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Daniel ben-Avraham

Publications and source records attributed to Daniel ben-Avraham.

72 records · Page 4Linked to original sources
arXiv2001

Continuously-variable survival exponent for random walks with movable partial reflectors

We study a one-dimensional lattice random walk with an absorbing boundary at the origin and a movable partial reflector. On encountering the reflector, at site x, the walker is reflected (with probability r) to x-1 and the reflector is simultaneously pushed to x+1. Iteration of the transition matrix, and asymptotic analysis of the probability generating function show that the critical exponent delta governing the survival probability varies continuously between 1/2 and 1 as r varies between 0 and 1. Our study suggests a mechanism for nonuniversal kinetic critical behavior, observed in models with an infinite number of absorbing configurations.

cond-mat.stat-mech↗
arXiv2001

Breakdown of the Internet under intentional attack

We study the tolerance of random networks to intentional attack, whereby a fraction p of the most connected sites is removed. We focus on scale-free networks, having connectivity distribution of P(k)~k^(-a) (where k is the site connectivity), and use percolation theory to study analytically and numerically the critical fraction p_c needed for the disintegration of the network, as well as the size of the largest connected cluster. We find that even networks with a<=3, known to be resilient to random removal of sites, are sensitive to intentional attack. We also argue that, near criticality, the average distance between sites in the spanning (largest) cluster scales with its mass, M, as sqrt(M), rather than as log_k M, as expected for random networks away from criticality. Thus, the disruptive effects of intentional attack become relevant even before the critical threshold is reached.

cond-mat.dis-nn↗
arXiv2001

The Solution of the Modified Helmholtz Equation in a Triangular Domain and an Application to Diffusion-Limited Coalescence

A new transform method for solving boundary value problems for linear and integrable nonlinear PDEs recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q_{xx}(x,y)+q_{yy}(x,y)-4b^2q(x,y)=0 in the triangular domain 0 < x < L-y < L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A + A <--> A, in the segment (-L/2,L/2), with traps at the edges.

cond-mat.stat-mech↗
arXiv2001

A Method of Intervals for the Study of Diffusion-Limited Annihilation, A + A --> 0

We introduce a method of intervals for the analysis of diffusion-limited annihilation, A+A -> 0, on the line. The method leads to manageable diffusion equations whose interpretation is intuitively clear. As an example, we treat the following cases: (a) annihilation in the infinite line and in infinite (discrete) chains; (b) annihilation with input of single particles, adjacent particle pairs, and particle pairs separated by a given distance; (c) annihilation, A+A -> 0, along with the birth reaction A -> 3A, on finite rings, with and without diffusion.

cond-mat.stat-mech↗
arXiv2000

Resilience of the Internet to random breakdowns

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k)=ck^-a. We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p_c, that need to be removed before the network disintegrates. We show that for a<=3 the transition never takes place, unless the network is finite. In the special case of the Internet (a=2.5), we find that it is impressively robust, where p_c is approximately 0.99.

cond-mat.dis-nn↗
arXiv2000

Large Scale Simulations of Two-Species Annihilation, A+B->0, with Drift

We present results of computer simulations of the diffusion-limited reaction process A+B->0, on the line, under extreme drift conditions, for lattices of up to 2^{27} sites, and where the process proceeds to completion (no particles left). These enormous simulations are made possible by the renormalized reaction-cell method (RRC). Our results allow us to resolve an existing controversy about the rate of growth of domain sizes, and about corrections to scaling of the concentration decay.

cond-mat.stat-mech↗
arXiv1999

Diffusion-Limited Coalescence and Annihilation in Random Media

We study the kinetics of diffusion-limited coalescence, A+A->A, and annihilation, A+A->0, in random media consisting of disconnected domains of reaction. Examples include excitons fusion and annihilation in porous matrices and along polymer chains. We begin with an exact analysis of A+A->A in a finite segment. This result is applied to coalescence in a random distribution of segment lengths, and the implications for coalescence and annihilation in percolation clusters and other confined geometries are then derived by means of scaling techniques.

cond-mat.stat-mech↗
arXiv1998

Generalized von Smoluchowski model of reaction rates, with reacting particles and a mobile trap

We study diffusion-limited coalescence, A+A<-->A, in one dimension, in the presence of a diffusing trap. The system may be regarded as a generalization of von Smoluchowski's model for reaction rates, in that: (a) it includes reactions between the particles surrounding the trap, and (b) the trap is mobile -- both considerations which render the model more physically relevant. As seen from the trap's frame of reference, the motion of the particles is highly correlated, because of the motion of the trap. An exact description of the long -time asymptotic limit is found using the IPDF method, and exploiting a "shielding" property of reversible coalescence that was discovered recently. In the case where the trap also acts as a source -- giving birth to particles -- the shielding property breaks down, but we find an "equivalence principle": Trapping and diffusion of the trap may be compensated by an appropriate rate of birth, such that the steady state of the system is identical with the equilibrium state in the absence of a trap.

cond-mat.stat-mech↗
arXiv1998

Inhomogeneous Steady States of Diffusion-Limited Coalescence, A+A<-->A

We study the steady state of diffusion-limited coalescence, A+A<-->A, in the presence of a trap and with a background drift. In one dimension this model can be analyzed exactly through the method of inter-particle distribution functions (IPDF). Because of the irreversible trap the steady state of the system is a non-equilibrium state. An interesting phase transition, controlled by the drift away from the trap, takes place: from a non-trivial steady state, when the drift is weak, to a trivial steady state (the vacuum), as the drift increases beyond some critical point. Surprisingly, regardless of the drift strength, the computed IPDF resembles that of an homogeneous equilibrium system, without the trap. We suggest that this is due to "shielding": the particle nearest to the trap shields the remaining particles from the effects of the trap. Finally, we compare the exact solution to that of a reaction-diffusion equation, and we determine the optimal values of the appropriate rate coefficients.

