Regular Series

Vol. 26 (1995), No. 6, pp. 977 – 1189

Heuristic versus Statistical Physics Approach to Optimization Problems

Acta Phys. Pol. B 26, 977 (1995)

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Optimization is a crucial ingredient of many calculation schemes in science and engineering. In this paper we assess several classes of methods: heuristic algorithms, methods directly relying on statistical physics such as the mean-field method and simulated annealing; and Hopfield-type neural networks and genetic algorithms partly related to statistical physics. We perform the analysis for three types of problems: (1) the Travelling Salesman Problem, (2) vector quantization, and (3) traffic control problem in multistage interconnection network. In general, heuristic algorithms perform better (except for genetic algorithms) and much faster but have to be specific for every problem. The key to improving the performance could be to include heuristic features into general purpose statistical physics methods.

Self Organized Criticality — Analytical Calculations and Open Problems

Acta Phys. Pol. B 26, 997 (1995)

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Some analytical calculations and results concerning self organized critical state in the sandpile-like cellular automata defined on the Bethe and square lattices are showed. The possibility of achieving a self organized critical state in nonconservative model systems is discussed.

Why the Mean Field Approach Fails in Case of Deterministic Homogeneous Dynamics on Square Lattice of Spins?

Acta Phys. Pol. B 26, 1009 (1995)

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The process of adjustment of local microdynamics to the spin relationships occurring from the lattice’s architecture of the system in consideration, is of the main interest of the note. Qualitative arguments are formulated to make this process responsible for the violation of the mean field theory approach.

Description of Microdomain Growth in Competitive 3D-Agglomerates

Acta Phys. Pol. B 26, 1021 (1995)

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A model of microdomain growth in three-dimensional systems like metals or ceramics is adapted to describe the growth kinetics and structure formation in competitive mass exchanging systems like biomembranes, liquid crystalline materials or polymers. The theory proposed assumes that the material in question can be partitioned into pieces (microdomains, clusters, grains) and concerns with modelling of the growth process in a time-dependent regime (i.e., when the so-called long tail kinetics is introduced). As a result, power and logarithmic laws of the average radius of the growing domain against time are obtained and some other probabilistic characteristics of the process are analyzed. An extension to disruption or defect processes in biosystems is presented. The approach developed can serve to elucidate some experimental results got e.g. for multilamellar lipid bilayers which till now are exclusively interpreted in terms of the Kolmogorov–Avrami model.

Reaction-Diffusion of Foreign Gas — Kinetic Theory Approach

Acta Phys. Pol. B 26, 1031 (1995)

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Diffusion and reaction of a foreign gas in a carrier gas is studied in a hydrodynamic regime by means of the Boltzmann–Lorentz equation. It is assumed that the reaction is relatively slow and can be treated as a perturbation. The hydrodynamic regime of the kinetic equation is derived with the use of the Resibois perturbative method. The diffusion coefficient and reaction rate constant are calculated in the third order approximation. The coefficients contain nonequilibrium corrections resulting from the deformation of the distribution function by chemical reaction. The nonequilibrium effect of thermally activated chemical reaction is calculated for models of reactive hard spheres. This influence can be significant if molecules of the foreign gas are much lighter than those of the carrier gas.

Lyapunov Instability of Squeezing (An Approach via Quantum Lyapunov Exponents)

Acta Phys. Pol. B 26, 1047 (1995)

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It is shown that the known instability of quadrature components of electric field during squeezing corresponds to two kinds of Lyapunov instability. One of them represents an instability of dynamics of averages and the other reflects a fundamental instability of the operator valued trajectories. The latter instability can be characterized by means of quantum characteristic exponents. The main result of the paper is the derivation of the correct Lyapunov exponent at the level of the Heisenberg picture. This shows that the quantum exponents correctly characterize properties of unstable dynamics of quantum observables.

Pattern Entropy a Tool for Nonlinear Dynamical Analysis of a Biological Nonstationary System: The Human Heart

Acta Phys. Pol. B 26, 1055 (1995)

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Tools for a nonlinear analysis of the dynamics of the rhythm of the human heart are discussed. Three-dimensional images in the phase space are formed by means of the Takens trajectory reconstruction method of 24-h sequences of time intervals between heart beats (RR intervals). Best projections of these images are sought and a surprising high symmetry is found for some types of pathology. The effects of filtering of arrhythmia on the symmetry is demonstrated. Images of RR intervals are also made in a time window of 100–400 beats and examples of such images preceding cardiac death are given. A new quantitative tool for the analysis of the local time degree of ordering of RR sequences — pattern entropy — is briefly discussed.

Introduction to Wetting Phenomena

Acta Phys. Pol. B 26, 1065 (1995)

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In these lectures the held of wetting phenomena is introduced from the point of view of statistical physics. The phase transition from partial to complete wetting is discussed and examples of relevant experiments in binary liquid mixtures are given. Cahn’s concept of critical-point wetting is examined in detail. Finally, a connection is drawn between wetting near bulk criticality and the universality classes of surface critical phenomena.

Mechanism of Long-Period Structural Modulations in A\(^{\prime }\)A\(^{\prime \prime }\)BX\(_4\) Compounds

Acta Phys. Pol. B 26, 1101 (1995)

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A simple lattice model with two Ising spins is proved to explain practically all structural phase transitions observed in almost 40 different A\(^{\prime }\)A\(^{\prime \prime }\)BX\(_4\) compounds. Ising variables describe four discrete orientational states of BX\(_4\) tetrahedra. Symmetrical nearest-neighbour interactions between spins stabilize crystallographic structures with up to four formula units per elementary cell. Longer-period, as well as incommensurate modulations of the order parameter, both along the hexagonal axis and in the perpendicular plane, originate either from symmetrical next-nearest-neighbour or, competitively, antisymmetrical nearest-neighbour interactions.

Non-Markovian Dichotomic Noises

Acta Phys. Pol. B 26, 1131 (1995)

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Two kinds of stochastic processes are discussed: explicitly non-markovian dichotomic noise with exponential damping of the memory, and implicitly non-markovian composite noise being a (linear and/or nonlinear) combination of several independent markovian dichotomic noises. The description of stochastic flows driven by such noises is given. To illustrate how the non-markovianity changes the behavior of the driven process, the evolution in time of the probability density \(P(x,t)\) describing the flow \(\dot X(t) = \xi (t)\) (the random telegraph process) driven by the non-markovian process \(\xi (t)\) is calculated and compared with that driven by markovian \(\xi (t)\). Among others, in the non-markovian case oscillations in \(P(x,t)\) are found, and the possibility of additional noise-induced transitions is indicated.

Quantum Transport in the 2D Electron Gas

Acta Phys. Pol. B 26, 1159 (1995)

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A quick introduction is provided to basic elements of quantum transport in a mesoscopic two-dimensional electron gas (2DEG). The introduction is designed for an audience with a background mostly in statistical physics.

Molecular Dynamics Simulations of a Propagating Chemical Wave Front

Acta Phys. Pol. B 26, 1177 (1995)

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The propagation of a chemical wave front in a nonhomogeneous system with a model reaction \(A + B \to A + A\) is simulated using periodically expanded molecular dynamics technique for reactive hard spheres. It is shown that for fast reactions the speed of a front does not depend on the rate constant as the standard, parabolic reaction-diffusion equation predicts and its value scaled by the square root of the rate constant \(k\) is an increasing function of \(k\). This phenomenon may be explained on the basis of extended irreversible thermodynamics if separated equations for the concentration of \(A\) and for the associated diffusive flow of \(A\) are considered.


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