Searcharxiv⌕ Search

arXiv subjects

Daniel ben-Avraham

Publications and source records attributed to Daniel ben-Avraham.

At least 55 records · Page 3Linked to original sources
arXiv2004

On the Relation between One-Species Diffusion-Limited Coalescence and Annihilation in One Dimension

The close similarity between the hierarchies of multiple-point correlation functions for the diffusion-limited coalescence and annihilation processes has caused some recent confusion, raising doubts as to whether such hierarchies uniquely determine an infinite particle system. We elucidate the precise relations between the two processes, arriving at the conclusion that the hierarchy of correlation functions does provide a complete representation of a particle system on the line. We also introduce a new hierarchy of probability density functions, for finding particles at specified locations and none in between. This hierarchy is computable for coalescence, through the method of empty intervals, and is naturally suited for questions concerning the ordering of particles on the line.

cond-mat.stat-mech↗
arXiv2004

Statistics of Cycles: How Loopy is your Network?

We study the distribution of cycles of length h in large networks (of size N>>1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h* ~ N^a. Our results suggest that h* and the exponent a might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte-Carlo sampling algorithm for approximately locating h* and reliably determining a. Our empirical results indicate that for small random scale-free nets of degree exponent g, a=1/(g-1), and a grows as the nets become larger.

cond-mat.dis-nn↗
arXiv2004

What is Special about Diffusion on Scale-Free Nets?

We study diffusion (random walks) on recursive scale-free graphs, and contrast the results to similar studies in other analytically soluble media. This allows us to identify ways in which diffusion in scale-free graphs is special. Most notably, scale-free architecture results in a faster transit time between existing nodes, when the network grows in size; and walks emanating from the most connected nodes are recurrent, despite the network's infinite dimension. We also find that other attributes of the graph, besides its scale-free distribution, have a strong influence on the nature of diffusion.

cond-mat.dis-nn↗
arXiv2004

How Famous is a Scientist? -- Famous to Those Who Know Us

Following a recent idea, to measure fame by the number of \Google hits found in a search on the WWW, we study the relation between fame (\Google hits) and merit (number of papers posted on an electronic archive) for a random group of scientists in condensed matter and statistical physics. Our findings show that fame and merit in science are linearly related, and that the probability distribution for a certain level of fame falls off exponentially. This is in sharp contrast with the original findings about WW II ace pilots, for which fame is exponentially related to merit (number of downed planes), and the probability of fame decays in power-law fashion. Other groups in our study show similar patterns of fame as for ace pilots.

cond-mat.other↗
arXiv2004

Velocity Distribution in a Viscous Granular Gas

We investigate the velocity relaxation of a viscous one-dimensional granular gas, that is, one in which neither energy nor momentum is conserved in a collision. Of interest is the distribution of velocities in the gas as it cools, and the time dependence of the relaxation behavior. A Boltzmann equation of instantaneous binary collisions leads to a two-peaked distribution with each peak relaxing to zero velocity as 1/t while each peak also narrows as 1/t. Numerical simulations of grains on a line also lead to a double-peaked distribution that narrows as 1/t. A Maxwell approximation leads to a single-peaked distribution about zero velocity with power-law wings. This distribution narrows exponentially. In either case, the relaxing distribution is not of Maxwell-Boltzmann form.

cond-mat.stat-mech↗
arXiv2004

Designer Nets from Local Strategies

We propose a local strategy for constructing scale-free networks of arbitrary degree distributions, based on the redirection method of Krapivsky and Redner [Phys. Rev. E 63, 066123 (2001)]. Our method includes a set of external parameters that can be tuned at will to match detailed behavior at small degree k, in addition to the scale-free power-law tail signature at large k. The choice of parameters determines other network characteristics, such as the degree of clustering. The method is local in that addition of a new node requires knowledge of only the immediate environs of the (randomly selected) node to which it is attached. (Global strategies require information on finite fractions of the growing net.)

cond-mat.dis-nn↗
arXiv2003

Efficient Immunization Strategies for Computer Networks and Populations

We present an effective immunization strategy for computer networks and populations with broad and, in particular, scale-free degree distributions. The proposed strategy, acquaintance immunization, calls for the immunization of random acquaintances of random nodes (individuals). The strategy requires no knowledge of the node degrees or any other global knowledge, as do targeted immunization strategies. We study analytically the critical threshold for complete immunization. We also study the strategy with respect to the susceptible-infected-removed epidemiological model. We show that the immunization threshold is dramatically reduced with the suggested strategy, for all studied cases.

cond-mat.dis-nn↗
arXiv2003

Self-Similarity in Random Collision Processes

Kinetics of collision processes with linear mixing rules are investigated analytically. The velocity distribution becomes self-similar in the long time limit and the similarity functions have algebraic or stretched exponential tails. The characteristic exponents are roots of transcendental equations and vary continuously with the mixing parameters. In the presence of conservation laws, the velocity distributions become universal.

cond-mat.stat-mech↗
arXiv2003

A Generalization of both the Method of Images and of the Classical Integral Transforms

A new method for the solution of initial-boundary value problems for evolution PDEs recently introduced by Fokas is generalised to multidimensions. Also the relation of this method with the method of images and with the classical integral transforms is discussed. The new method is easy to implement, yet it is applicable to problems for which the classical approaches apparently fail. As illustrative examples, initial-boundary value problems for the diffusion-convection equation in one and higher dimensions, as well as for the linearised Korteweg-de Vries equation with the space variables on the half-line are solved. The suitability of the new method for the analysis of the the long-time asymptotics is ellucidated.

cond-mat↗
arXiv2003

Mean-field solution of the parity-conserving kinetic phase transition in one dimension

