arXiv subjects

E. Ben-Naim

Publications and source records attributed to E. Ben-Naim.

At least 73 records · Page 4Linked to original sources

arXiv2005

Dynamics of Social Diversity

We introduce and solve analytically a model for the development of disparate social classes in a competitive population. Individuals advance their fitness by competing against those in lower classes, and in parallel, individuals decline due to inactivity. We find a phase transition from a homogeneous, single-class society to a hierarchical, multi-class society. In the latter case, a finite fraction of the population still belongs to the lower class, and the rest of the population is in the middle class, on top of which lies a tiny upper class. While the lower class is static and poor, the middle class is upwardly mobile.

cond-mat.stat-mech↗

arXiv2005

Percolation with Multiple Giant Clusters

We study the evolution of percolation with freezing. Specifically, we consider cluster formation via two competing processes: irreversible aggregation and freezing. We find that when the freezing rate exceeds a certain threshold, the percolation transition is suppressed. Below this threshold, the system undergoes a series of percolation transitions with multiple giant clusters ("gels") formed. Giant clusters are not self-averaging as their total number and their sizes fluctuate from realization to realization. The size distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~ k^{-3}. We propose freezing as a practical mechanism for controlling the gel size.

cond-mat.stat-mech↗

arXiv2004

Stationary states and energy cascades in inelastic gases

We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transfered from large velocity scales to small velocity scales. These steady-states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) ~ v^{-sigma}. The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high velocities. We propose that these states may be realized experimentally in driven granular systems.

cond-mat.soft↗

arXiv2004

Opinion dynamics: rise and fall of political parties

We analyze the evolution of political organizations using a model in which agents change their opinions via two competing mechanisms. Two agents may interact and reach consensus, and additionally, individual agents may spontaneously change their opinions by a random, diffusive process. We find three distinct possibilities. For strong diffusion, the distribution of opinions is uniform and no political organizations (parties) are formed. For weak diffusion, parties do form and furthermore, the political landscape continually evolves as small parties merge into larger ones. Without diffusion, a pattern develops: parties have the same size and they possess equal niches. These phenomena are analyzed using pattern formation and scaling techniques.

cond-mat.stat-mech↗

arXiv2004

Winning quick and dirty: the greedy random walk

As a strategy to complete games quickly, we investigate one-dimensional random walks where the step length increases deterministically upon each return to the origin. When the step length after the kth return equals k, the displacement of the walk x grows linearly in time. Asymptotically, the probability distribution of displacements is a purely exponentially decaying function of |x|/t. The probability E(t,L) for the walk to escape a bounded domain of size L at time t decays algebraically in the long time limit, E(t,L) ~ L/t^2. Consequently, the mean escape time ~ L ln L, while ~ L^{2n-1} for n>1. Corresponding results are derived when the step length after the kth return scales as k^alpha$ for alpha>0.

cond-mat.stat-mech↗

arXiv2004

Kinetic Theory of Random Graphs: from Paths to Cycles

Structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically. At the gelation point, the typical length of paths and cycles, l, scales with the component size k as l ~ k^{1/2}. Dynamic and finite-size scaling laws for the behavior at and near the gelation point are obtained. Finite-size scaling laws are verified using numerical simulations.

cond-mat.stat-mech↗

arXiv2004

A Tractable Complex Network Model based on the Stochastic Mean-field Model of Distance

Much recent research activity has been devoted to empirical study and theoretical models of complex networks (random graphs) with three qualitative features: power-law degree distribution, local clustering of edges, and small diameter. We point out a new (in this context) platform for such models -- the stochastic mean-field model of distance -- and within this platform study a simple two-parameter proportional attachment model. The model is mathematicallly natural, permits a wide variety of explicit calculations, has the desired three qualitative features, and fits the complete range of degree scaling exponents and clustering parameters; in these respects it compares favorably to existing models.

cond-mat.dis-nn↗

arXiv2004

Unicyclic Components in Random Graphs

The distribution of unicyclic components in a random graph is obtained analytically. The number of unicyclic components of a given size approaches a self-similar form in the vicinity of the gelation transition. At the gelation point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a result, the total number of unicyclic components grows logarithmically with the system size.

cond-mat.stat-mech↗

arXiv2004

Random Geometric Series

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.

cond-mat.stat-mech↗

arXiv2004

Size of Outbreaks Near the Epidemic Threshold

The spread of infectious diseases near the epidemic threshold is investigated. Scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained. The maximal size of an outbreak n_* scales as N^{2/3} with N the population size. This scaling law implies that the average outbreak size scales as N^{1/3}. Moreover, the maximal and the average duration of an outbreak grow as t_* ~ N^{1/3} and ~ ln N, respectively.

