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Analysis of experimental kinetic dependences and results of the computer modeling of pairwise interactions in the heterogeneous system have shown that a character of its fractal property variation in the evolution process would be different at distinguished temperature ranges and dependent on the proportion of reagent concentrations and their initial distribution type.
For a large class of inhomogeneous interacting particle systems (IPS) on a lattice we develop a rigorous method for mapping them onto homogeneous IPS. Our novel approach provides a direct way of obtaining the statistical properties of such inhomogeneous systems by studying the far simpler homogeneous systems. In the cases when the latter can be solved exactly our method yields an exact solution for the statistical properties of an inhomogeneous IPS. This approach is illustrated by studies of three of IPS, namely those with particles of different sizes, or with varying (between particles) maximal velocities, or accelerations.
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
We establish a matrix generalization of the ultradiscrete fourth Painlevé equation (ud-PIV). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints imposed by the requirement of a consistent evolution of the systems. The ultradiscrete limit of these systems yields coupled multicomponent ultradiscrete systems that generalize ud-PIV. The dynamics, irreducibility, and integrability of the matrix valued ultradiscrete systems are studied.
We investigate dynamics of the cellular automaton rule 142. This rule possesses additive invariant of the second order, namely it conserves the number of blocks 10. Rule 142 can be alternatively described as an operation on a binary string in which we simultaneously flip all symbols which have dissenting right neighbours. We show that the probability of having a dissenting neighbour can be computed exactly using the fact that the surjective rule 60 transforms rule 142 into rule 226. We also demonstrate that the conservation of the number of 10 blocks implies that these blocks move with speed -1 or stay in the same place, depending on the state of the preceding site. At the density of blocks 10 equal to 0.25, the rule 142 exhibits a phenomenon similar to the jamming transitions occurring in discrete models of traffic flow.
The advantage of Cellular Potts Model (CPM) is due to its ability for introducing cell-cell interaction based on the well known statistical model i.e. the Potts model. On the other hand, Lattice gas Cellular Automata (LGCA) can simulate movement of cell in a simple and correct physical way. These characters of CPM and LGCA have been combined in a reaction-diffusion frame to simulate the dynamic of avascular cancer growth on a more physical basis.The cellular automaton is evolved on a square lattice on which in the diffusion step tumor cells (C) and necrotic cells (N) propagate in two dimensions and in the reaction step every cell can proliferate, be quiescent or die due to the apoptosis and the necrosis depending on its environment. The transition probabilities in the reaction step have been calculated by the Glauber algorithm and depend on the KCC, KNC, and KNN (cancer-cancer, necrotic-cancer, and necrotic-necrotic couplings respectively). It is shown the main feature of the cancer growth depends on the choice of magnitude of couplings and the advantage of this method compared to other methods is due to the fact that it needs only three parameters KCC, KNC and KNN which are based on the well known physical ground i.e. the Potts model.
The critical points of the 3-states two-layer Potts model on square lattice for different interlayer couplings (Kx, Ky,and Kz) are calculated with high precision using probabilistic cellular automata with Glauber algorithm, where Kx and Ky are the nearest-neighbor interactions within each layer in the x and y directions, respectively and Kz is the interlayer coupling. The obtained results are 0.726, 0.807, 0.928, 0.987 and 1.00 for the cases Kz per Kx = 1, 0.5, 0.1, 0.01 and 0.001, respectively. Then, the critical curve has been constructed for this model.
We performed a numerical study on random Boolean networks with power-law rank outdegree distributions to find local structural cause for the emergence of high or low degree of coherence in binary state variables. The degree of randomness and coherence of the binary sequence are measured by entropy and mutual information, depending on local structure that consists of a node with a highly connected, called hub and its upstream nodes, and types of Boolean functions for the nodes. With a large number of output connections from a hub, the effects of Boolean function on the hub are more prominent. The local structures that give larger entropy tend to give rise to larger mutual information. Based on the numerical results and structural conditions we derived a time-independent transmission characteristic function of state variables for the local structures. We obtained good relationships between the numerical and analytical results, which indicate that dynamical properties from the whole networks can be inferred from the differences in the local structures.
