Nonlinear Sciences

We systematically study the boundaries of one-dimensional, 2-color cellular automata depending on 4 cells, begun from simple initial conditions. We determine the exact growth rates of the boundaries that appear to be reducible. Morphic words characterize the reducible boundaries. For boundaries that appear to be irreducible, we apply curve-fitting techniques to compute an empirical growth exponent and (in the case of linear growth) a growth rate. We find that the random walk statistics of irreducible boundaries exhibit surprising regularities and suggest that a threshold separates two classes. Finally, we construct a cellular automaton whose growth exponent does not exist, showing that a strict classification by exponent is not possible.

We introduce density dependence of the cell size in cellular-automaton models for traffic flow, which allows a more precise correspondence between real-world phenomena and what observed in simulation. Also, we give an explicit calibration of the particle density particularly for the asymmetric simple exclusion process with some update rules. We thus find that the present method is valid in that it reproduces a realistic flow-density diagram.

We present a systematic study of capillary filling for a binary fluid by using mesoscopic a lattice Boltzmann model describing a diffusive interface moving at a given contact angle with respect to the walls. We compare the numerical results at changing the ratio the typical size of the capillary, H, and the wettability of walls. Numerical results yield quantitative agreement with the Washburn law in all cases, provided the channel lenght is sufficiently larger then the interface width. We also show that in the initial stage of the filling process, transient behaviour induced by inertial effects are under control in our lattice Boltzmann equation and in good agreement with the phenomenology of capillary filling. Finally, at variance with multiphase LB simulations, velocity and pressure profiles evolve under the sole effect of capillary drive all along the channel.

We present a systematic study of capillary filling for multi-phase flows by using mesoscopic lattice Boltzmann models describing a diffusive interface moving at a given contact angle with respect to the walls. We compare the numerical results at changing the density ratio between liquid and gas phases and the ratio between the typical size of the capillary and the interface width. It is shown that numerical results yield quantitative agreement with the Washburn law when both ratios are large, i.e. as the hydrodynamic limit of a infinitely thin interface is approached. We also show that in the initial stage of the filling process, transient behaviour induced by inertial effects and ``vena contracta'' mechanisms, may induce significant departure from the Washburn law. Both effects are under control in our lattice Boltzmann equation and in good agreement with the phenomenology of capillary filling.

We discuss example of an elementary cellular automaton for which the density of ones decays toward its limiting value as a power of the number of iterations n . Using the fact that this rule conserves the number of blocks 10 and that preimages of some other blocks exhibit patterns closely related to patterns observed in rule 184, we derive expressions for the number of n -step preimages of all blocks of length 3. These expressions involve Catalan numbers, and together with basic properties of iterated probability measures they allow us to to compute the density of ones after n iterations, as well as probabilities of occurrence of arbitrary block of length smaller or equal to 3.

Emergence, the phenomena where a system's micro-scale dynamics facilitate the development of non-trivial, informative higher scales, has become a foundational concept in modern sciences, tying together fields as diverse as physics, biology, economics, and ecology. Despite it's apparent universality and the considerable interest, historically researchers have struggled to provide a rigorous, formal definition of emergence that is applicable across fields. Recent theoretical work using information theory and network science to formalize emergence in state-transition networks (causal emergence) has provided a promising way forward, however the relationship between this new framework and other well-studied system dynamics is unknown. In this study, we apply causal emergence analysis to two well-described dynamical systems: the 88 unique elementary cellular automata and the continuous Rossler system in periodic, critical, and chaotic regimes. We find that emergence, as well as its component elements (determinism, degeneracy, and effectiveness) vary dramatically in different dynamical regimes in sometimes unexpected ways. We conclude that the causal emergence framework provides a rich new area of research to explore both to theoreticians and natural scientists in many fields.

We study the classification of cellular-automaton update rules into Wolfram's four classes. We start with the notion of the input entropy of a spatiotemporal block in the evolution of a cellular automaton, and build on it by introducing two novel entropy measures, one that is also based on inputs to the cells, the other based on state transitions by the cells. Our two new entropies are both targeted at the classification of update rules by parallel machines, being therefore mindful of the necessary communications requirements; we call them cell-centric input entropy and cell-centric transition entropy to reflect this fact. We report on extensive computational experiments on both one- and two-dimensional cellular automata. These experiments allow us to conclude that the two new entropies possess strong discriminatory capabilities, therefore providing valuable aid in the classification process.

We construct a two dimensional Cellular Automata based model for the description of pedestrian dynamics. Wide range of complicated pattern formation phenomena in pedestrian dynamics are described in the model, e.g. lane formation, jams in a counterflow and egress behavior. Mean-field solution of the densely populated case and numerical solution of the sparsely populated case are provided. This model has the potential to describe more flow phenomena.

The classical dynamical systems model of continuous stirred tank reactors (CSTR) in which a first order chemical reaction takes place is reformulated in terms of stochastic cellular automata by extending previous works of Seyborg (1997) and Neuforth (2000) by including the feed flow of chemical reactants. We show that this cellular automata procedure is able to simulate the dilution rate and the mixing process in the CSTR, as well as the details of the heat removal due to the jacket. The cellular automata approach is expected to be of considerable applicability at any industrial scales and especially for any type of microchemical systems

A small-world cellular automaton network has been formulated to simulate the long-range interactions of complex networks using unconventional computing methods in this paper. Conventional cellular automata use local updating rules. The new type of cellular automata networks uses local rules with a fraction of long-range shortcuts derived from the properties of small-world networks. Simulations show that the self-organized criticality emerges naturally in the system for a given probability of shortcuts and transition occurs as the probability increases to some critical value indicating the small-world behaviour of the complex automata networks. Pattern formation of cellular automata networks and the comparison with equation-based reaction-diffusion systems are also discussed