Table of contents

Complex systems represent one of the richest and more fascinating fields of current scientific research. The reason behind this is the important role that the properties of complex systems and materials play in a variety of different but overlapping areas in physics, chemistry, biology, mathematics, and social sciences, like medicine and economy. Such unusually broad research field is, therefore, of primary interest nowadays in pure science and technology. The role of statistical physics in this new field of complex systems has been present since its onset and it has been accelerating recently. Methods developed for studying ordering phenomena in simple systems have been generalized for application to more complex forms of matter (polymers, biological macromolecules, glasses, etc) and complex processes (e.g. chaos, turbulence, economy, jamming, biological processes). In particular, many different phenomena (considered in the past to belong to separate research fields) have now a common description. Pillars of such a description are the concepts of scaling and universality. The International Conference on `Scaling Concepts and Complex Systems' (a satellite meeting of STATPHYS21) was devoted to give an overview on recent developments around these two concepts. The Conference took place in Merida, Yucatan, Mexico, in July 9-14 2001. The meeting was held in the Gordon Conference style and was attended by about 100 scientists, it covered a large variety of theoretical and experimental research topics of current interest in complex systems and materials. The meeting consisted of a total number of about 40 invited and contributed talks and a poster session. The topics covered included: scaling behaviour, supra-molecular systems, aggregation, aggregation kinetics, growth mechanisms, disordered systems, soft condensed matter (polymers, biological polymers, bio-colloids, gels, colloids, membranes and interfacial phenomena), granular matter, phase separation and out-of-equilibrium dynamics, non-linear dynamics, chaos, turbulence and chaotic dynamics. The present issue contains a substantial number of the invited and contributed talks presented at the meeting. We made an effort to arrange these papers with an order similar to that of presentation during the meeting. It is our pleasure to thank the scientific committee, all the speakers, the session chairs and all participants who contributed to the success of the conference. We are grateful to the Bonino-Pulejo Foundation (Messina-Italy), and to the President On. Nino Calarco, for the Patronage and the enthusiastic support. Our thanks goes also for the Messina University, the INFM (Istituto Nazionale per la Fisica della Materia, Italy), the Consejo Nacional de Ciencia y Tecnología (CONACyT, Mexico) and the Universidad Nacional Autonoma de Mexico (UNAM). The Conference was sponsored by the INFM-Sec.C, CONACyT, UNAM, the Bonino-Pulejo Foundation which contributed financial support to participants and to the publication of the present issue. We are grateful to them for the support. Last, but not the least, we express our warmest gratitude to all the members of the local organizing committee for their assistance and for the work spent in organizing this meeting and especially to Professor~Alberto Robledo for his valuable advice.

This paper discusses some of the similarities between work being done by economists and by computational physicists seeking to contribute to economics. We also mention some of the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge. In particular, we review two such new results. Specifically, we discuss the two newly discovered scaling results that appear to be `universal', in the sense that they hold for widely different economies as well as for different time periods: (i) the fluctuation of price changes of any stock market is characterized by a probability density function, which is a simple power law with exponent -4 extending over 10<sup>2</sup> standard deviations (a factor of 10<sup>8</sup> on the y-axis); this result is analogous to the Gutenberg-Richter power law describing the histogram of earthquakes of a given strength; (ii) for a wide range of economic organizations, the histogram that shows how size of organization is inversely correlated to fluctuations in size with an exponent ≈0.2. Neither of these two new empirical laws has a firm theoretical foundation. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behaviour of the response function at the critical point (zero magnetic field) leads to large fluctuations. We discuss a curious `symmetry breaking' for values of Σ above a certain threshold value Σ_c; here Σ is defined to be the local first moment of the probability distribution of demand Ω - the difference between the number of shares traded in buyer-initiated and seller-initiated trades. This feature is qualitatively identical to the behaviour of the probability density of the magnetization for fixed values of the inverse temperature.

