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Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations. We consider a set of N independent and identically distributed random variables {X_{1},X_{2},...,X_{N}} whose common distribution has a parameter Y (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter Y, the X_{i} variables are independent and we call them conditionally independent and identically distributed. However, once integrated over the distribution of the parameter Y, the X_{i} variables get strongly correlated yet retain a solvable structure for various observables, such as for the sum and the extremes of X_{i}^{'}s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where N particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via simultaneous resetting to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.

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We study a one-dimensional gas of N Brownian particles that diffuse independently, but are simultaneously reset to the origin at a constant rate r. The system approaches a nonequilibrium stationary state with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the kth rightmost particle, and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realization of this resetting gas using optical traps.

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Among all types of corruption, police corruption is probably the one that most directly hurts society, as those trusted with protecting the people either side with the criminals that victimize the citizens, or are themselves, criminals. However, both corruption and its effects are very difficult to measure quantitatively other than by perception surveys, but the perception that citizens have of this phenomenon may be different from reality. Using a simple agent-based model, we analyze the effect on crime rates as a result of both corruption and the perception of corruption within law-enforcement corporations. Our results show a phase transition in which crime can propagate across the population even when the majority of police officers are honest. We find that one of the parameters that strongly controls crime incidence is the probability that regular citizens become criminals. In contrast, other actions, such as arresting crime lords, or the amount of crime-associated money that is confiscated, have little impact on the long-term crime incidence. Our results suggest that in addition to combating corruption within law-enforcement institutions, to further reduce the incidence of crime, policymakers should strive to restore confidence in these institutions and the justice system.

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We study the first passage probability and mean number of sites visited by a discrete persistent random walker on a lattice in one and two dimensions. This is performed by using the multistate formulation of the process. We obtain explicit expressions for the generating functions of these quantities. To evaluate these expressions, we study the system in the strongly persistent limit. In the one-dimensional case, we recover the behavior of the continuous one-dimensional persistent random walk (telegrapher process). In two dimensions we obtain an explicit expression for the probability distribution in the strongly persistent limit, however, the Laplace transforms required to evaluate the first-passage processes could only be evaluated in the asymptotic limit corresponding to long times in which regime we recover the behavior of normal two-dimensional diffusion.

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Since the pioneering work by Vicsek and his collaborators on the motion of self-propelled particles, most of the subsequent studies have focused on the onset of ordered states through a phase transition driven by particle density and noise. Usually, the particles in these systems are placed within periodic boundary conditions and interact via short-range velocity alignment forces. However, when the periodic boundaries are eliminated, letting the particles move in open space, the system is not able to organize into a coherently moving group since even small amounts of noise cause the flock to break apart. While the phase transition has been thoroughly studied, the conditions to keep the flock cohesive in open space are still poorly understood. Here we extend the Vicsek model of collective motion by introducing long-range alignment interactions between the particles. We show that just a small number of these interactions is enough for the system to build up long lasting ordered states of collective motion in open space and in the presence of noise. This finding was verified for other models in addition to the Vicsek one, suggesting its generality and revealing the importance that long-range interactions can have for the cohesion of the flock.

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We study the effect of introducing altruistic agents in a Schelling-like model of residential segregation. We find that even an infinitesimal proportion of altruists has dramatic catalytic effects on the collective utility of the system. Altruists provide pathways that move the system away from the suboptimal equilibrium it would reach if the system included only egoist agents, allowing it to reach the optimal steady state.

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Stochastic resonance is a phenomenon in which noise enhances the response of a system to an input signal. The brain is an example of a system that has to detect and transmit signals in a noisy environment, suggesting that it is a good candidate to take advantage of stochastic resonance. In this work, we aim to identify the optimal levels of noise that promote signal transmission through a simple network model of the human brain. Specifically, using a dynamic model implemented on an anatomical brain network (connectome), we investigate the similarity between an input signal and a signal that has traveled across the network while the system is subject to different noise levels. We find that non-zero levels of noise enhance the similarity between the input signal and the signal that has traveled through the system. The optimal noise level is not unique; rather, there is a set of parameter values at which the information is transmitted with greater precision, this set corresponds to the parameter values that place the system in a critical regime. The multiplicity of critical points in our model allows it to adapt to different noise situations and remain at criticality.

