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arXiv1997

Dynamic Predictions from Time Series Data- An Artificial Neural Network Approach

A hybrid approach, incorporating concepts of nonlinear dynamics in artificial neural networks (ANN), is proposed to model time series generated by complex dynamic systems. We introduce well known features used in the study of dynamic systems - time delay $τ$ and embedding dimension $d$ - for ANN modelling of time series. These features provide a theoretical basis for selecting the optimal size for the number of neurons in the input layer. The main outcome for the number of neurons in the input layer. The main outcome of the new approach for such problems is that to a large extent it defines the ANN architecture and leads to better predictions. We illustrate our method by considering computer generated periodic and chaotic time series. The ANN model developed gave excellent quality of fit for the training and test sets as well as for iterative dynamic predictions for future values of the two time series. Further, computer experiments were conducted by introducing Gaussian noise of various degrees in the two time series, to simulate real world effects. We find rather surprising results that upto a limit introduction of noise leads to a smaller network with good generalizing capability.

comp-gas↗

arXiv1998

Computations on Nondeterministic Cellular Automata

The work is concerned with the trade-offs between the dimension and the time and space complexity of computations on nondeterministic cellular automata. It is proved, that 1). Every NCA $\Cal A$ of dimension $r$, computing a predicate $P$ with time complexity T(n) and space complexity S(n) can be simulated by $r$-dimensional NCA with time and space complexity $O(T^{\frac{1}{r+1}} S^{\frac{r}{r+1}})$ and by $r+1$-dimensional NCA with time and space complexity $O(T^{1/2} +S)$. 2) For any predicate $P$ and integer $r>1$ if $\Cal A$ is a fastest $r$-dimensional NCA computing $P$ with time complexity T(n) and space complexity S(n), then $T= O(S)$. 3). If $T_{r,P}$ is time complexity of a fastest $r$-dimensional NCA computing predicate $P$ then $T_{r+1,P} &=O((T_{r,P})^{1-r/(r+1)^2})$, $T_{r-1,P} &=O((T_{r,P})^{1+2/r})$. Similar problems for deterministic CA are discussed.

comp-gas↗

arXiv1998

Lindenmayer systems as a model of computations

LS is a particular type of computational processes simulating living tissue. They use an unlimited branching process arising from the simultaneous substitutions of some words instead of letters in some initial word. This combines the properties of cellular automata and grammars. It is proved that 1) The set of languages, computed in a polynomial time on such LS that all replacing words are not empty, is exactly NP- languages. 2) The set of languages, computed in a polynomial time on arbitrary LS, contains the polynomial hierarchy. 3) The set of languages, computed in a polynomial time on a nondeterministic version of LS, strictly contains the set of languages, computed in a polynomial time on Turing Machines with a space complexity $n^a$, where $a$ is positive integer. In particular, the last two results mean that Lindenmayer systems may be even more powerful tool of computations than nondeterministic Turing Machine.

comp-gas↗

arXiv1998

Effect of Boundary Conditions on Cellular Automata that Classify Density

The properties of two-state nearest-neighbour cellular automata (CA) that are capable of density classification are discussed. It is shown that these CA actually conserve the total density, rather than merely classifying it. This is also the criterion for any CA simulation of DFT. The effect of boundary and periodicity conditions upon the evolution of such CA are elaborated by considering linear and cyclic lattices and boundary conditions consistent with the conservation criterion. In a bounded linear lattice, it is possible to achieve a configuration with a single 01 and no 10 domain wall, or vice versa, but this is not possible in a cyclic lattice, where these two domain walls have to appear in pairs. This determines the final stable state of the automaton.

comp-gas↗

arXiv1998

Generalized Thermal Lattice Gases

We show how to employ thermal lattice gas models to describe non-equilibrium phenomena. This is achieved by relaxing the restrictions of the usual micro-canonical ensemble for these models via the introduction of thermal ``demons'' in the style of Creutz. Within the Lattice Boltzmann approximation, we then derive general expressions for the usual transport coefficients of such models, in terms of the derivatives of their equilibrium distribution functions. To illustrate potential applications, we choose a model obeying Maxwell-Boltzmann statistics, and simulate Rayleigh-Bénard convection with a forcing term and a temperature gradient, both of which are continuously variable.

comp-gas↗

arXiv1998

Lattice-Boltzmann Simulations of Fluid Flows in MEMS

The lattice Boltzmann model is a simplified kinetic method based on the particle distribution function. We use this method to simulate problems in MEMS, in which the velocity slip near the wall plays an important role. It is demonstrated that the lattice Boltzmann method can capture the fundamental behavior in micro-channel flow, including velocity slip and nonlinear pressure drop along the channel. The Knudsen number dependence of the position of the vortex center and the pressure contour in micro-cavity flows is also demonstrated.