We propose a simple statistical mechanics model for the study of the dynamics of gelling systems. It is based on percolation and bond-fluctuation dynamics for the bond vectors: we study the critical viscoelastic properties and the relaxation patterns in the case of irreversible gelation and the results obtained are discussed by means of scaling arguments. By introducing the idea of a finite lifetime for the bonds and by simply tuning this timescale, the model can be made to present very different dynamics and relaxation patterns corresponding to different gelling phenomenology.

The evolution and the statistical properties of an infinite gravitating system represent an interesting and widely investigated subject of research. In cosmology, the standard approach is based on equations of hydrodynamics. In this paper, we analyse the problem from a different perspective, which is usually neglected. We focus our attention on the fact that at small scale the distribution is point-like, or granular, and not fluid-like. The basic result is that the discrete nature of the system is a fundamental ingredient in understanding its evolution. The initial configuration is a Poisson distribution in which the distribution of forces is governed by the Holtsmark function. Computer simulations show that the structure formation corresponds to the shift of the granularity from small to large scales. We also present a simple cellular automaton model that reproduces this phenomenon.

The propagation of elastic perturbations in magneto-rheological suspensions is studied theoretically and experimentally. Under the application of a magnetic field, these systems acquire a fibrillar fractal structure formed by clusters. In systems in that condition, two low-frequency sound propagation modes have been observed. In both of them, the speed of sound depends on the intensity of the applied field. We discuss the statistical fractal properties of the cluster structure and, on this basis, we calculate the speed of sound for both of the low-frequency modes. This theoretical approach provides a good quantitative agreement with the experimental results.

We investigate the decay dynamics of the ballistic annihilation process, where particles following ballistic trajectories annihilate at each collision. In the framework of a Boltzmann equation, a Gaussian assumption for the velocity distribution is used to obtain analytical predictions for the decay exponents of the density and kinetic energy. In dimensions 1 and 2, these predictions are shown to be in good agreement with the complementary results of Monte Carlo and molecular dynamics simulations.

We study numerically and analytically a simple off-lattice model of scalar harmonic vibrations by means of Euclidean random matrix theory. Since the spectrum of this model shares the most puzzling spectral features with the high-frequency domain of glasses (non-Rayleigh broadening of the Brillouin peak, boson peak and secondary peak), Euclidean random matrix theory provides a single and fairly simple theoretical framework for their explanation.

Computer simulation investigations of two model fluids interacting through short-range forces are presented. In one case, the chosen potential parameters make the model suitable to represent C₆₀ fullerenes. For such a system, Monte Carlo calculations of the free energy are performed in order to determine the solid-liquid coexistence line and the whole phase diagram. In the other case, the potential is adapted to model the interaction between globular proteins in aqueous solutions, by obtaining a system whose phase diagram is known to have only metastable liquid-vapour equilibrium. We report on a previous study of such a fluid, concerning extensive molecular dynamics simulations of the crystallization process, and discuss the related results in the present context. The peculiar features of the phase behaviour of the two model systems, as well as their sensitive dependence on the potential properties, are also documented.

The stability of a one-component model system interacting through an isotropic potential with an attractive part and a softened core is investigated through integral equations and molecular dynamics simulation. The `penetrability' of the soft core makes it possible for the system to pass from an expanded liquid structure at intermediate densities to a more compact one at high densities.

Water is characterized by a density anomaly whose origin is a matter of debate. Theoretical works have shown that two of the proposed explanations, the second-critical-point hypothesis and the singularity-free scenario, have the same microscopic origin, but arise from different choices of parameters, such as the hydrogen bond strength or geometry. We consider a Hamiltonian model proposed by Sastry et al that supports the singularity-free scenario and was solved in an approximation where the intra-molecular interactions are neglected. We show that, by including these interactions, the second critical point is recovered, elucidating the differences in the mechanisms at the origin of the two interpretations.