A two-offspring branching annihilating random walk model, with finite reaction rates, is studied in one-dimension. The model exhibits a transition from an active to an absorbing phase, expected to belong to the $DP2$ universality class embracing systems that possess two symmetric absorbing states, which in one-dimensional systems, is in many cases equivalent to parity conservation. The phase transition is studied analytically through a mean-field like modification of the so-called {\it parity interval method}. The original method of parity intervals allows for an exact analysis of the diffusion-controlled limit of infinite reaction rate, where there is no active phase and hence no phase transition. For finite rates, we obtain a surprisingly good description of the transition which compares favorably with the outcome of Monte Carlo simulations. This provides one of the first analytical attempts to deal with the broadly studied DP2 universality class.

cond-mat.stat-mech↗
arXiv2003

Variable survival exponents in history-dependent random walks: hard movable reflector

We review recent studies demonstrating a nonuniversal (continuously variable) survival exponent for history-dependent random walks, and analyze a new example, the hard movable partial reflector. These processes serve as a simplified models of infection in a medium with a history-dependent susceptibility, and for spreading in systems with an infinite number of absorbing configurations. The memory may take the form of a history-dependent step length, or be the result of a partial reflector whose position marks the maximum distance the walker has ventured from the origin. In each case, a process with memory is rendered Markovian by a suitable expansion of the state space. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) \sim t^{-delta}, where δvaries with the parameters of the model. We report new results for a hard partial reflector, i.e., one that moves forward only when the walker does. When the walker tries to jump to the site R occupied by the reflector, it is reflected back with probability r, and stays at R with probability 1-r; only in the latter case does the reflector move (R \to R+1). For this model, delta = 1/2(1-r), and becomes arbitrarily large as r approaches 1. This prediction is confirmed via iteration of the transfer matrix, which also reveals slowly-decaying corrections to scaling.

cond-mat.stat-mech↗
arXiv2003

Geographical Embedding of Scale-Free Networks

A method for embedding graphs in Euclidean space is suggested. The method connects nodes to their geographically closest neighbors and economizes on the total physical length of links. The topological and geometrical properties of scale-free networks embedded by the suggested algorithm are studied both analytically and through simulations. Our findings indicate dramatic changes in the embedded networks, in comparison to their off-lattice counterparts, and call into question the applicability of off-lattice scale-free models to realistic, everyday-life networks.

cond-mat↗
arXiv2003

Large-Scale Simulations of Diffusion-Limited n-Species Annihilation

We present results from computer simulations for diffusion-limited $n$-species annihilation, $A_i+A_j\to0$ $(i,j=1,2,...,n;i\neq j)$, on the line, for lattices of up to $2^{28}$ sites, and where the process proceeds to completion (no further reactions possible), involving up to $10^{15}$ time steps. These enormous simulations are made possible by the renormalized reaction-cell method (RRC). Our results suggest that the concentration decay exponent for $n$ species is $\a(n)=(n-1)/2n$ instead of $(2n-3)/(4n-4)$, as previously believed, and are in agreement with recent theoretical arguments \cite{tauber}. We also propose a scaling relation for $Δ$, the correction-to-scaling exponent for the concentration decay; $c(t)\sim t^{-\a}(A+Bt^{-Δ})$.

cond-mat.soft↗
arXiv2002

Asymptotic analysis of a random walk with a history-dependent step length

We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and steps of length n when entering a region that has never been visited. The process provides a simplified model of spreading in systems with an infinite number of absorbing configurations. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) \sim t^{-delta}, with delta = v/2n. Our expression for the decay exponent is in agreement with results obtained via numerical iteration of the transition matrix.

cond-mat.stat-mech↗
arXiv2002

Compact directed percolation with movable partial reflectors

We study a version of compact directed percolation (CDP) in one dimension in which occupation of a site for the first time requires that a "mine" or antiparticle be eliminated. This process is analogous to the variant of directed percolation with a long-time memory, proposed by Grassberger, Chate and Rousseau [Phys. Rev. E 55, 2488 (1997)] in order to understand spreading at a critical point involving an infinite number of absorbing configurations. The problem is equivalent to that of a pair of random walkers in the presence of movable partial reflectors. The walkers, which are unbiased, start one lattice spacing apart, and annihilate on their first contact. Each time one of the walkers tries to visit a new site, it is reflected (with probability r) back to its previous position, while the reflector is simultaneously pushed one step away from the walker. Iteration of the discrete-time evolution equation for the probability distribution yields the survival probability S(t). We find that S(t) \sim t^{-delta}, with delta varying continuously between 1/2 and 1.160 as the reflection probability varies between 0 and 1.

cond-mat.stat-mech↗
arXiv2002

Scale-free Networks on Lattices

We suggest a method for embedding scale-free networks, with degree distribution P(k) k^-lambda, in regular Euclidean lattices. The embedding is driven by a natural constraint of minimization of the total length of the links in the system. We find that all networks with lambda>2 can be successfully embedded up to an (Euclidean) distance xi which can be made as large as desired upon the changing of an external parameter. Clusters of successive chemical shells are found to be compact (the fractal dimension is d_f=d), while the dimension of the shortest path between any two sites is smaller than one: d_min=(lambda-2)/(lambda-1-1/d), contrary to all other known examples of fractals and disordered lattices.

cond-mat.dis-nn↗
arXiv2001

Correlation Functions for Diffusion-Limited Annihilation, A + A -> 0

The full hierarchy of multiple-point correlation functions for diffusion-limited annihilation, A + A -> 0, is obtained analytically and explicitly, following the method of intervals. In the long time asymptotic limit, the correlation functions of annihilation are identical to those of coalescence, A + A -> A, despite differences between the two models in other statistical measures, such as the interparticle distribution function.

cond-mat.stat-mech↗