Mendeli H. Vainstein

Random mobility and spatial structure often enhance cooperation

The effects of an unconditional move rule in the spatial Prisoners Dilemma, Snowdrift and Stag Hunt games are studied. Spatial structure by itself is known to modify the outcome of many games when compared with a randomly mixed population, sometimes promoting, sometimes inhibiting cooperation. Here we show that random dilution and mobility may suppress the inhibiting factors of the spatial structure in the Snowdrift game, while enhancing the already larger cooperation found in the Prisoners dilemma and Stag Hunt games.

Disordered environments in spatial games.

The Prisoners dilemma is the main game theoretical framework in which the onset and maintainance of cooperation in biological populations is studied. In the spatial version of the model, we study the robustness of cooperation in heterogeneous ecosystems in spatial evolutionary games by considering site diluted lattices. The main result is that, due to disorder, the fraction of cooperators in the population is enhanced. Moreover, the system presents a dynamical transition at rho*, separating a region with spatial chaos from one with localized, stable groups of cooperators.

Entropic stochastic resonance.

We present a novel scheme for the appearance of stochastic resonance when the dynamics of a Brownian particle takes place in a confined medium. The presence of uneven boundaries, giving rise to an entropic contribution to the potential, may upon application of a periodic driving force result in an increase of the spectral amplification at an optimum value of the ambient noise level. The entropic stochastic resonance, characteristic of small-scale systems, may constitute a useful mechanism for the manipulation and control of single molecules and nanodevices.

Khinchin theorem and anomalous diffusion.

A recent Letter [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, (Dover, New York, 1949)] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a generalized Langevin equation the Khinchin theorem holds.

Spatial social dilemmas: Dilution, mobility and grouping effects with imitation dynamics

We present an extensive, systematic study of the Prisoner’s Dilemma and Snowdrift games on a square lattice under a synchronous, noiseless imitation dynamics. We show that for both the occupancy of the network and the (random) mobility of the agents there are intermediate values that may increase the amount of cooperators in the system and new phases appear. We analytically determine the transition lines between these phases and compare with the mean field prediction and the observed behavior on a square lattice. We point out which are the more relevant microscopic processes that entitle cooperators to invade a population of defectors in the presence of mobility and discuss the universality of these results.

Non-exponential relaxation for anomalous diffusion

We study the relaxation process in normal and anomalous diffusion regimes for systems described by a generalized Langevin equation (GLE). We demonstrate the existence of a very general correlation function which describes the relaxation phenomena. Such function is even; therefore, it cannot be an exponential or a stretched exponential. However, for a proper choice of the parameters, those functions can be reproduced within certain intervals with good precision. We also show the passage from the non-Markovian to the Markovian behaviour in the normal diffusion regime. For times longer than the relaxation time, the correlation function for anomalous diffusion becomes a power law for broad-band noise.

Modeling of Droplet Evaporation on Superhydrophobic Surfaces.

When a drop of water is placed on a rough surface, there are two possible extreme regimes of wetting: the one called Cassie-Baxter (CB) with air pockets trapped underneath the droplet and the one called the Wenzel (W) state characterized by the homogeneous wetting of the surface. A way to investigate the transition between these two states is by means of evaporation experiments, in which the droplet starts in a CB state and, as its volume decreases, penetrates the surfaces grooves, reaching a W state. Here we present a theoretical model based on the global interfacial energies for CB and W states that allows us to predict the thermodynamic wetting state of the droplet for a given volume and surface texture. We first analyze the influence of the surface geometric parameters on the droplets final wetting state with constant volume and show that it depends strongly on the surface texture. We then vary the volume of the droplet, keeping the geometric surface parameters fixed to mimic evaporation and show that the drop experiences a transition from the CB to the W state when its volume reduces, as observed in experiments. To investigate the dependency of the wetting state on the initial state of the droplet, we implement a cellular Potts model in three dimensions. Simulations show very good agreement with theory when the initial state is W, but it disagrees when the droplet is initialized in a CB state, in accordance with previous observations which show that the CB state is metastable in many cases. Both simulations and the theoretical model can be modified to study other types of surfaces.

Percolation and cooperation with mobile agents: Geometric and strategy clusters

We study the conditions for persistent cooperation in an off-lattice model of mobile agents playing the Prisoners Dilemma game with pure, unconditional strategies. Each agent has an exclusion radius r(P), which accounts for the population viscosity, and an interaction radius r(int), which defines the instantaneous contact network for the game dynamics. We show that, differently from the r(P)=0 case, the model with finite-sized agents presents a coexistence phase with both cooperators and defectors, besides the two absorbing phases, in which either cooperators or defectors dominate. We provide, in addition, a geometric interpretation of the transitions between phases. In analogy with lattice models, the geometric percolation of the contact network (i.e., irrespective of the strategy) enhances cooperation. More importantly, we show that the percolation of defectors is an essential condition for their survival. Differently from compact clusters of cooperators, isolated groups of defectors will eventually become extinct if not percolating, independently of their size.

Gaussian noise and time-reversal symmetry in nonequilibrium Langevin models.

We show that in driven systems the Gaussian nature of the fluctuating force and time reversibility are equivalent properties. This result together with the potential condition of the external force drastically restricts the form of the probability distribution function, which can be shown to satisfy time-independent relations. We have corroborated this feature by explicitly analyzing a model for the stretching of a polymer and a model for a suspension of noninteracting Brownian particles in steady flow.

Entropy, non-ergodicity and non-Gaussian behaviour in ballistic transport

Ballistic transportation introduces new challenges in the thermodynamic properties of a gas of particles. For example, violation of mixing, ergodicity and of the fluctuation-dissipation theorem may occur, since all these processes are connected. In this work, we obtain results for all ranges of diffusion, i.e., both for subdiffusion and superdiffusion, where the bath is such that it gives origin to a colored noise. In this way we obtain the skewness and the non-Gaussian factor for the probability distribution function of the dynamical variable. We put particular emphasis on ballistic diffusion, and we demonstrate that in this case, although the second law of thermodynamics is preserved, the entropy does not reach a maximum and a non-Gaussian behavior occurs. This implies the non-applicability of the central limit theorem.