We describe the modifications undergone by phase transitions associated with a planar surface when the system is confined by the presence of a second surface parallel to the first. The inhomogeneous states are studied within the Landau density functional theory, which imparts scaling properties to the order parameter profiles. We distinguish between two situations, identical and symmetrically opposed surface fields, and provide three illustrations: (i) a liquid crystal at the nematic-isotropic transition, (ii) a racemic mixture of enantiomeric species and (iii) a ternary molecular fluid mixture close to a consolute point.

We discuss the ideal glass transition for two types of potential model of attractive colloidal systems, i.e. the square-well system and the Yukawa hard-sphere fluid. We use the framework of the ideal mode-coupling theory and we mostly focus our attention on the nature of the singularities predicted by the theory. We also study the phenomena that arise by varying the range of the attraction, since this parameter has been identified as one of the key parameters in colloidal systems.

The numerical turbulence experiments conducted by Gotoh et al are analysed with high precision with the help of formulae for the scaling exponents of the velocity structure function and for the probability density function of the velocity fluctuations. These formulae are derived by the present authors, with the multifractal aspect based on statistics constructed using generalized measures of entropy, i.e., the extensive Rényi entropy or the non-extensive Tsallis entropy. It is shown, explicitly, that there exist two scaling regions, i.e., the upper scaling region with larger separations which may correspond to the scaling range observed by Gotoh et al and the lower scaling region with smaller separations which is a new scaling region, extracted for the first time by the present systematic analysis. These scaling regions are divided by a definite length approximately of the order of the Taylor microscale, which may correspond to the crossover length proposed by Gotoh et al as the low end of the scaling range (i.e., the upper scaling region).

We investigate the role of noise in the nonlinear relaxation of two ecosystems described by generalized Lotka-Volterra equations in the presence of multiplicative noise. Specifically we study two cases: (i) an ecosystem with two interacting species in the presence of periodic driving; (ii) an ecosystem with a great number of interacting species with random interaction matrix. We analyse the interplay between noise and periodic modulation for case (i) and the role of the noise in the transient dynamics of the ecosystem in the presence of an absorbing barrier in case (ii). We find that the presence of noise is responsible for the generation of temporal oscillations and for the appearance of spatial patterns in the first case. In the other case we obtain the asymptotic behaviour of the time average of the ith population and discuss the effect of the noise on the probability distributions of the population and of the local field.

We study a model that consists of two identical unidirectionally coupled one-dimensional arrays of chaotic phase oscillators. The time series (TS) of the distance between the arrays is analysed. The probability distribution functions (PDFs) of these distances typically display tails with power-law dependence. The autocorrelation function and the PDF of the laminar phases of these TS depend strongly on the stroboscopic section.

Often the current mode coupling theory (MCT) of glass transitions is compared with mean field theories. We explore this possible correspondence. After showing a simple-minded derivation of MCT with some difficulties we give a concise account of our toy model developed to gain more insight into MCT. We then reduce this toy model by adiabatically eliminating rapidly varying velocity-like variables to obtain a Fokker-Planck equation for the slowly varying density-like variables where the diffusion matrix can be singular. This gives room for non-ergodic stationary solutions of the above equation.

Detrended fluctuation analysis is used to test the performance of global climate models. We study the temperature data simulated by seven leading models for the greenhouse gas forcing only (GGFO) scenario and test their ability to reproduce the universal scaling (persistence) law found in the real records for four sites on the globe: (i) New York, (ii) Brookings, (iii) Tashkent and (iv) Saint Petersburg. We find that the models perform quite differently for the four sites and the data simulated by the models lack the universal persistence found in the observed data. We also compare the scaling behaviour of this scenario with that of the control run where the CO₂ concentration is kept constant. Surprisingly, from the scaling point of view, the simple control run performs better than the more sophisticated GGFO scenario. This comparison indicates that the variation of the greenhouse gases affects not only trends but also fluctuations.