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We present an agent based model of a single asset financial market that is capable of replicating most of the non-trivial statistical properties observed in real financial markets, generically referred to as stylized facts. In our model agents employ strategies inspired on those used in real markets, and a realistic trade mechanism based on a double auction order book. We study the role of the distinct types of trader on the return statistics: specifically, correlation properties (or lack thereof), volatility clustering, heavy tails, and the degree to which the distribution can be described by a log-normal. Further, by introducing the practice of "profit taking", our model is also capable of replicating the stylized fact related to an asymmetry in the distribution of losses and gains.

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We present a statistical analysis of music scores from different composers using detrended fluctuation analysis (DFA). We find different fluctuation profiles that correspond to distinct autocorrelation structures of the musical pieces. Further, we reveal evidence for the presence of nonlinear autocorrelations by estimating the DFA of the magnitude series, a result validated by a corresponding study of appropriate surrogate data. The amount and the character of nonlinear correlations vary from one composer to another. Finally, we performed a simple experiment in order to evaluate the pleasantness of the musical surrogate pieces in comparison with the original music and find that nonlinear correlations could play an important role in the aesthetic perception of a musical piece.

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We consider a one-dimensional system with particles having either positive or negative velocity, and these particles annihilate on contact. Diffusion is superimposed on the ballistic motion of the particle. The annihilation may represent a reaction in which the two particles yield an inert species. This model has been the subject of previous work, in which it was shown that the particle concentration decays faster than either the purely ballistic or the purely diffusive case. We report on previously unnoticed behavior for large times when only one of the two species remains, and we also unravel the underlying fractal structure present in the system. We also consider in detail the case in which the initial concentration of right-going particles is 1/2+ɛ, with ɛ≠0. It is shown that remarkably rich behavior arises, in which two crossover times are observed as ɛ→0.

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Searching, defined for the purpose of the present study as the displacement of an individual to locate resources, is a fundamental behavior of all mobile organisms. In humans this behavior underlies many aspects of everyday life, involving cognitive processes such as sustained attention, memory and inhibition. We explored the performance of 36 treatment-free children diagnosed with attention-deficit hyperactivity disorder (ADHD) and 132 children from a control school sample on the ecologically based ball-search field task (BSFT), which required them to locate and collect golf balls in a large outdoor area. Children of both groups enjoyed the task and were motivated to participate in it. However, performance showed that ADHD-diagnosed subjects were significantly less efficient in their searching. We suggest that the BSFT provides a promising basis for developing more complex ecologically-derived tests that might help to better identify particular cognitive processes and impairments associated with ADHD.

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Since December 2006, more than a thousand cities in México have suffered the effects of the war between several drug cartels, amongst themselves, as well as with Mexican armed forces. Sources are not in agreement about the number of casualties of this war, with reports varying from 30 to 100 thousand dead; the economic and social ravages are impossible to quantify. In this work we analyze the official report of casualties in terms of the location and the date of occurrence of the homicides. We show how the violence, as reflected by the number of casualties, has increased over time and spread across the country. Next, based on the correlations between cities in the changes of the monthly number of casualties attributed to organized crime, we construct a narco-war network where nodes are the affected cities and links represent correlations between them. We find that close geographical distance between violent cities does not imply a strong correlation amongst them. We observe that the dynamics of the conflict has evolved in short-term periods where a small core of violent cities determines the main theatre of the war at each stage. This kind of analysis may also help to describe the emergence and propagation of gang-related violence waves.

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We consider the statistical mechanics of a small gaseous system subject to a constant external field. As is well known, in the canonical ensemble, that the system (i) obeys a barometric formula for the density profile, and (ii) the kinetic temperature is independent of height, even when the system is small. We show here that in the microcanonical ensemble the kinetic temperature of the particles affected by the field is not constant with height, but that rather, generally speaking, it decreases with a gradient of order 1/N. Even more, if we have a mixture of two species, one which is influenced by the field and the other which is not, we find that the two species' kinetic temperatures are generally different, even at the same height. These facts are shown in detail by studying a simple mechanical model: a Lorentz Gas where particles and spinning disks interact and the particles are subjected to a constant external force. In the microcanonical ensemble, the kinetic temperature of the particles is indeed found to vary with height; the disks' kinetic temperature, on the other hand, is height-independent, and thus, differs from that of the particles with which they interact.

