Local structure theory: Calculation on hexagonal arrays,and interaction of rule and lattice

Local structure theory calculations⁷ are applied to the study of cellular automata on the two-dimensional hexagonal lattice. A particular hexagonal lattice rule denoted (3422) is considered in detail. This rule has many features in common with Conway'sLife. The local structure theory captures many of the statistical properties of this rule; this supports hypotheses raised by a study ofLife itself⁽⁶⁾. As inLife, the state of a cell under (3422) depends only on the state of the cell itself and the sum of states in its neighborhood at the previous time step. This property implies that evolution rules which operate in the same way can be studied on different lattices. The differences between the behavior of these rules on different lattices are dramatic. The mean field theory cannot reflect these differences. However, a generalization of the mean field theory, the local structure theory, does account for the rule-lattice interaction.     

Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state _σi(t)∈{0,1}${\sigma }_i(t)\in \{0,1\}$ of a cell i   does not only depend on the states in its local neighborhood at time t-1$t-1$, but also on the memory of its own past states _σi(t-2),_σi(t-3),…,_σi(t-τ),…${\sigma }_i(t-2),{\sigma }_i(t-3),\dots ,{\sigma }_i(t-\tau ),\dots$ . We assume that the weight of this memory decays proportionally to τ^-α${\tau }^{-\alpha }$, with α?0$\alpha ?0$ (the limit α→∞$\alpha \to \infty$ corresponds to the usual CA). Since the memory function is summable for α>1$\alpha >1$ and nonsummable for 0?α?1$0?\alpha ?1$, we expect pronounced changes of the dynamical behavior near α=1$\alpha =1$. This is precisely what our simulations exhibit, particularly for the time evolution of the Hamming distance H   of initially close trajectories. We typically expect the asymptotic behavior H(t)∝t^1/(1-q)$H(t)\propto t^{1/(1-q)}$, where q   is the entropic index associated with nonextensive statistical mechanics. In all cases, the function q(α)$q(\alpha )$ exhibits a sensible change at α?1$\alpha ?1$. We focus on the class II rules 61, 99 and 111. For rule 61, q=0$q=0$ for 0?α?_αc?1.3$0?\alpha ?{\alpha }_c?1.3$, and q<0$q<0$ for α>_αc$\alpha >{\alpha }_c$, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size N   indicate that the range of the power-law regime for H(t)$H(t)$ typically diverges ∝N^z$\propto N^z$ with 0?z?1$0?z?1$.     

Simulation of the nonlinear vibration of a string using the Cellular Automata based on the reflection rule

In this study, the nonlinear dynamic responses of a string are simulated using the Cellular Automata method based on the reflection rule. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. A new treatment of the dynamic time step is proposed for the Cellular Automata method considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic are simulated using the Cellular Automata method. A typical resonance curve can be obtained. Second, the dynamic responses of a string with nonlinear characteristic are simulated using the proposed method. Some characteristic types of vibration can be obtained. It is concluded that the linear and nonlinear dynamic responses of a string may be obtained by simulation using the Cellular Automata method.     

敏感换道对下匝道系统交通流的影响

在NaSh模型基础上,充分考虑驾驶员在匝口指示牌的诱导作用下驾驶方式的变化,定义了车辆在匝口上游的换道、直行驾驶规则,提出了敏感换道的元胞自动机下匝道交通流模型. 通过计算机数值模拟,结果表明：敏感换道过程能减少直行车道上的转出车辆比例,对非必要的换道行为有明显的抑制作用,且随敏感换道区长度增加,该作用越明显;匝口提示位置并非越长越好,系统转出车辆比例越小,系统所需的最佳敏感换道区长度越短. 工程设计中根据转出车辆比例选取一个适宜距离安放匝口指示牌,能有效增加系统流量和临界加入概率. 关键词： 交通流 元胞自动机 驾驶方式 计算机数值模拟     

Molar-mass dependence of the lamellar thickness in symmetric diblock copolymers

Summary A homologous series of nine symmetric poly(styrene)-block-poly(butadieme) diblock copolymers was synthesized and characterized. The scaling of the lamellar thickness,D, with the chain lengthN(DαN ^δ ) was investigated with small-angle X-ray scattering. Different sample preparation methods were used: annealing, solvent-casting and shear alignment. We find that the lamellar thickness and thus the value of the exponent δ depends on the preparation method used,i.e. care must be taken to ensure that the samples investigated are in thermodynamic equilibrium. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

Statistical mechanics of probabilistic cellular automata

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ^d and translation-invariant Gibbs states for a related Hamiltonian on ( ^d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Toom's proof of nonergodicity of a certain class of PCAs.     

A percolation approach to the Kauffman model

A study is made of the spreading of damage in the random but deterministic Kauffman model on the square lattice with the spreading from one edge of the lattice. The critical value of the parameterp _c above which the system becomes chaotic is found to bep _c0.298. The possibility of suppression of the chaotic phase by noise is also studied. It is found that forpp _c, an extremely large noise levelg>0.99 is required, if possible at all.On leave from Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China.     