Results of an extensive Monte Carlo (MC) study on both single and many semiflexible charged chains with excluded volume (EV) are summarized. The model employed has been tailored to mimic wormlike micelles in solution. Simulations have been performed at different ionic strengths of added salt, charge densities, chain lengths and volume fractions Φ, covering the dilute to concentrated regime. At infinite dilution the scattering functions can be fitted by the same fitting functions as for uncharged semiflexible chains with EV, provided that an electrostatic contribution b_el is added to the bare Kuhn length. The scaling of b_el is found to be more complex than the Odijk-Skolnick-Fixman predictions, and qualitatively compatible with more recent variational calculations. Universality in the scaling of the radius of gyration is found if all lengths are rescaled by the total Kuhn length. At finite concentrations, the simple model used is able to reproduce the structural peak in the scattering function S(q) observed in many experiments, as well as other properties of polyelectrolytes (PELs) in solution. Universal behaviour of the forward scattering S(0) is established after a rescaling of Φ. MC data are found to be in very good agreement with experimental scattering measurements with equilibrium PELs, which are giant wormlike micelles formed in mixtures of nonionic and ionic surfactants in dilute aqueous solution, with added salt.

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole-Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak-Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.

In this brief review I will discuss criticality in strongly correlated fluids. Unlike the molecules of simple fluids, which interact through short-ranged isotropic potentials, particles of strongly correlated fluids usually interact through long-ranged forces of Coulomb or dipolar form. While for simple fluids the mechanism of phase separation into liquid and gas was elucidated by van der Waals more than a century ago, the universality class of strongly correlated fluids, or in some cases even the existence of liquid-gas phase separation, remains uncertain.

The mechanical properties of polymer network structures depend mostly on the network rigidity. In very weakly cross-linked systems, entanglements also play a predominant role in the mechanical properties, especially hardness; and the contribution of entanglements competes with the contribution of the network rigidity. The entanglement and the network rigidity both depend on chain length and its distribution. In this work, the chain length distribution was affected by adding the cross-linker in late periods. The chain distribution was expressed by a mathematical equation using probabilistic concepts. Hardness was expressed in terms of a chain distribution probability function. It was modified to include the effect of the chain alignment for the polymers that have alignment. It was found that scaling relations existed between (i) the cross-linker concentration `C', (ii) the molecular weight of the prepolymer that gives the maximum hardness (MW<sub>max</sub> ), (iii) a parameter `k' that appears in the mathematical equation, and (iv) a parameter `g' that shows the contribution of chain alignment to hardness. Combined scalings with simple numerical powers were also found among these parameters. The combined scaling k×MW_max ~C^-1 holds true for both polymethyl methacrylate and polystyrene. It seems to be a universal relation.

We consider the question of whether a two-dimensional hard-disc fluid has a first-order transition from the liquid state to the solid state as in the three-dimensional melting-crystallization transition or whether one has two subsequent continuous transitions, from the liquid to the hexatic phase and then to the solid phase, as proposed by Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY). Monte Carlo (MC) simulations of the fluid that study the growth of the bond orientational correlation length, and of the crystal are discussed.

The emphasis is on a recent consistency test of the KTHNY renormalization group (RG) scenario, where MC simulations are used to estimate the bare elastic constants and dislocation fugacities in the solid, as a function of density, which then are used as starting values for the RG flow. This approach was validated earlier for the XY model as well.

We present the results of a computational model for colloidal aggregation that considers the Brownian motion, sedimentation and deposition experienced by the colloidal particles and clusters. Among our results, we find that for intermediate strengths of downward gravitational drift, the aggregation crosses over from diffusion-limited colloidal aggregation to another type of aggregation with a higher cluster fractal dimension, D_f. We also get a critical gelation concentration that is higher by several orders of magnitude than for the non-drifting case. We also found a speeding up followed by a slowing down of the aggregation rate and an algebraically decaying cluster size distribution. As the drift strength becomes much higher, the new fractal dimension is reduced due to the anisotropy of the clusters, becoming more elongated along the vertical direction. We finally present the scaling shown by the cluster size distribution for all the drift strengths studied.

