that the vehicles will change lane more frequently when traffic is inhomogeneous with different types of vehicle or with an accident car.     

A mathematical model of carbon dioxide flooding with hydrate formation

The injection of carbon dioxide into a reservoir that contains methane and water in a free state is investigated. A mathematical model of this process is proposed that suggests the formation of the CO₂ hydrate on the surface of the phase transition separating regions of methane and carbon dioxide. The conditions on the interface are derived, and an asymptotic solution of the problem is found. Critical diagrams are obtained that define parameter ranges in which there is full or partial transition of gaseous carbon dioxide to a hydrate state.     

Fundamental diagram in traffic flow of mixed vehicles on multi-lane highway

We study the fundamental diagram for traffic flow of vehicular mixture on a multi-lane highway. We present the car-following model of multi-lane traffic in which slow and fast vehicles flow with changing lanes. We investigate the traffic states of the vehicular mixture under the periodic boundary. Two values of the current appear at a density and two current curves are obtained. Vehicles move with changing lanes in the traffic state of high current, while vehicles move without changing lanes in the traffic state of low current. They depend on the density, the fraction of slow vehicles, and the initial condition. In the high-current curve, the jamming transition between the free flow and the jammed state occurs at a low density. The fundamental diagrams (current-density diagrams) are shown for the single-lane, two-lane, three-lane, and four-lane traffics.