q-bio.PE↗

arXiv2003

Finite Size Fluctuations in Interacting Particle Systems

Fluctuations may govern the fate of an interacting particle system even on the mean-field level. This is demonstrated via a three species cyclic trapping reaction with a large, yet finite number of particles, where the final number of particles N_f scales logarithmically with the system size N, N_f ~ ln N. Statistical fluctuations, that become significant as the number of particles diminishes, are responsible for this behavior. This phenomenon underlies a broad range of interacting particle systems including in particular multispecies annihilation processes.

cond-mat.stat-mech↗

arXiv2003

Leadership Statistics in Random Structures

The largest component (``the leader'') in evolving random structures often exhibits universal statistical properties. This phenomenon is demonstrated analytically for two ubiquitous structures: random trees and random graphs. In both cases, lead changes are rare as the average number of lead changes increases quadratically with logarithm of the system size. As a function of time, the number of lead changes is self-similar. Additionally, the probability that no lead change ever occurs decays exponentially with the average number of lead changes.

cond-mat.stat-mech↗

arXiv2003

Exchange Driven Growth

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: I) Growth: Clusters grow indefinitely; II) Gelation: All mass is transformed into an infinite gel in a finite time; and III) Instant Gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) ~ exp(-x^{2-ν}), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x)\sim x^{-5}. In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T\sim [\ln N]^{-(ν-2)}, in the large system size limit N\to\infty. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.

cond-mat.stat-mech↗

arXiv2003

Shattering Transitions in Collision-Induced Fragmentation

We investigate the kinetics of nonlinear collision-induced fragmentation. We obtain the fragment mass distribution analytically by utilizing its travelling wave behavior. The system undergoes a shattering transition in which a finite fraction of the mass is lost to infinitesimal fragments (dust). The nature of the shattering transition depends on the fragmentation process. When the larger of the two colliding fragments splits, the transition is discontinuous and the entire mass is transformed into dust at the transition point. When the smaller fragment splits, the transition is continuous with the dust gaining mass steadily on the account of the fragments. At the transition point, the fragment mass distribution diverges algebraically for small masses, c(m) ~ m^-a, with a=1.20191.

cond-mat.stat-mech↗

arXiv2003

The Inelastic Maxwell Model

Dynamics of inelastic gases are studied within the framework of random collision processes. The corresponding Boltzmann equation with uniform collision rates is solved analytically for gases, impurities, and mixtures. Generally, the energy dissipation leads to a significant departure from the elastic case. Specifically, the velocity distributions have overpopulated high energy tails and different velocity components are correlated. In the freely cooling case, the velocity distribution develops an algebraic high-energy tail, with an exponent that depends sensitively on the dimension and the degree of dissipation. Moments of the velocity distribution exhibit multiscaling asymptotic behavior, and the autocorrelation function decays algebraically with time. In the forced case, the steady state velocity distribution decays exponentially at large velocities. An impurity immersed in a uniform inelastic gas may or may not mimic the behavior of the background, and the departure from the background behavior is characterized by a series of phase transitions.

cond-mat.stat-mech↗

arXiv2002

Bifurcations and Patterns in Compromise Processes

We study an opinion dynamics model in which agents reach compromise via pairwise interactions. When the opinions of two agents are sufficiently close, they both acquire the average of their initial opinions; otherwise, they do not interact. Generically, the system reaches a steady state with a finite number of isolated, noninteracting opinion clusters (``parties''). As the initial opinion range increases, the number of such parties undergoes a periodic sequence of bifurcations. Both major and minor parties emerge, and these are organized in alternating pattern. This behavior is illuminated by considering discrete opinion states.

cond-mat.stat-mech↗

arXiv2002

Spontaneous spirals in vibrated granular chains

We present experimental measurements on the spontaneous formation of compact spiral structures in vertically-vibrated granular chains. Under weak vibration, when the chain is quasi two-dimensional and self-avoiding, spiral structures emerge from generic initial configurations. We compare geometrical characteristics of the spiral with that of an ideal tight spiral. Globally, the spiral undergoes a slow rotation such that to keep itself wound, while internally, fast vibrational modes are excited along the backbone with transverse oscillations dominating over longitudinal ones. Spirals have an extremely small volume in phase space, and hence, their formation demonstrates how nonequilibrium dynamics can result in a nonuniform sampling of phase space.

cond-mat.soft↗

arXiv2002

Growth and Structure of Stochastic Sequences

We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences leads to a wide variety of behaviors ranging from stretched exponential to log-normal to algebraic growth. Interestingly, the set of all possible sequence values has an intricate structure.

cond-mat.stat-mech↗