We demonstrate how texture logs computed from multifractal analysis of dipmeter microresistivity signals can be used for characterizing geological formations (lithofacies) in combination with conventional well logs. In particular, we show that the generalized dimension D(1) (entropy dimension) can be considered as a heterogeneity index providing information on the spatial distribution (amounts of clustering) of heterogeneities in geological sediments. In addition, D(1) logs provide complementary information, compared to the conventional GR-log. This is illustrated by comparing core images of two intervals with similar GR responses and differing D(1) responses. Moreover, the method is equally valid if applied to the texture parameter D₁(1) computed by generalized multifractal analysis. In this way, we propose a tool for extracting textural information from well logging signals, which provides valuable information suitable for integration with data obtained from conventional well logging tools.

The avalanche properties of models that exhibit `self-organized criticality' (SOC) are still mostly awaiting theoretical explanations. A recent mapping (Alava M and Lauritsen K B 2001 Europhys. Lett. 53 569) of many sand-pile models to interface depinning is presented first, to provide an understanding of how to reach the SOC ensemble and the differences between this ensemble and the usual depinning scenario. In order to derive the SOC avalanche exponents from those of the depinning critical point, a geometric description of the quenched landscape, in which the `interface' measuring the integrated activity moves, is considered. It turns out that there are two main alternatives concerning the scaling properties of the SOC ensemble. These are outlined in one dimension in the light of scaling arguments and numerical simulations of a sand-pile model which is in the quenched Edwards-Wilkinson universality class.

The three-dimensional strongly screened vortex glass model is studied numerically using methods from combinatorial optimization. We focus on the effect of disorder strength on the ground state and find the existence of a disorder-driven normal-to-superconducting phase transition. The transition turns out to be a geometrical phase transition with percolating vortex loops in the ground state configuration. We determine the critical exponents and provide evidence for a new universality class of correlated percolation.

The state of a two-dimensional random resistor network, resulting from the simultaneous evolutions of two competing biased percolations, is studied in a wide range of bias values. Monte Carlo simulations show that when the external current I is below the threshold value for electrical breakdown, the network reaches a steady state with nonlinear current-voltage characteristics. The properties of this nonlinear regime are investigated as a function of different model parameters. A scaling relation is found between ⟨R⟩/⟨R⟩₀ and I/I₀, where ⟨R⟩ is the average resistance, ⟨R⟩₀ the linear regime resistance and I₀ the threshold value for the onset of nonlinearity. The scaling exponent is found to be independent of the model parameters. A similar scaling behaviour is also found for the relative variance of resistance fluctuations. These results compare well with resistance measurements in composite materials performed in the Joule regime up to breakdown.

It has recently been suggested that the property of isostaticity of the contact network of a frictionless polydisperse granular packing in the limit of low applied pressure is responsible for some of the anomalous static behaviour of packings. In this paper we discuss the fact that, on disordered isostatic networks, displacement-displacement and stress-stress static Green functions are described by coupled random multiplicative processes and thus have a truncated power-law distribution, with a cut-off that grows exponentially with distance. The expectation values of Green functions on these systems differ from observed averages by an exponentially large factor unless the number of samples over which averages are taken is exponentially large. Thus predicted averages will seldom be observed in experiments. If the external pressure is increased sufficiently, excess contacts are created, the packing becomes hyperstatic, and the above-mentioned anomalous properties disappear because Green functions now have a bounded distribution. Thus the low-pressure, isostatic, limit is a critical point where the Green function distribution becomes scale-free. This criticality is induced by multiplicative noise.

The transmission of forces through a disordered granular system is studied by means of a geometrical-topological approach that reduces the granular packing into a set of layers. This layered structure constitutes the skeleton through which the force chains set up. Given the granular packing, and the region where the force is applied, such a skeleton is uniquely defined. Within this framework, we write an equation for the transmission of the vertical forces that can be solved recursively layer by layer. We find that a special class of analytical solutions for this equation are Lévi-stable distributions. We discuss the link between criticality and fragility and we show how the disordered packing naturally induces the formation of force chains and arches. We point out that critical regimes, with power law distributions, are associated with the roughness of the topological layers, whereas fragility is associated with local changes in the force network induced by local granular rearrangements or by changes in the applied force. The results are compared with recent experimental observations in particulate matter and with computer simulations.