comp-gas↗

arXiv1998

Cellular Automaton Model for Immunology of Tumor Growth

The stochastic discrete space-time model of an immune response on tumor spreading in a two-dimensional square lattice has been developed. The immunity-tumor interactions are described at the cellular level and then transferred into the setting of cellular automata (CA). The multistate CA model for system, in which all statesoflattice sites, composing of both immune and tumor cells populations, are the functions of the states of the 12 nearest neighbors. The CA model incorporates the essential featuresof the immunity-tumor system. Three regimes of neoplastic evolution including metastatic tumor growth and screen effect by inactive immune cells surrounding a tumor have been predicted.

comp-gas↗

arXiv1998

Crystalline Computation

A cellular automaton is a deterministic and exactly computable dynamical system which mimics certain fundamental aspects of physical dynamics such as spatial locality and finite entropy. CA systems can be constructed which have additional attributes that are basic to physics: systems which are exactly invertible at their finest scale, which obey exact conservation laws, which support the evolution of arbitrary complexity, etc. In this paper, we discuss techniques for bringing CA models closer to physics, and some of the interesting consequences of doing so.

comp-gas↗

arXiv1999

Lattice Gases and Cellular Automata

We review the class of cellular automata known as lattice gases, and their applications to problems in physics and materials science. The presentation is self-contained, and assumes very little prior knowledge of the subject. Hydrodynamic lattice gases are emphasized, and non-lattice-gas cellular automata -- even those with physical applications -- are not treated at all. We begin with a review of lattice gases as the term is understood in equilibrium statistical physics. We then discuss the various methods that have been proposed to simulate hydrodynamics with a lattice gas, leading up to the 1986 discovery of a lattice gas for the isotropic Navier-Stokes equations. Finally, we discuss variants of lattice-gas models that have been used for the simulation of complex fluids.

comp-gas↗

arXiv1999

Construction of Molecular Dynamics Like Cellular Automata Models for Simulation of Compressible Fluid Dynamic Systems

This study aims at finding a method for constructing molecular dynamics like models using the formalism of cellular automata for fast simulation of fluid dynamic systems (including compressible phenomena). In as much as the results indicate, the attempt is successful. A systematic method for constructing cellular automata models of fluid dynamic systems is discovered and proposed following a review of the existing developments. The considerations required for constructing such models for fluid dynamic systems consisting of particles with arbitrary interaction potentials and existing over arbitrary spatial lattices are outlined. The method is illustrated by constructing a model for simulation of systems of particles moving with unit speed along the links of an underlying square spatial lattice. Using this model, two two-dimensional systems are simulated and studied for a number of model and system parameters. For almost all the model and system parameters, the results are found to be in complete agreement with the available theoretical predictions. For some model parameters, results show (expected) departure from theoretical predictions which is explained.

comp-gas↗

arXiv2000

How fast does Langton's ant move?

The automaton known as `Langton's ant' exhibits a dynamical transition from a disordered phase to an ordered phase where the particle dynamics (the ant) produces a regular periodic pattern (called `highway'). Despite the simplicity of its basic algorithm, Langton's ant has remained a puzzle in terms of analytical description. Here I show that the highway dynamics obeys a discrete equation where from the speed of the ant ($c={\sqrt 2}/52$) follows exactly.

cond-mat.stat-mech↗

arXiv2001

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring: II. The Charged System and the Two-component Burgers Equations

We further study the stochastic model discussed in Ref.[2] in which positive and negative particles diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. We extend the analysis of this model to the case when the densities of the charged particles are not the same. The mean-field equations describing the model are coupled nonlinear differential equations that we call the two-component Burgers equations. We find roundabout weak solutions of these equations. These solutions are used to describe the properties of the stationary states of the stochastic model. The values of the currents and of various two-point correlation functions obtained from Monte-Carlo simulations are compared with the mean-field results. Like in the case of equal densities, one finds a pure phase, a mixed phase and a disordered phase.

cond-mat.stat-mech↗

arXiv2002

Evolution of Cooperation in a Spatial Prisoner's Dilemma

We investigate the spatial distribution and the global frequency of agents who can either cooperate or defect. The agent interaction is described by a deterministic, non-iterated prisoner's dilemma game, further each agent only locally interacts with his neighbors. Based on a detailed analysis of the local payoff structures we derive critical conditions for the invasion or the spatial coexistence of cooperators and defectors. These results are concluded in a phase diagram that allows to identify five regimes, each characterized by a distinct spatiotemporal dynamics and a corresponding final spatial structure. In addition to the complete invasion of defectors, we find coexistence regimes with either a majority of cooperators in large spatial domains, or a minority of cooperators organized in small non-stationary domains or in small clusters. The analysis further allowed a verification of computer simulation results by Nowak and May (1993). Eventually, we present simulation results of a true 5-person game on a lattice. This modification leads to non-uniform spatial interactions that may even enhance the effect of cooperation. Keywords: Prisoner's dilemma; cooperation; spatial 5-person game

