MODELING COMPLEX SYSTEMS: Sixth Granada Lectures on Computational Physics

Modeling active memory: Experiment, theory and simulation
View Description Hide DescriptionNeurophysiological experiments on cognitively performing primates are described to argue that strong evidence exists for localized, nonergodic (stimulus specific) attractor dynamics in the cortex. The specific phenomena are delay activity distributionsenhanced spikerate distributions resulting from training, which we associate with working memory. The anatomy of the relevant cortex region and the physiological characteristics of the participating elements (neural cells) are reviewed to provide a substrate for modeling the observed phenomena. Modeling is based on the properties of the integrateandfire neural element in presence of an input current of Gaussian distribution. Theory of stochastic processes provides an expression for the spike emission rate as a function of the mean and the variance of the current distribution. Meanfield theory is then based on the assumption that spike emission processes in different neurons in the network are independent, and hence the input current to a neuron is Gaussian. Consequently, the dynamics of the interacting network is reduced to the computation of the mean and the variance of the current received by a cell of a given population in terms of the constitutive parameters of the network and the emission rates of the neurons in the different populations. Within this logic we analyze the stationary states of an unstructured network, corresponding to spontaneous activity, and show that it can be stable only if locally the net input current of a neuron is inhibitory. This is then tested against simulations and it is found that agreement is excellent down to great detail. A confirmation of the independence hypothesis. On top of stable spontaneous activity, keeping all parameters fixed, training is described by (Hebbian) modification of synapses between neurons responsive to a stimulus and other neurons in the modulesynapses are potentiated between two excited neurons and depressed between an excited and a quiescent neuron. Meanfield theory is then extended to take into account the existence of functionally diverse neural populations, due to learning and to simulation. It then predicts that beyond a critical value of synaptic potentiation, ergodicity is broken and a multiplicity of stable attractors, representing the particular stimulus presented, apart from spontaneous activity that continues to be stable and is provoked by stimuli very different from the set used in training. Also these results agree well with simulations.

Complex adaptive systems and the evolution of reciprocation
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On the role of external constraints in a spatially extended evolutionary prisoner’s dilemma game
View Description Hide DescriptionWe study the emergency of mutual cooperation in evolutionary prisoner’s dilemma games when the players are located on a square lattice. The players can choose one of the three strategies: cooperation (C), defection (D) or “tit for tat” (T), and their total payoffs come from games with the nearest neighbors. During the random sequential updates the players adopt one of their neighboring strategies if the chosen neighbor has higher payoff. We compare the effect of two types of external constraints added to the Darwinian evolutionary processes. In both cases the strategy of a randomly chosen player is replaced with probability P by another strategy. In the first case, the strategy is replaced by a randomly chosen one among the two others, while in the second case the new strategy is always C. Using generalized meanfield approximations and Monte Carlo simulations the strategy concentrations are evaluated in the stationary state for different strength of external constraints characterized by the probability P.

A model for predation pressure in colonial birds
View Description Hide DescriptionDifferent explanations have been proposed for the existence of colonial breeding behavior in birds, but field studies offer no conclusive results. We analyze the interplay between learning habits and predation pressure by means of numerical simulations. Our analysis suggests that extremely simple learning processes and equally simplistic models of predation pressure lead to the formation of stable colonies.

A dynamical thermostat approach to financial asset price dynamics
View Description Hide DescriptionA dynamical price formation model for financial assets is presented. It aims to capture the essence of speculative trading where mispricings of assets are used to make profits. It is shown that together with the incorporation of the concept of risk aversion of agents the model is able to reproduce several key characteristics of financial price series. The approach is contrasted to the conventional view of price formation in financial economics.

Introduction to mathematical modeling of earthquakes
View Description Hide DescriptionWe first overview statistics and kinematics, necessities in modeling of earthquakes. In statistics, sizefrequency distributions of earthquakes and temporal changes of aftershock activities are main subjects. We pay attention not only to powerlaw behaviors but to nonpowerlaw behaviors as well. In particular, comparison of the two sizefrequency distributions, the one by GutenbergRichter and the other based on the characteristic earthquake scheme which assumes periodic generation of earthquakes similar in size, is important from a viewpoint of earthquake prediction. Kinematically, a framework is presented which treats earthquakes as generation of dislocations, discontinuities in the displacement over fault surfaces. As static models, we discuss percolation models on a tree and a twodimensional square lattice. Here the sizefrequency generally decays exponentially or stretched exponentially as earthquakes become large, and does algebraically (GutenbergRichter law) only at the critical point. Time dependent problem is discussed using cellular automaton models. One of the main concerns here is whether power laws in sizefrequency distribution are realized at stationary states. We observe this property, selforganized criticality, is shown only by models close the original sand pile model by Bak et al. Physical processes are included by using elements of blocks and springs. Power laws as well as nonpower laws are allowed as stationary sizefrequency distributions. In order to account for decay of aftershock activities, it is necessary to introduce some relaxation mechanisms. To take into full account the kinematics of earthquakes, the dislocation picture, we need to stack springs and blocks threedimensionally. A continuum version is presented to study a case of a subducting plate, where earthquakes occur following a characteristic earthquake scheme.