We review the role of dynamical effects and the important interplay of off-equilibrium and equilibrium phenomena in the physics of magnetic and transport properties of vortex matter in type-II superconductors. More specifically, we discuss the unifying framework of these phenomena emerging in the context of a recently introduced model for vortices and their important deep relations with other glass formers.

We write equations of motion for density variables that are equivalent to Newton's equations. We then propose a set of trial equations parametrized by two unknown functions to describe the exact equations. These are chosen to best fit the exact Newtonian equations. Following established ideas, we choose to separate these trial functions into a set representing integrable motions of density waves, and a set containing all effects of non-integrability. The density waves are found to have the dispersion of sound waves, and this ensures that the interactions between the independent waves are minimized. Furthermore, it transpires that the static structure factor is fixed by this minimum condition to be the solution of the Yvon-Born-Green equation. The residual interactions between density waves are explicitly isolated in their Newtonian representation and expanded by choosing the dominant objects in the phase space of the system, that can be represented by a dissipative term with memory and a random noise. This provides a mapping between deterministic and stochastic dynamics. Imposing the fluctuation-dissipation theorem allows us to calculate the memory kernel. We write exactly the expression for it, following two different routes, i.e. using explicitly Newton's equations, or instead, their implicit form, that must be projected onto density pairs, as in the development of the well established mode coupling theory. We compare these two ways of proceeding, showing the necessity to enforce a new equation of constraint for the two schemes to be consistent. Thus, while in the first `Newtonian' representation a simple Gaussian approximation for the random process leads easily to the mean spherical approximation for the statics and to MCT for the dynamics of the system, in the second case higher levels of approximation are required to have a fully consistent theory.

Starting from the atomic coordinates of proteins, by modern surface calculation programs the exact surface topography of proteins can be visualized in terms of dot surface points. In combination with tricky hydration algorithms, based on the hydration numbers of individual amino acid (AA) residues and appropriate selection criteria, this may be used for advanced modelling studies including a fine tuning of the input parameters for such computer simulations. `Bead modelling' can be used for both the individual AA residues and individual water molecules placed at preferred positions on the protein envelope. Problems of special concern are connected with the properties and the localization of the water molecules bound to the protein surface. Qualified assumptions regarding number, density/volume, position and behaviour of the water molecules at the protein-water interface are required.

Structural relaxations in AOT (sodium bis(2-ethylhexyl) sulphosuccinate) and lecithin reverse micelles are investigated by means of dielectric relaxation and conductivity measurements. The different behaviours exhibited by the two systems are interpreted in terms of the different kinds of interaction between the water and the surfactant molecules. In the case of lecithin, the application of an external electric field induces the establishing of some electrorheological structure. The temperature dependence of the observed electrorheological effects agrees with the hypothesis of a structural arrangement consisting in a percolated network of branched cylindrical micelles. The experimental results are compared with other literature data and discussed within the framework of the current theories.

The distribution of an electrolyte solution close to a charge macroion is investigated theoretically. A local density functional form based on one component plasma theory is proposed to incorporate the correlations between the microions into Poisson-Boltzmann theory. Our results are compared with those obtained by the mean-field approximation.

We study theoretically and experimentally the thermal convection in long tilted fractures filled with a porous material (porous layer) embedded in an impermeable solid and saturated with a fluid. The solid is subjected to a constant, vertical temperature gradient and has thermal conductivity larger than that of the saturated porous layer. We discuss different cases of interest in terms of the fracture aspect ratio and the fracture-to-solid conductivity ratio. Analytical expressions for the temperature and velocity profiles of the flow in the porous layer are worked out for low-Rayleigh-number flows.