Simply modified GKL density classifiers that reach consensus faster

The two-state Gacs-Kurdyumov-Levin (GKL) cellular automaton has been a staple model in the study of complex systems due to its ability to classify binary arrays of symbols according to their initial density. We show that a class of modified GKL models over extended neighborhoods, but still involving only three cells at a time, achieves comparable density classification performance but in some cases reach consensus more than twice as fast. Our results suggest the time to consensus (relative to the length of the CA) as a complementary measure of density classification performance.     

Exponential convergence of Toom's probabilistic cellular automata

We obtain cluster expansions for small random perturbations of deterministic Toom's automata in the one-dimensional case. Exponential convergence follows. Analyticity of invariant measures is examined as well as the simplest multidimensional case.     

On the nature of a*kak and the emergence of the Born rule

This paper is intended to show that a review in the concept of the game theoretical utility, the revised utility to be applied to the definition of the utility of a wave function representing an object subsystem relative to its observer subsystem, both within an isolated system, leads to the emergence of Max Born's rule as a profit under a von Neumann good measure game.     

A note on inherent replication properties of local cellular automata transition functions

Local transition functions of elementary cellular automata show different tendencies to replicate parts of a configuration in a later generation. This is seen as regularities in the time-space diagram. This replication depends on both the configuration and the local transition function. A possibility to isolate the influence of the local transition function is shown.     

A stochastic cellular automata model of tautomer equilibria

ABSTRACT

Many chemical substances, including drugs and biomolecules, exist in solution not as a single species, but as a collection of tautomers and related species. Importantly, each of these species is an independent compoundwith its own specific biochemical and physicochemical properties. The species interconvert in a dynamic and often complicated manner, making modelling the overall species composition difficult. Agent-based cellular automata models are uniquely suited to meet this challenge, allowing the equilibria to be simulated using simple rulesand at the same time capturing the inherent stochasticity of the natural phenomenon. In the present example a stochastic cellular automata model is employed to simulate the tautomer equilibria of 9-anthrone and 9-anthrol in the presence of their common anion. The observed K_E of the 9-anthrone ? 9-anthrol tautomerisation along with the measured tautomer pK_a values were used to model the equilibria at pH values 4, 7 and 10. At pH 4 and 7, the anthrone comprises >99% of the total species population, while at pH 10the anthrone and the anion each represent just under half of the total population. The advantages of the cellular automata approach over the customary coupled differential equation approach are discussed.     

A theoretical model for uni-directional ant trails

A theoretical model of uni-directional ant traffic, motivated by the motion of ants in trail is proposed. Two different type of ants, one of which smells very well and the other does not, are considered. The flux of ants in this model is investigated as functions of the probability of evaporation rate of pheromone. The obtained results indicate that the mean velocity of the ants varies non-monotonically with their density. In addition, it is observed that phase transition in the flux and the mean velocity vs. density occurs at certain density for a fixed evaporation rate. The effective hopping probability is investigated as well depending on the evaporation rate of pheromone. It is worth to note that the proposed model can be generalized for vehicular traffic on freeways.     

Initiation of damage in the Kauffman model

An alternate method is presented for initiating damage to the Kauffman model. A numerical study suggests that this modification increases the efficiency of generating damage clusters without changing their fractal properties.     

Born’s rule as signature of a superclassical current algebra

We present a new tool for calculating the interference patterns and particle trajectories of a double-, three- and N$N$-slit system on the basis of an emergent sub-quantum theory developed by our group throughout the last years. The quantum itself is considered as an emergent system representing an off-equilibrium steady state oscillation maintained by a constant throughput of energy provided by a classical zero-point energy field. We introduce the concept of a “relational causality” which allows for evaluating structural interdependences of different systems levels, i.e. in our case of the relations between partial and total probability density currents, respectively. Combined with the application of 21st century classical physics like, e.g., modern nonequilibrium thermodynamics, we thus arrive at a “superclassical” theory. Within this framework, the proposed current algebra directly leads to a new formulation of the guiding equation which is equivalent to the original one of the de Broglie–Bohm theory. By proving the absence of third order interferences in three-path systems it is shown that Born’s rule is a natural consequence of our theory. Considering the series of one-, double-, or, generally, of N$N$-slit systems, with the first appearance of an interference term in the double slit case, we can explain the violation of Sorkin’s first order sum rule, just as the validity of all higher order sum rules. Moreover, the Talbot patterns and Talbot distance for an arbitrary N$N$-slit device can be reproduced exactly by our model without any quantum physics tool.     

Fractal dimension at the phase transition of inhomogeneous cellular automata

For random binary mixtures of cellular automata in the square lattice, calculations are made of the fractal dimensions associated with the damage spreading and the propagation time of damage at the transition to chaos. Two rules are mixed and universalities of these quantities are sought with respect to change of the rules.     

Kink movements and percolation in the binary additive cellular automaton

We show that under the Bernoulli initial condition two kinks in the cellular automaton (CA) 18/256 will annihilate each other with probability one. It turns out that there is an equivalent statement in terms of percolation in the simple binary additive CA. Namely, under the Bernoulli initial condition, l's do not percolate in the binary additive CA.