The fractal properties of internet
View Description Hide DescriptionIn this paper we show that the Internet web, from a user’s perspective, manifests robust scaling properties of the type where n is the size of the basin connected to a given point, P represents the density of probability of finding a basin of size n connected and is a characteristic universal exponent. The connection between users and providers are studied and modeled as branches of a world spanning tree. This scalefree structure is the result of the spontaneous growth of the web, but is not necessarily the optimal one for efficient transport. We introduce an appropriate figure of merit and suggest that a planning of few big links, acting as information highways, may noticeably increase the efficiency of the net without affecting its robustness.

Etching of random solids: Hardening dynamics and selforganized fractality
View Description Hide DescriptionWhen a finite volume of an etching solution comes in contact with a disordered solid, a complex dynamics of the solidsolution interface develops. Since only the weak parts are corroded, the solid surface hardens progressively. If the etchant is consumed in the chemical reaction, the corrosion dynamics slows down and stops spontaneously leaving a fractal solid surface, which reveals the latent percolation criticality hidden in any random system. Here we introduce and study, both analytically and numerically, a simple model for this phenomenon. In this way we obtain a detailed description of the process in terms of percolation theory. In particular we explain the mechanism of hardening of the surface and connect it to Gradient Percolation.

Generating powerlaw tails in probability distributions
View Description Hide DescriptionWe study how the presence of correlations in physical variables contributes to the form of probability distributions. We investigate a process with correlations in the variance generated by (i) a Gaussian or (ii) a truncated Lévy distribution. For both (i) and (ii), we find that due to the correlations in the variance, the process “dynamically” generates powerlaw tails in the distributions, whose exponents can be controlled through the way the correlations in the variance are introduced. For (ii), we find that the process can extend a truncated distribution beyond the truncation cutoff, which leads to a crossover between a Lévy stable power law and the present “dynamicallygenerated” power law. We show that the process can explain the crossover behavior recently observed in the stock index.

Sandpiles and absorbingstate phase transitions: Recent results and open problems
View Description Hide DescriptionWe review some recent results on the relations between sandpiles and a class of absorbing state phase transitions. We use the concept of fixed energy sandpiles (FES), in which external driving and dissipation are absent. FES are shown to exhibit an absorbing state transition with critical properties coinciding with those of the corresponding sandpile model. We propose a set of Langevin equations capturing the relevant features of this transition. These equations characterize the universality class of systems with an infinite number of absorbing states and a static conserved field coupled to the order parameter. Different models in this class are identified, and strong evidence is presented showing that the Manna sandpile, as well as some other stochastic sandpiles, belong in this universality class. Finally some open problems and questions are discussed.

Queuing in critical sandpile automata
View Description Hide DescriptionWe simulate queues of activity in a directed sandpile automation in dimensions by adding grains with rate r at the top row. Duration t of elementary jobs (avalanches) is given exactly by the “heavy tail” distribution for large t. In practice the durations are limited either by the system size or by dissipation at defects (concentration c). We study numerically and analytically tail behavior of the distributions of busy periods and energy dissipated in the queue. Depending on the relative ratio of driving and dissipation rates, the distribution of intermittent queue activities exhibit longrange correlations and multifractal scaling properties.

Blocking, persistence and scaling in the 2d Ising model
View Description Hide DescriptionThe nonequilibrium dynamics of the disordered 2d Ising model is studied numerically. Evidence is found for “blocking” in both the bonddiluted and the ±J models. For strong bonddilution the residual persistence decays exponentially to zero at large times. For the ±J model it is found to decay algebraically. Our results indicate that persistence is highly nonuniversal.