In this paper we study the problem of the existence of a least-action principle for invertible, second-order dynamical systems, discrete in time and space. We show that, when the configuration space is finite, a least-action principle does not exist for such systems. We dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold, and those not of finite type for which it is possible to construct a least-action principle. We also show how to recover an action by restriction of the phase space of certain second-order discrete dynamical systems. We provide numerous examples to illustrate each of these results.
We examined the effects of inhomogeneity on the dynamics and structural properties using Boolean networks. Two different power-law rank outdegree distributions were embedded to determine the role of hubs. The degree of randomness and coherence of the binary sequence in the networks were measured by entropy and mutual information, depending on the number of outdegrees and types of Boolean functions for the hub. With a large number of outdegrees, the path length from the hub reduces as well as the effects of Boolean function on the hub are more prominent. These results indicate that the hubs play important roles in networks' dynamics and structural properties. By comparing the effect of the skewness of the two different power-law rank distributions, we found that networks with more uniform distribution exhibit shorter average path length and higher event probability of coherence but lower degree of coherence. Networks with more skewed rank distribution have complementary properties. These results indicate that highly connected hubs provide an effective route for propagating their signals to the entire network.
Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using Monte-Carlo simulations. We show that this phenomenon is a second-order phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics.
We present results of cellular automata based investigations of rotating spiral autowaves in a nonequilibrium excitable medium which models three-level paramagnetic microwave phonon laser (phaser). The computational model is described in arXiv:cond-mat/0410460v2 and arXiv:cond-mat/0602345v1 . We have observed several new scenarios of self-organization, competition and dynamical stabilization of rotating spiral autowaves under conditions of cross-relaxation between three-level active centers. In particular, phenomena of inversion of topological charge, as well as processes of regeneration and replication of rotating spiral autowaves in various excitable media were revealed and visualized for mesoscopic-scale areas of phaser-type active systems, which model real phaser devices.
Ownership and trade emerge from anarchy as evolutionary stable strategies. In these evolutionary game models, ownership status provides an endogenous asymmetrizing criterion enabling cheaper resolution of property conflicts.
An elementary analytical fluid flow is composed by a geometric domain, a list of analytical constraints and by the function which depends on the physical properties, as Reynolds number, of the considered fluid. For this object, notions of laminar or weakly turbulent behavior are described using a simple mathematical model.
The paper describes a possible physical characterization for the definition of elementary fluid flow. As consequence, an analytical expansion based on Reynolds number powers for the velocity field is shown in the weakly turbulent case.
Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.
We apply the methods of lattice field theories to the quantization of cellular automata. We discuss the quantization of five main categories of cellular automata: bosonic, fermionic, supersymmetric, spin and quantum dot using path integral and operator formalisms of lattice field theories. We show that the quantization of supersymmetric cellular automata is related to recently discussed string bit models of Thorn and Bergman and represents a link of cellular automata theory to fundamental physics. We discuss spin and quantum dot cellular automata for their importance in experimental realizations and their use in quantum computation. Previous studies of quantum cellular automata utilize the wave function values as cell contents and the discretized linear Dirac equation as an update equation. We show that our approach to the quantization of fermionic cellular automata includes this utilization as a field equation, and in addition allows for nonlinearity through lattice field interactions.
In recent work [quant-ph/0405174] by Schumacher and Werner was discussed an abstract algebraic approach to a model of reversible quantum cellular automata (CA) on a lattice. It was used special model of CA based on partitioning scheme and so there is a question about quantum CA derived from more general, standard model of classical CA. In present work is considered an approach to definition of a scheme with "history", valid for quantization both irreversible and reversible classical CA directly using local transition rules. It is used language of vectors in Hilbert spaces instead of C*-algebras, but results may be compared in some cases. Finally, the quantum lattice gases, quantum walk and "bots" are also discussed briefly.