Showing papers in "Physica A-statistical Mechanics and Its Applications in 1995"

The thermodynamics of fractals revisited with wavelets

Abstract: The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions D q and the f ( α ) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.

Non-extensive statistical mechanics and generalized Fokker-Planck equation

Abstract: A non-linear, generalized Fokker-Planck (GFP) equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by Tsallis in his generalization of Statistical Mechanics. In the case of a constant or linearly varying drift, the time dependent solutions of the GFP equation are seen to obey a suitable form of the celebrated H-theorem. The temporal changes of Tsallis' entropies are seen to be given in terms of the Fisher's information measure. Particular time-dependent solutions of the GFP equation of the maximum (Tsallis') entropy form are obtained.

A thermodynamic theory of supercooled liquids

Abstract: A novel thermodynamic theory of the properties of supercooled liquids as they get glassy is presented. It is based on the postulated existence of a narrowly avoided thermodynamic phase transition at a temperature T ∗ ⩾ T m , where Tm is the melting point, and the “avoidance” is due to geometric frustration. We show that as a consequence two large emergent length scales develop at temperatures less than T ∗ , and we also show that this picture is consistent with appropriate statistical mechanical models. A theoretical expression is obtained which permits collapse of the viscosity versus temperature of all known glass-formers onto a single master-curve.

Depletion force in colloidal systems

Abstract: The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. We calculate the interaction between large spheres (of radius R) in a dilute solution of mutually avoiding small spheres (of diameter σ ⪡ R and volume fraction φ), to third order in φ. In addition to the well-known attractive force for 0 < h < σ, we find a repulsive barrier at larger separations, and beyond that a secondary minimum. Except for unusually large size ratios (perhaps abetted by relatively high density φ), these features of the interaction potential are too small, compared to kBT, for kinetic stabilization (arising from the barrier) or flocculation into the secondary minimum, to be widespread, although such effects are possible in principle. For feasible size ratios, the same features should have observable consequences for the radial distribution function of the large spheres. Such effects can be viewed as precursors, at low density, of liquidlike structuring (solvation forces) expected at higher φ. Our third order calculation gives satisfactory agreement with a recent computer simulation at moderate density and size ratio (2R/σ = 10; φ = π/15).

Measuring correlations in symbol sequences

Abstract: The paper is devoted to relations between correlation functions and mutual information. It is shown that, in sequences over an alphabet of λ symbols, statistical dependences are measured by (λ − 1)2 independent parameters. However, not all of them can be determined by autocorrelation functions. Appropriate sets of correlation functions (including crosscorrelations) are introduced, which allow the detection of all dependences. The results are exemplified for binary, ternary, and quaternary symbol sequences. As an application, it is discussed that a nonuniform codon usage in protein-coding DNA sequences introduces periodic correlations even at distances in the order of 1000 base pairs.

Non-extensive thermostatistics: brief review and comments

Abstract: We briefly review, with regard to physical applications, the present status of the recently introduced non-extensive thermostatistics characterized by the entropic index q (q = 1 corresponds to standard, extensive, Boltzmann-Gibbs thermostatistics). In addition to that, we comment on (i) how meta-equilibrium-equilibrium crossovers may occur as a function of time, (ii) spin glasses and the replica trick, and (iii) the theory of perceptions.

Simulating deformations of granular solids under shear

Abstract: We present MD simulations of shear cells of a densily packed array of polygonal grains in two dimensions. We study the dependence of the shear force on the shear velocity. Generally shear hardening is observed. The dependence of the dilatancy on the shear velocity is measured. We also apply the method to simulate two tectonic plates moving under shear. We find a Gutenberg-Richter-like law with a b-value of (1.02 ± 0.06) in good agreement with geophysical measurements. Shear bands of widths dependent on the confining pressure and the shear velocity are observed.

Damage spreading and critical exponents for “model A” Ising dynamics

Abstract: Using damage spreading and heat bath dynamics, we study the Ising model in 2 and 3 dimensions with non-conservative dynamics. Our algorithm differs in some important points from previous ones, which makes it rather efficient. We give estimates for the exponent z which seem to be the most precise published so far (2.172 ± 0.006 for d = 2, 2.032 ± 0.004 for d = 3). We also give precise estimates of the exponent θ′ introduced by Janssen et al. (Z. Phys. B 73 (1989) 539) and of analogous but in principle independent exponents. We find surprisingly that some of the latter agree with θ′, and give an explanation for this.