cond-mat.stat-mech↗
arXiv1998

Fisher Waves in the Diffusion-Limited Coalescence Process A+A<-->A

Fisher waves have been studied recently in the specific case of diffusion-limited reversible coalescence, A+A<-->A, on the line. An exact analysis of the particles concentration showed that waves propagate from a stable region to an unstable region at constant speed, just as in Fisher's "mean-field" theory; but also that the wave front fails to retain its initial shape and instead it broadens with time. Our present analysis encompasses the full hierarchy of multiple-point density correlation functions, and thus it provides a complete exact description of the same system. We find that as the wave propagates, the particles in the stable phase remain distributed exactly as in their initial (equilibrium) state. On the other hand, the leading particle---the one at the edge of the wave---advances as a biased random walk, rather than simply linearly with time. Thus the shape of the wave remains actually constant, but it is the "noisy" propagation of the wave's edge that causes its apparent broadening.

cond-mat.stat-mech↗
arXiv1998

Complete Exact Solution of Diffusion-Limited Coalescence, A + A -> A

Some models of diffusion-limited reaction processes in one dimension lend themselves to exact analysis. The known approaches yield exact expressions for a limited number of quantities of interest, such as the particle concentration, or the distribution of distances between nearest particles. However, a full characterization of a particle system is only provided by the infinite hierarchy of multiple-point density correlation functions. We derive an exact description of the full hierarchy of correlation functions for the diffusion-limited irreversible coalescence process A + A -> A.

cond-mat.stat-mech↗
arXiv1998

Diffusion-Limited Coalescence, A+A<-->A, with a Trap

We study diffusion-limited coalescence, A+A<-->A$, in one dimension, and derive an exact solution for the steady state in the presence of a trap. Without the trap, the system arrives at an equilibrium state which satisfies detailed balance, and can therefore be analyzed by classical equilibrium methods. The trap introduces an irreversible element, and the stationary state is no longer an equilibrium state. The exact solution is compared to that of a reaction-diffusion equation --- the habitual approximation method of choice. The exact solution allows us to determine the rate coefficients in the reaction-diffusion equation, without appealing to renormalization group techniques.

cond-mat.stat-mech↗
arXiv1995

Universality Class of Two-Offspring Branching Annihilating Random Walks

We analyze a two-offspring Branching Annihilating Random Walk ($n=2$ BAW) model, with finite annihilation rate. The finite annihilation rate allows for a dynamical phase transition between a vacuum, absorbing state and a non-empty, active steady state. We find numerically that this transition belongs to the same universality class as BAW's with an even number of offspring, $n\geq 4$, and that of other models whose dynamic rules conserve the parity of the particles locally. The simplicity of the model is exploited in computer simulations to obtain various critical exponents with a high level of accuracy.

cond-mat↗
arXiv1995

Inter-Particle Distribution Functions for One-Species Diffusion-Limited Annihilation, A+A->0

Diffusion-limited annihilation, $A+A\to 0$, and coalescence, $A+A\to A$, may both be exactly analyzed in one dimension. While the concentrations of $A$ particles in the two processes bear a simple relation, the inter-particle distribution functions (IPDF) exhibit remarkable differences. However, the IPDF is known exactly only for the coalescence process. We obtain the IPDF for the annihilation process, based on the Glauber spin approach and assuming that the IPDF's of nearest-particle pairs are statistically independent. This assumption is supported by computer simulations. Our analysis sheds further light on the relationship between the annihilation and the coalescence models.

cond-mat↗
arXiv1995

Two-Species Annihilation with Drift: A Model with Continuous Concentration-Decay Exponents

We propose a model for diffusion-limited annihilation of two species, $A+B\to A$ or $B$, where the motion of the particles is subject to a drift. For equal initial concentrations of the two species, the density follows a power-law decay for large times. However, the decay exponent varies continuously as a function of the probability of which particle, the hopping one or the target, survives in the reaction. These results suggest that diffusion-limited reactions subject to drift do not fall into a limited number of universality classes.

cond-mat↗
arXiv1994

Diffusion-Limited Coalescence with Finite Reaction Rates in One Dimension

We study the diffusion-limited process $A+A\to A$ in one dimension, with finite reaction rates. We develop an approximation scheme based on the method of Inter-Particle Distribution Functions (IPDF), which was formerly used for the exact solution of the same process with infinite reaction rate. The approximation becomes exact in the very early time regime (or the reaction-controlled limit) and in the long time (diffusion-controlled) asymptotic limit. For the intermediate time regime, we obtain a simple interpolative behavior between these two limits. We also study the coalescence process (with finite reaction rates) with the back reaction $A\to A+A$, and in the presence of particle input. In each of these cases the system reaches a non-trivial steady state with a finite concentration of particles. Theoretical predictions for the concentration time dependence and for the IPDF are compared to computer simulations. P. A. C. S. Numbers: 82.20.Mj 02.50.+s 05.40.+j 05.70.Ln

cond-mat↗
arXiv1994

Propagation and Extinction in Branching Annihilating Random Walks

We investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite density wave, or extinction may occur, in which the number of particles vanishes in the long-time limit. The number parity conserving case where 2-offspring are produced in each branching event can be solved exactly for unit reaction probability, from which qualitative features of the transition between propagation and extinction, as well as intriguing parity-specific effects are elucidated. An approximate analysis is developed to treat this transition for general BAW processes. A scaling description suggests that the critical exponents which describe the vanishing of the particle density at the transition are unrelated to those of conventional models, such as Reggeon Field Theory. P. A. C. S. Numbers: 02.50.+s, 05.40.+j, 82.20.-w

cond-mat↗