cond-mat.stat-mech↗

arXiv2004

Size-Independent Non-Equilibrium Fluctuations

A local quantum phenomenon that gives rise to generic for all surface reactions macroscopic fluctuations is studied. The issue is viewed with respect to the necessary conditions for a long-term stable evolution of any natural and artificial system. It is shown that global coupling of the local fluctuations is necessary for providing a long-term stability of the system. A successful coupling mechanism is achieved on the grounds of new assumptions about the Hamiltonian response to certain perturbations. The coupling mechanism acts towards a global synchronisation, i.e. to a coherent response of the excited species to any further perturbation. It is proven that the synchronisation is a scale-free process that has universal properties, e.g. it is insensitive to the chemical identity of the reacting species and to the particularities of the surface reaction. Its hallmark is that the global adsorption rate exhibits permanent temporal variations whose amplitude is independent of the system size. The presence of these fluctuations fundamentally changes the temporal behavior of the system, namely it becomes pulse-like both on the quantum and the macro-level. The pulse-like behavior gives rise to a persistent continuous band at the quantum spectra whose major properties are: (I) it does not correspond to any real radiation; (ii) its presence is insensitive to the particularities of the system and the incident radiation; (iii) its shape and the infrared edge are typical for the -type noise. These properties give rise to its name: alias -type noise.

cond-mat.stat-mech↗

arXiv2003

Common and different features between the behavior of the chaotic dynamical systems and the 1/f^alpha(f) noise

The major goal of the present paper is to find out the manifestation of the boundedness of fluctuations. Two different subjects are considered: (i) an ergodic Markovian process associated with a new type of large scaled fluctuations at spatially homogeneous reaction systems; (ii) simulated dynamical systems that posses strange attractors. Their common property is that the fluctuations are bounded. It is found out that the mathematical description of the stochasity at both types of systems is identical. Then, it is to be expected that it exhibits certain common features whose onset is the stochasticity, namely: (i) The power spectrum of a time series of length $T$ comprises a strictly decreasing band that uniformly fits the shape $1/f^α(f)$ where $α(1/T)=1$ and $α(f)$ strictly increases to the value $α(\inf)=p$ ($p>2$) as $f$ approaches infinity. Practically, at low frequencies this shape is $1/f$-like with high accuracy because the deviations of the non-constant exponent $α(f)$ from 1 are very small and become even smaller as the frequency tends to 1/T. The greatest advantage of the shape $1/α(f)$ is that it ensures a finite variance of the fluctuations. (ii) It is found out that the structure of a physical and strange attractor is identical and they are non-homogeneous. (iii) The Kolmogorov entropy is finite.

cond-mat.stat-mech↗

arXiv2004

Application of Thermodynamics to the Reduction of Data Generated by a Non-Standard System

In traditional thermodynamical and statistical-mechanical approaches one has (some) detailed knowledge of the principles governing the microdynamics of a system. However in many instances we may not have a Hamiltonian or good information about the degrees of freedom (or, in a roughly complementary sense, the redundancies in observed data), but on the other hand we have access to a preponderance of raw data. The development outlined here is an attempt to apply the principles of thermodynamics and statistical mechanics generically to such cases.

cond-mat.stat-mech↗

arXiv2005

A Computational Study of Rotating Spiral Waves and Spatio-Temporal Transient Chaos in a Deterministic Three-Level Active System

Spatio-temporal dynamics of a deterministic three-level cellular automaton (TLCA) of Zykov-Mikhailov type (Sov. Phys. - Dokl., 1986, Vol.31, No.1, P.51) is studied numerically. Evolution of spatial structures is investigated both for the original Zykov-Mikhailov model (which is applicable to, for example, Belousov-Zhabotinskii chemical reactions) and for proposed by us TLCA, which is a generalization of Zykov-Mikhailov model for the case of two-channel diffusion. Such the TLCA is a minimal model for an excitable medium of microwave phonon laser, called phaser (D. N. Makovetskii, Tech. Phys., 2004, Vol.49, No.2, P.224; cond-mat/0402640). The most interesting observed forms of TLCA dynamics are as follows: (a) spatio-temporal transient chaos in form of highly bottlenecked collective evolution of excitations by rotating spiral waves (RSW) with variable topological charges; (b) competition of left-handed and right-handed RSW with unexpected features, including self-induced alteration of integral effective topological charge; (c) transient chimera states, i.e. coexistence of regular and chaotic domains in TLCA patterns; (d) branching of TLCA states with different symmetry which may lead to full restoring of symmetry of imperfect starting pattern. Phenomena (a) and (c) are directly related to phaser dynamics features observed earlier in real experiments at liquid helium temperatures on corundum crystals doped by iron-group ions. ACM: F.1.1, I.6, J.2; PACS:05.65.+b, 07.05.Tp, 82.20.Wt

cond-mat.other↗

arXiv2004

The failure of the master equation for the reactive systems

Two crucial for the breakdown of the master equation arguments are put forward. The first one is related to the violence of a fundamental requirement to the notion of state (thermodynamical) variable, namely: a state variable is defined provided it is insensitive to the particularities of the spatio-temporal configurations upon which the averaging over the dynamical variables proceeds. The second one is related to a ubiquitous divergence of the scattering length in the low-energy limit. In turn, it makes the rates of all the elementary processes divergent as well. Though radically novel viewpoints to the low-energy limit and to the evolution ensure the boundedness of the rates and hold the notion of a state variable available, the master equation remains inappropriate.

cond-mat.stat-mech↗