Statistical properties of DNA sequences

Abstract: We review evidence supporting the idea that the DNA sequence in genes containing non-coding regions is correlated, and that the correlation is remarkably long range--indeed, nucleotides thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene. We resolve the problem of the "non-stationarity" feature of the sequence of base pairs by applying a new algorithm called detrended fluctuation analysis (DFA). We address the claim of Voss that there is no difference in the statistical properties of coding and non-coding regions of DNA by systematically applying the DFA algorithm, as well as standard FFT analysis, to every DNA sequence (33301 coding and 29453 non-coding) in the entire GenBank database. Finally, we describe briefly some recent work showing that the non-coding sequences have certain statistical features in common with natural and artificial languages. Specifically, we adapt to DNA the Zipf approach to analyzing linguistic texts. These statistical properties of non-coding sequences support the possibility that non-coding regions of DNA may carry biological information.

Phase diagram of a model of self-organizing hierarchies

Abstract: We introduce a simple model of self-organizing hierarchies in animal societies which relies on a basic positive feedback mechanism reinforcing the ability of a given individual to win or lose in a hierarchical interaction, depending on how many times it won or lost in previous interactions. If a forgetting strength is included, which determines the rate at which events in the past are forgotten and no longer influence the force of an individual, subcritical or supercritical bifurcations in the formation of the hierarchical structure are observed as the density ϱ of individuals is varied. The nature of the transition is shown to depend on a parameter η, analogous to the inverse of a temperature, defining the amount of determinism in the outcomes of the fights. We therefore observe a dynamical tricritical point in the ϱ-η plane.

Equilibrium behaviour of quantum Ising spin glass

Abstract: A phenomenological theory of the ordered phase of short-range quantum Ising spin glass is developed in terms of droplet excitations, and presented in detail. These excitations have free energies that provene from an interplay between the classical excitation energies ϵL, with a broad distribution whose characteristic magnitude grows with length scale L as Lθ, and quantum tunneling rates ΓL, decreasing faster than exponentially with L. At temperature T = 0, the equal-time spatial correlations due to quantum fluctuations do not show the power-law decay that occurs for T > 0 due to thermal fluctuations, but ather an exponential decay of Ornstein-Zernicke type. At finite, but still very small T, there exists a crossover length scale L ∗ (T), set by ħΓ L∗(T)⋍ kBT, below which the droplets behave quantum-mechanically and above which the behaviour is essentially classical. As for the classical spin glass, only a small fraction of droplets are thermally active at low T; it is shown that many of the low-T static properties are dominated by the thermally active droplets at the crossover length L ∗ (T) . The uniform static linear susceptibility is found to diverge at T = 0 below the lower critical dimension, d l , and to be finite above d l . The static nonlinear susceptibility diverges in all dimensions, d. The zero temperature linear ac susceptibility χ(ω) is dominated by droplets at a length scale such that ω is of order the characteristic frequency of the quantum system. The behaviour near to the quantum critical point is discussed within a conventional scaling framework if it is approached at strictly zero temperature as well as from finite T. Implications of the existence of Griffiths singularities at the critical point and the disordered phase are pointed out: In the disordered phase, Griffiths singularities dominate the low-T specific heat and the long-time correlations.

On the theory of concentrated hard-sphere suspensions

Abstract: A systematic theory for the dynamics of hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is presented. A generalized diffusion equation is derived for concentrated suspensions. The volume fraction (φ) dependence of the short- and long-time self-diffusion coefficients are thus explored from a unifying point of view. The long-range hydrodynamic interactions due to the Oseen tensor are shown to play a crucial role in both coefficients, while the short-range hydrodynamic interactions just lead to corrections. The importance of the correlation effects between particles due to the long-range hydrodynamic interactions is also stressed. The nonlocal correlation effect is an important factor, leading to the behavior of the long-time self-diffusion coefficient (DSL) as DSL ∼ (1 − φ/φ0)2 near the volume fraction of φ0 = 0.5718. The direct interactions are also found to be drastically reduced by the short-range hydrodynamic interactions.

Symmetry constraint of MKdV equations by binary nonlinearization

Abstract: A symmetry constraint for MKdV integrable hierarchy is presented by binary nonlinearization. The spatial part and the temporal parts of the Lax pairs and the adjoint Lax pairs of MKdV equations are all constrained as finite dimensional Liouville integrable Hamiltonian systems, whose integrals of motion are explicitly given. In terms of the proposed symmetry constraint, MKdV equations are decomposed into two finite-dimensional Liouville integrable constrained systems and thus a kind of separation of variables for MKdV equations is established.

