Weakly Nonextensive Thermostatistics and the Ising Model with Long--range Interactions

We introduce a nonextensive entropic measure $S_{\chi}$ that grows like $N^{\chi}$, where $N$ is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some $N$-body systems endowed with long-range interactions described by $r^{-\alpha}$ interparticle potentials. The power law (weakly nonextensive) behavior exhibited by $S_{\chi}$ is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and the (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized $q$-entropies. The functional $S_{\chi}$ is parametrized by the real number $\chi \in[1,2]$ in such a way that the standard logarithmic entropy is recovered when $\chi=1$ >. We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., $S_{\chi}$ possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since $S_{\chi}$ is nonextensive. For $1<\chi<2$, the entropy $S_{\chi}$ becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions.Comment: LaTeX file, 20 pages, 7 figure

Dynamical mechanism of anticipating synchronization in excitable systems

We analyze the phenomenon of anticipating synchronization of two excitable systems with unidirectional delayed coupling which are subject to the same external forcing. We demonstrate for different paradigms of excitable system that, due to the coupling, the excitability threshold for the slave system is always lower than that for the master. As a consequence the two systems respond to a common external forcing with different response times. This allows to explain in a simple way the mechanism behind the phenomenon of anticipating synchronization.Comment: 4 pages including 7 figures. Submitted for publicatio

Discrete--time ratchets, the Fokker--Planck equation and Parrondo's paradox

Parrondo's games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature. Here we review two recent approaches, based on the Fokker-Planck equation, that rigorously establish the connection between Parrondo's games and a physical model known as the flashing Brownian ratchet. This gives rise to a new set of Parrondo's games, of which the original games are a special case. For the first time, we perform a complete analysis of the new games via a discrete-time Markov chain (DTMC) analysis, producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs.Comment: 17 pages, 5 figure

A Model of Intra-seasonal Oscillations in the Earth atmosphere

We suggest a way of rationalizing an intra-seasonal oscillations (IOs) of the Earth atmospheric flow as four meteorological relevant triads of interacting planetary waves, isolated from the system of all the rest planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of IOs both in the North and South Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.Comment: 4 pages, 2 figs, Submitted to PR

Predict-prevent control method for perturbed excitable systems

We present a control method based on two steps: prediction and prevention. For prediction we use the anticipated synchronization scheme, considering unidirectional coupling between excitable systems in a master-slave configuration. The master is the perturbed system to be controlled, meanwhile the slave is an auxiliary system which is used to predict the master's behavior. We demonstrate theoretically and experimentally that an efficient control may be achieved.Comment: 4 pages, 5 figure

Diffusing opinions in bounded confidence processes

We study the effects of diffusing opinions on the Deffuant et al. model for continuous opinion dynamics. Individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside an interval centered around the present opinion. We show that diffusion induces an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion clusters are formed, although with some opinion spread inside them. If the diffusion jumps are not large, clusters coalesce, so that weak diffusion favors opinion consensus. A master equation for the process described above is presented. We find that the master equation and the Monte-Carlo simulations do not always agree due to finite-size induced fluctuations. Using a linear stability analysis we can derive approximate conditions for the transition between opinion clusters and the disordered state. The linear stability analysis is compared with Monte Carlo simulations. Novel interesting phenomena are analyzed

Analytical and numerical study of the non-linear noisy voter model on complex networks

We study the noisy voter model using a specific non-linear dependence of the rates that takes into account collective interaction between individuals. The resulting model is solved exactly under the all-to-all coupling configuration and approximately in some random networks environments. In the all-to-all setup we find that the non-linear interactions induce "bona fide" phase transitions that, contrary to the linear version of the model, survive in the thermodynamic limit. The main effect of the complex network is to shift the transition lines and modify the finite-size dependence, a modification that can be captured with the introduction of an effective system size that decreases with the degree heterogeneity of the network. While a non-trivial finite-size dependence of the moments of the probability distribution is derived from our treatment, mean-field exponents are nevertheless obtained in the thermodynamic limit. These theoretical predictions are well confirmed by numerical simulations of the stochastic process