Monte Carlo study of the relaxation of the metastable phases of a mixed Ising ferrimagnet
View Description Hide DescriptionWe present preliminary results from a Monte Carlo study of the metastable decay of a mixed Ising ferrimagnetic model were spins S that can take values ±1, 0 and spins σ that can take values ±1/2, are on the two sublattices of a two dimensional square lattice. We study the relaxation of the system magnetization, starting from an initial metastable configuration in which the magnetization is opposite to the applied field. The relaxation kinetic is simulated by a heatbath algorithm with random updates. We find that the decay time of the metastable phase depends strongly on the applied field and the size of the system. We have identified three different regimes, depending on the relaxation mechanism by which the system decays: the singledroplet regime for weak fields, and the multidroplet and strongfield regimes. We calculated the crossover fields between the different regimes. The field and the size dependence of the relaxation process are analyzed in terms of the droplet theory of homogeneous nucleation.

Bound states in the 3d Ising and models
View Description Hide DescriptionWe discuss the spectrum of the three dimensional Ising model and field theory in the broken symmetry phase. In this phase the effective potential between the elementary quanta is attractive and bound states of two or more of them may exist. We give theoretical and numerical evidence for the existence of these bound states. Looking in particular at the Ising model we discuss, by means of low temperature expansion techniques, the general pattern of the bound states spectrum.

Accurate results for Ising models from large order cluster variation method
View Description Hide DescriptionThe cluster variation method is a powerful hierarchy of meanfieldlike approximations for Isinglike lattice models, which is known to give results that are particularly accurate at high and/or low temperature and converge to the exact ones as the size of the clusters taken into account gets larger. Here we show how these properties can be exploited to obtain nonclassical, quite accurate estimates of quantities characterizing the critical behavior of the ordinary Ising model and the ground state of the triangular Ising antiferromagnet.

Correlation length of the simple cubic layer Potts model
View Description Hide DescriptionThe correlation length of the simple cubic semiinfinite lattice Potts model is numerically studied by using the eigenvalues of the transfer matrix. From the intersection of a correlation length curve family, the critical coupling parameter for the simple cubic layer lattice is calculated.

Improved estimator of disorder observables in qstate Potts model
View Description Hide DescriptionIn most lattice models the direct numerical evaluation of disorder observables is very noisy, owing to the nonlocal nature of these operators. It is pointed out that in the models where an updating cluster algorithm is implementable, like for instance in the qstate Potts model, one can define an estimator based on the linking properties of the clusters which produces a dramatic improvement of the measure of disorder operators of large size near criticality.

Kinetic Ising model with modified Kawasaki dynamics: Firstorder transition
View Description Hide DescriptionThe nonequilibrium phase diagram of a twodimensional kinetic Ising model with competing Glauber and modified Kawasaki dynamics is studied by using dynamical meanfield approximations and Monte Carlo simulations. In contrast to the previous meanfield calculation and Monte Carlo result the improved dynamical meanfield approximations predict firstorder phase transition when the influence of Kawasaki dynamics becomes dominant. Extensive Monte Carlo simulations support the emergence of a dynamical tricritical point that is in agreement with earlier observations in similar nonequilibrium models.

Recent results on driven lattice gases
View Description Hide DescriptionThe standard field theoretic approach to the driven lattice gas model and a new recently proposed one are briefly reviewed. We comment on the singular nature of the infinite driving limit and on the role of the particle current term.

Nonequilibrium molecular dynamics simulations: Techniques and applications
View Description Hide DescriptionThese lectures notes are devoted to the presentation of properties of matter in nonequilibrium conditions by using the techniques of NonEquilibrium Molecular Dynamics. As a first application, results on the simulation of shock waves in solids are presented. Shock propagation can be modeled by a direct Molecular Dynamics simulation. The shocked state is compared to a modeling through a dynamics constrained by the Hugoniot relations with a uniaxial compression: this new technique, named Hugoniotstat, allows to compute directly the shocked state properties without having to make a full study of the shock. The next application is concerned with properties of granular fluids. Those systems are always in a nonequilibrium state with energy flowing from an external forcing towards the internal elastic energy of the grains. Constantenergy simulation of inelastic hard disks are performed by a rescaling of the velocities at every dissipative collisions, a method we call Sotostat. This technique is used to compare flow properties with predictions from continuum equations. Moreover the chaotic indicators of the system behavior, the Lyapunov spectrum, are computed and the influence of the dissipativity of the grains on the spectrum is discussed.