Long-range correlations between letters and sentences in texts

Abstract: We mapped three long texts to random walks and calculated several correlation measures as Holder exponents, higher-order cumulants and power spectra. By means of computer experiments we have found that shuffling on/or below the sentence level generates strings showing no anomalous diffusion, no higher-order cumulants and no power spectra with 1/fδ-shape. In this way we have shown that the long correlations reflected in these measures are not based on correlations inside sentences but reflect the large-scale structure beyond the sentence level.

Pattern formation during the electrodeposition of a silver-antimony alloy

Abstract: It is described how during the electrodeposition of a silver-antimony alloy patterns can develop in the deposit. The patterns occur as moving bands, target patterns, and spiral waves, and the characterized using various types of microscopy. Forthe patterns to develop, the hydrodynamic conditions close to the electrode appear to be of prime importance. The wave characteristics are discussed in relation with recent studies on patterns on (single crystal) catalyst surface and convection-induced chemical instabilities.

Irreducible memory function for dissipative stochastic systems with detailed balance

Abstract: We show quite generally that the memory function for purely dissipative stochastic systems with detailed balance can be reduced further and can be expressed in terms of the irreducible memory function. This generalizes the similar finding for the Smoluchowski equation. The irreducible memory function is intimately connected with the lifetime renormalization, and is shown to provide a useful starting point for introducing approximations such as the mode coupling approximation.

A general isomorphism approach to thermodynamic and transport properties of binary fluid mixtures near critical points

Abstract: A general isomorphism approach to critical phenomena in binary fluid mixtures that may exhibit complex critical-line behavior is developed by relating the two relevant scaling fields to linear combinations of three physical field variables. These physical field variables are related to the temperature and chemical potentials of the two components. The proposed approach includes crossover from vapor-liquid critical behavior to liquid-liquid critical behavior and incorporates also the critical behavior near other special points on critical loci. It is shown that the key factor which determines the apparent behavior of the thermodynamic and transport properties of near-critical mixtures is the shape of the critical locus. The number of system-dependent coefficients that determine the asymptotic critical behavior is elucidated. The choice of zero-points of entropy and energy in binary mixtures is also discussed. The approach provides a powerful tool for predicting thermodynamic and transport properties of fluid mixtures in the critical region.

A Statistical Approach to Vehicular Traffic

Abstract: A two-dimensional cellular automaton is introduced to model the flow and jamming of vehicular traffic in cities. Each site of the automaton represents a crossing where a finite number of cars can wait approaching the crossing from each of the four directions. The flow of cars obeys realistic traffic rules. We investigate the dependence of the average velocity of cars on the global traffic density. At a critical threshold for the density the average velocity reduces drastically caused by jamming. For the low-density regime we provide analytical results which agree with the numerical results.

Self-diffusion and ‘visited’ surface in the droplet condensation problem (breath figures)

Abstract: We study experimentally and theoretically new aspects of condensation, growth and coalescence of droplets on a substrate. We address in particular the dynamics of a ‘marked’ droplet which undergoes coalescences with neighbouring droplets. We find that the number of coalescences of the droplet grows as log t (t is time) and that the traveled distance increases as 〈R(t)〉, the average radius of the droplets at time t. The fraction of the surface which was never covered by any droplet (an important quantity for applications, such as drug spreading or surface decontamination), decays as t−θ, showing that completion occurs only slowly. Heuristic arguments, accurate numerical simulations on simplified models (which neglect polydispersity) and an exact solution reported elsewhere [A. Bray, B. Derrida and C. Godreche, Europhys. Lett. 27 (1994) 175] strongly support these findings, and show that this power law decay is a generic feature, common to many different situations. Finally, the contour of the ensemble of ‘dry’ sites appears fractal with dimension d ƒ ⋍ 1.22 , an experimental result not reproduced by the simplified models.

Theoretical foundation of macroscopic traffic models

Abstract: On the basis of the elementary laws of individual driver behavior concerning the acceleration and interaction of vehicles a gas-kinetic traffic model is constructed. This yields theoretical relations for the fundamental diagram and the equilibrium variance-density relation, but above all it allows the derivation of macroscopic traffic equations. These build a hierarchy of non-closed equations which can be closed by different zeroth-order approximations. In this way one obtains the traffic equations of Lighthill and Whitham or those of Phillips. Alternatively one can derive Euler-like traffic equations which are proposed here since they include additional dynamical equations for the velocity variance. The new variance equation is able to describe the empirically observed increase of variance directly before a traffic jam develops.

Infinite-range Ising ferromagnet: Thermodynamic limit within Generalized Statistical Mechanics

Abstract: We first discuss, for a variety of similar systems, the physical need for departure from Boltzmann-Gibbs statistical mechanics and thermodynamics. Then, we numerically discuss the infinite-range spin-1/2 Ising ferromagnet within the recently generalized statistical mechanics (canonical ensemble). Through the specific heat, we exhibit (for the first time, as far as we know, for an interacting system) that the thermodynamic limit is well defined.

