Stochastic Effects in Physical Systems

A tutorial review is given of some developments and applications of stochastic processes from the point of view of the practicioner physicist. The index is the following: 1.- Introduction 2.- Stochastic Processes 3.- Transient Stochastic Dynamics 4.- Noise in Dynamical Systems 5.- Noise Effects in Spatially Extended Systems 6.- Fluctuations, Phase Transitions and Noise-Induced Transitions.Comment: 93 pages, 36 figures, LaTeX. To appear in Instabilities and Nonequilibrium Structures VI, E. Tirapegui and W. Zeller,eds. Kluwer Academi

Zealots in the mean-field noisy voter model

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: apart from the original herding processes, voters may change their states because of an intrinsic, noisy in origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasi-consensus state, where most of the voters share the same opinion, to a one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil the new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding/zealotry acting together in the voter model yields not a trivial mixture of the scenarios with the two mechanisms acting alone: it represents a situation where the global-local (noise-herding) competitions is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasi--consensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases ...Comment: 13 pages, 15 figure

Resistance to learning and the evolution of cooperation

In many evolutionary algorithms, crossover is the main operator used in generating new individuals from old ones. However, the usual mechanism for generating offsprings in spatially structured evolutionary games has to date been clonation. Here we study the effect of incorporating crossover on these models. Our framework is the spatial Continuous Prisoner's Dilemma. For this evolutionary game, it has been reported that occasional errors (mutations) in the clonal process can explain the emergence of cooperation from a non-cooperative initial state. First, we show that this only occurs for particular regimes of low costs of cooperation. Then, we display how crossover gets greater the range of scenarios where cooperative mutants can invade selfish populations. In a social context, where crossover involves a general rule of gradual learning, our results show that the less that is learnt in a single step, the larger the degree of global cooperation finally attained. In general, the effect of step-by-step learning can be more efficient for the evolution of cooperation than a full blast one.Evolutionary games, Continuous prisoner's dilemma, Spatially structured, Crossover, Learning

First-passage distributions for the one-dimensional Fokker-Planck equation

We present an analytical framework to study the first-passage (FP) and first-return (FR) distributions for the broad family of models described by the one-dimensional Fokker-Planck equation in finite domains, identifying general properties of these distributions for different classes of models. When in the Fokker-Planck equation the diffusion coefficient is positive (nonzero) and the drift term is bounded, as in the case of a Brownian walker, both distributions may exhibit a power-law decay with exponent -3/2 for intermediate times. We discuss how the influence of an absorbing state changes this exponent. The absorbing state is characterized by a vanishing diffusion coefficient and/or a diverging drift term. Remarkably, the exponent of the Brownian walker class of models is still found, as long as the departure and arrival regions are far enough from the absorbing state, but the range of times where the power law is observed narrows. Close enough to the absorbing point, though, a new exponent may appear. The particular value of the exponent depends on the behavior of the diffusion and the drift terms of the Fokker-Planck equation. We focus on the case of a diffusion term vanishing linearly at the absorbing point. In this case, the FP and FR distributions are similar to those of the voter model, characterized by a power law with exponent -2. As an illustration of the general theory, we compare it with exact analytical solutions and extensive numerical simulations of a two-parameter voter-like family models. We study the behavior of the FP and FR distributions by tuning the importance of the absorbing points throughout changes of the parameters. Finally, the possibility of inferring relevant information about the steady-sate probability distribution of a model from the FP and FR distributions is addressed.Comment: 17 pages, 8 figure

Life After Amputation: A Case Study

In the Philippines, the National Council on Disability made a survey revealing a prevalence of43.367 disabled who lost one or both legs and/or feet. This study explored the challengesencountered and coping mechanisms an amputee manifests, and the assistance that they getfrom their expected support system. This case study was anchored on Dorothea Orem’s SelfCare Theory of Nursing. Four purposively sampled amputee informants were interviewed andobserved. Primary and secondary data were gathered and triangulation with the relatives wasdone to ensure the validity and depth of the results. Data were encoded and analyzed usingthematic analysis. Results revealed that the informants experienced physical, psychological,emotional, socio-economic, and spiritual challenges. Physical challenges involved mobilityproblem and lack of gait balance, physical deformity, adjustment to the new body, alteredphysical appearance, a sedentary behavior, and phantom pain. Psychological challengesinvolved suicidal tendency, loss of libido, self-pity, and depression. Emotional challengesincluded fear and hopelessness. Socio-economic challenges involved the development of antisocial behavior, dissociative behavior, fear of losing a job, problem with money, and fear ofrejection. Spiritual challenges involved loss of faith. Coping mechanism included support fromfamily and friends, mastery of gait and balance, proper practice in using assistive devices,hastened adjustment to the new body, wearing of prosthesis, medication and mobility, and trustin God. Although they get strong support from the family and friends, there is very limitedassistance from the government and the community. There is a need to increase familyawareness in anticipating the needs of the amputees. Likewise, full support must be given tothem. Health education campaign may be formulated by the local health leaders and provisionof assistive devices and equipment to achieve the equalization and opportunities for personswith disabilities may also be done

Absorbing transition in a coevolution model with node and link states in an adaptive network: Network fragmentation transition at criticality

We consider a general model in which there is a coupled dynamics of node states and links states in a network. This coupled dynamics coevolves with dynamical changes of the topology of the network caused by a link rewiring mechanism. Such coevolution model features the interaction of the local dynamics of node and link states with the nonlocal dynamics of link-rewiring in a random network. The coupled dynamics of the states of the nodes and the links produces by itself an absorbing phase transition which is shown to be robust against the link rewiring mechanism. However, the dynamics of the network gives rise to significant physical changes, specially in the limit in which some links do not change state but are always rewired: First a network fragmentation occurs at the critical line of the absorbing transition, and only at this line, so that fragmentation is a manifestation of criticality. Secondly, in the active phase of the absorbing transition, finite-size fluctuations take the system to a single network component consensus phase, while other configurations are possible in the absence of rewiring. In addition, this phase is reached after a survival time that scales linearly with system size, while the survival time scales exponentially with system size when there is no rewiring. A social interpretation of our results contribute to the description of processes of emergence of social fragmentation and polarization

Partisan Voter Model: Quasi-stationary properties and noise-induced transitions

We revisit the Partisan Voter Model (PVM) reporting analytical results for the quasi-stationary distribution, exit probabilities and fixation times. Similarly to the Noisy Voter Model (NVM) we introduce a Noisy version of the PVM (NPVM). We find that the finite size noise induced transition of the NVM is modified in the NPVM so that there exists a new intermediate phase

Absorbing phase transition in the coupled dynamics of node and link states in random networks

We present a stochastic dynamics model of coupled evolution for the binary states of nodes and links in a complex network. In the context of opinion formation node states represent two possible opinions and link states a positive or negative relation. Dynamics proceeds via node and link state update towards pairwise satisfactory relations in which nodes in the same state are connected by positive links or nodes in different states are connected by negative links. By a mean-field rate equations analysis and Monte Carlo simulations in random networks we find an absorbing phase transition from a dynamically active phase to an absorbing phase. The transition occurs for a critical value of the relative time scale for node and link state updates. In the absorbing phase the order parameter, measuring global order, approaches exponentially the final frozen configuration. Finite size effects are such that in the absorbing phase the final configuration is reached in a characteristic time that scales logarithmically with system size, while in the active phase, finite-size fluctuation take the system to a frozen configuration in a characteristic time that grows exponentially with system size. There is also a finite-size topological transition associated with group splitting in the network of these final frozen configurations