Phase transitions in systems of self-propelled agents and related network models

An important characteristic of flocks of birds, school of fish, and many similar assemblies of self-propelled particles is the emergence of states of collective order in which the particles move in the same direction. When noise is added into the system, the onset of such collective order occurs through a dynamical phase transition controlled by the noise intensity. While originally thought to be continuous, the phase transition has been claimed to be discontinuous on the basis of recently reported numerical evidence. We address this issue by analyzing two representative network models closely related to systems of self-propelled particles. We present analytical as well as numerical results showing that the nature of the phase transition depends crucially on the way in which noise is introduced into the system.Comment: Four pages, four figures. Submitted to PR

L\'evy-like behavior in deterministic models of intelligent agents exploring heterogeneous environments

Many studies on animal and human movement patterns report the existence of scaling laws and power-law distributions. Whereas a number of random walk models have been proposed to explain observations, in many situations individuals actually rely on mental maps to explore strongly heterogeneous environments. In this work we study a model of a deterministic walker, visiting sites randomly distributed on the plane and with varying weight or attractiveness. At each step, the walker minimizes a function that depends on the distance to the next unvisited target (cost) and on the weight of that target (gain). If the target weight distribution is a power-law, $p(k)\sim k^{-\beta}$, in some range of the exponent $\beta$, the foraging medium induces movements that are similar to L\'evy flights and are characterized by non-trivial exponents. We explore variations of the choice rule in order to test the robustness of the model and argue that the addition of noise has a limited impact on the dynamics in strongly disordered media.Comment: 15 pages, 7 figures. One section adde

Static Pairwise Annihilation in Complex Networks

We study static annihilation on complex networks, in which pairs of connected particles annihilate at a constant rate during time. Through a mean-field formalism, we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections. This general formalism, which is exact for disordered networks, is applied to Kronecker, Erd\"os-R\'enyi (i.e. Poisson) and scale-free networks. We compare our theoretical results with extensive numerical simulations obtaining excellent agreement. Although the mean-field approach applies in an exact way neither to ordered lattices nor to small-world networks, it qualitatively describes the annihilation dynamics in such structures. Our results indicate that the higher the connectivity of a given network element, the faster it annihilates. This fact has dramatic consequences in scale-free networks, for which, once the ``hubs'' have been annihilated, the network disintegrates and only isolated sites are left.Comment: 7 Figures, 10 page

Metastability in Markov processes

We present a formalism to describe slowly decaying systems in the context of finite Markov chains obeying detailed balance. We show that phase space can be partitioned into approximately decoupled regions, in which one may introduce restricted Markov chains which are close to the original process but do not leave these regions. Within this context, we identify the conditions under which the decaying system can be considered to be in a metastable state. Furthermore, we show that such metastable states can be described in thermodynamic terms and define their free energy. This is accomplished showing that the probability distribution describing the metastable state is indeed proportional to the equilibrium distribution, as is commonly assumed. We test the formalism numerically in the case of the two-dimensional kinetic Ising model, using the Wang--Landau algorithm to show this proportionality explicitly, and confirm that the proportionality constant is as derived in the theory. Finally, we extend the formalism to situations in which a system can have several metastable states.Comment: 30 pages, 5 figures; version with one higher quality figure available at http://www.fis.unam.mx/~dsanders

The Reaction-Diffusion Front for A+B→∅A+B \to\emptyset in One Dimension

We study theoretically and numerically the steady state diffusion controlled reaction $A+B\rightarrow\emptyset$, where currents $J$ of $A$ and $B$ particles are applied at opposite boundaries. For a reaction rate $\lambda$, and equal diffusion constants $D$, we find that when $\lambda J^{-1/2} D^{-1/2}\ll 1$ the reaction front is well described by mean field theory. However, for $\lambda J^{-1/2} D^{-1/2}\gg 1$, the front acquires a Gaussian profile - a result of noise induced wandering of the reaction front center. We make a theoretical prediction for this profile which is in good agreement with simulation. Finally, we investigate the intrinsic (non-wandering) front width and find results consistent with scaling and field theoretic predictions.Comment: 11 pages, revtex, 4 separate PostScript figure

Refined Simulations of the Reaction Front for Diffusion-Limited Two-Species Annihilation in One Dimension

Extensive simulations are performed of the diffusion-limited reaction A$+$B$\to 0$ in one dimension, with initially separated reagents. The reaction rate profile, and the probability distributions of the separation and midpoint of the nearest-neighbour pair of A and B particles, are all shown to exhibit dynamic scaling, independently of the presence of fluctuations in the initial state and of an exclusion principle in the model. The data is consistent with all lengthscales behaving as $t^{1/4}$ as $t\to\infty$. Evidence of multiscaling, found by other authors, is discussed in the light of these findings.Comment: Resubmitted as TeX rather than Postscript file. RevTeX version 3.0, 10 pages with 16 Encapsulated Postscript figures (need epsf). University of Geneva preprint UGVA/DPT 1994/10-85

Transport Properties of the Diluted Lorentz Slab

We study the behavior of a point particle incident from the left on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.Comment: 14 pages of text in LaTeX, 7 figures in Postscrip

Localisation Transition of A Dynamic Reaction Front

We study the reaction-diffusion process $A+B\to \emptyset$ with injection of each species at opposite boundaries of a one-dimensional lattice and bulk driving of each species in opposing directions with a hardcore interaction. The system shows the novel feature of phase transitions between localised and delocalised reaction zones as the injection rate or reaction rate is varied. An approximate analytical form for the phase diagram is derived by relating both the domain of reactants $A$ and the domain of reactants $B$ to asymmetric exclusion processes with open boundaries, a system for which the phase diagram is known exactly, giving rise to three phases. The reaction zone width $w$ is described by a finite size scaling form relating the early time growth, relaxation time and saturation width exponents. In each phase the exponents are distinct from the previously studied case where the reactants diffuse isotropically.Comment: 13 pages, latex, uses eps

The statistics of diffusive flux

We calculate the explicit probability distribution function for the flux between sites in a simple discrete time diffusive system composed of independent random walkers. We highlight some of the features of the distribution and we discuss its relation to the local instantaneous entropy production in the system. Our results are applicable both to equilibrium and non-equilibrium steady states as well as for certain time dependent situations.Comment: 12 pages, 1 figur