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We consider the distribution of the duration time, the time elapsed since it began, of a diffusion process given its present position, under the assumption that the process began at the origin. For unbiased diffusion, the distribution does not exist (it is identically zero) for one- and two-dimensional systems. We find the explicit expression for the distribution for three and higher dimensions and discuss the behavior of the duration time statistics: we find that the expected duration time exists only for dimensions five and higher, whereas the variance becomes finite for seven dimensions and above. We then turn to the case of biased diffusion. The drift velocity introduces a new time scale and the resulting statistics arise from the interplay of the diffusive time scale and the drift time scale. For these systems all the moments exist and explicit expressions are presented and discussed for the expected duration time and its variance for all dimensions.

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We study the properties of the dynamical phase transition occurring in neural network models in which a competition between associative memory and sequential pattern recognition exists. This competition occurs through a weighted mixture of the symmetric and asymmetric parts of the synaptic matrix. Through a generating functional formalism, we determine the structure of the parameter space at non-zero temperature and near saturation (i.e., when the number of stored patterns scales with the size of the network), identifying the regions of high and weak pattern correlations, the spin-glass solutions, and the order-disorder transition between these regions. This analysis reveals that, when associative memory is dominant, smooth transitions appear between high correlated regions and spurious states. In contrast when sequential pattern recognition is stronger than associative memory, the transitions are always discontinuous. Additionally, when the symmetric and asymmetric parts of the synaptic matrix are defined in terms of the same set of patterns, there is a discontinuous transition between associative memory and sequential pattern recognition. In contrast, when the symmetric and asymmetric parts of the synaptic matrix are defined in terms of independent sets of patterns, the network is able to perform both associative memory and sequential pattern recognition for a wide range of parameter values.

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The problem of determining the joint probability distributions for correlated random variables with prespecified marginals is considered. When the joint distribution satisfying all the required conditions is not unique, the "most unbiased" choice corresponds to the distribution of maximum entropy. The calculation of the maximum-entropy distribution requires the solution of rather complicated nonlinear coupled integral equations, exact solutions to which are obtained for the case of Gaussian marginals; otherwise, the solution can be expressed as a perturbation around the product of the marginals if the marginal moments exist.

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We analyze order-disorder phase transitions driven by noise that occur in two kinds of network models closely related to the self-propelled model proposed by Vicsek [Phys. Rev. Lett. 75, 1226 (1995)] to describe the collective motion of groups of organisms. Two different types of noise, which we call intrinsic and extrinsic, are considered. The intrinsic noise, the one used by Vicsek in their original work, is related to the decision mechanism through which the particles update their positions. In contrast, the extrinsic noise, later introduced by Grégoire and Chaté [Phys. Rev. Lett. 92, 025702 (2004)], affects the signal that the particles receive from the environment. The network models presented here can be considered as mean-field representations of the self-propelled model. We show analytically and numerically that, for these two network models, the phase transitions driven by the intrinsic noise are continuous, whereas the extrinsic noise produces discontinuous phase transitions. This is true even for the small-world topology, which induces strong spatial correlations between the network elements. We also analyze the case where both types of noise are present simultaneously. In this situation, the phase transition can be continuous or discontinuous depending upon the amplitude of each type of noise.

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Scale-free foraging patterns are widespread among animals. These may be the outcome of an optimal searching strategy to find scarce, randomly distributed resources, but a less explored alternative is that this behaviour may result from the interaction of foraging animals with a particular distribution of resources. We introduce a simple foraging model where individual primates follow mental maps and choose their displacements according to a maximum efficiency criterion, in a spatially disordered environment containing many trees with a heterogeneous size distribution. We show that a particular tree-size frequency distribution induces non-Gaussian movement patterns with multiple spatial scales (Lévy walks). These results are consistent with field observations of tree-size variation and spider monkey (Ateles geoffroyi) foraging patterns. We discuss the consequences that our results may have for the patterns of seed dispersal by foraging primates.

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From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of pi generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e., when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as t ln t. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e., power-law growth with an exponent larger than . This behavior cannot be accounted for quantitatively by a simple continuous-time random walk model. Instead, we argue that anomalous diffusion correlates with the existence of families of propagating periodic orbits. Finally we show that when a configuration with parallel scatterers is approached there is a crossover from normal to anomalous diffusion, with the diffusion coefficient exhibiting a power-law divergence.

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We present a definition for metastable states applicable to arbitrary finite state Markov processes satisfying detailed balance. In particular, we identify a crucial condition that distinguishes metastable states from other slow decaying modes and which allows us to show that our definition has several desirable properties similar to those postulated in the restricted ensemble approach. The intuitive physical meaning of this condition is simply that the total equilibrium probability of finding the system in the metastable state is negligible.