The distance between Zipf plots

Abstract: We study the Zipf plots describing the occurrence of different elements in a given group as a function of their rank. We define a “distance” between two Zipf plots characterizing the differences between the two groups. We apply the distance concept on groups of words contained in books. Our results suggest that the distance between books written by the same author is smaller than the distance between books written by different authors.

The super long-time decay of velocity fluctuations in a two-dimensional fluid

Abstract: We have calculated the velocity autocorrelation function for a tracer particle in a model two-dimensional fluid. The fluid was represented by a lattice Boltzmann equation with imposed fluctuations. By choosing a low Boltzmann diffusion coefficient for the tracer, the diverging contribution to the diffusion coefficient can be made to exceed the Boltzmann value even at short times. We were thus able to find evidence for the renormalized, or ‘super long-time’, decay of the VACF in a two-dimensional fluid. We find quantitative evidence for the 1/t√ln(t) decay predicted by theory.

Pattern selection and localized structures in reaction-diffusion systems

Abstract: We present some theoretical concepts that have been used in the study of chemical disspative structures together with a brief description of recent experimental work on Turing patterns and their interaction with travelling waves.

Phase diagram of a liquid crystal model: A computer simulation study

Abstract: Continuous potential models used in computer simulation of liquid crystals are reviewed, paying special attention to the anisotropic Gay-Berne model. This potential model, though phenomenological in nature, has a physical background and is able to reproduce mesogenic behaviour. Results from molecular dynamics simulations are presented here showing the different phases appearing in Gay-Berne fluids made up by molecules with axial ratio 3:1. The effect of the attractive forces in stabilizing the orientationally ordered phases is also studied by performing simulations for a WCA-type Gay-Berne fluid. The dynamics of the fluid is examined, especially the process in the vicinity of the isotropic-nematic transition region.

Damage spreading and critical exponents for ‘model A’ Ising dynamics

Abstract: Recent simulations by Grassberger gave very precise estimates of the dynamical critical exponent z via damage spreading techniques. We confirm these results using a slightly altered method and we examine larger systems up to 18161 2 and 695 2 giving z = 2.18 ± 0.02 and 2.04 ± 0.01, respectively.

Forces on the walls and stagnation zones in a hopper filled with granular material

Abstract: We measure the pressure and the shear forces acting on the walls of an outflowing hopper using Molecular Dynamics simulations. We find very strong fluctuations which for large opening angles follow a power spectrum but have white noise for smaller angles. We also calculate the shape of the stagnation zones that appear during funnel flow and compare it to the experimentally observed ones.

Spreading analysis and finite-size scaling study of the critical behavior of a forest fire model with immune trees

Abstract: In a forest fire model with immune trees (FFMIT) recently proposed by Drossel and Schwabl (Physica A 199 (1993) 183), the sites of the lattice are either empty, a tree or a burning tree, and at each time step the system is updated in parallel according to the following rules: (i) A burning tree becomes an empty site, (ii) trees grow with probability p from empty sites, and (iii) a green tree becomes a burning tree with probability (1-g) if at least one next neighbor is burning. Fixing an arbitrary grow probability (p) and starting with a small immunity (g), increments of g causes the fire density of the steady state to decrease until the fire becomes irreversibly extinguished at a certain critical point of coordinates {pc, gc}. The set of critical points defines a critical curve gc (p) which, in the L = ∞ limit, divides the {p, g}-plane in two regions: a steady state with fire fronts for g

Simulation of rotating drum experiments using non-circular particles

Abstract: We investigate the flow of granular material in a rotating cylinder numerically using molecular dynamics in two dimensions. The particles are described by a new model which allows to simulate geometrically complicated shaped grains. The results of the simulation agree significantly better with experiments than the results which are based on circular particles.

Viscous fingering in complex fluids

Abstract: Viscous fingers form when in a thin linear channel a fluid pushes a more viscous fluid. The instability of the interface results from a competition between viscous and capillary forces. We show here by acting on the viscosity or the surface tension by means of surfactants or polymers that the instability can be modified drastically. For the two different systems, unlike in the classical system, the width of the finger can go through a minimum and increases with increasing velocity before settling at a plateau value larger than half the channel width. A numerical resolution of the relevant hydrodynamic equations reveals that these large deviations from the classical result can be interpreted in terms of a velocity dependent dynamic interfacial tension for the surfactant system and viscosity for the polymer solution.