arXiv subjects

P. L. Krapivsky

Publications and source records attributed to P. L. Krapivsky.

At least 289 records · Page 16Linked to original sources

arXiv1997

Stationary Velocity Distributions in Traffic Flows

We introduce a traffic flow model that incorporates clustering and passing. We obtain analytically the steady state characteristics of the flow from a Boltzmann-like equation. A single dimensionless parameter, R=c_0v_0t_0 with c_0 the concentration, v_0 the velocity range, and 1/t_0 the passing rate, determines the nature of the steady state. When R<<1, uninterrupted flow with single cars occurs. When R>>1, large clusters with average mass ~ R^α form, and the flux is J ~ R^{-γ}. The initial distribution of slow cars governs the statistics. When P_0(v) ~ v^μ as v->0, the scaling exponents are γ=1/(μ+2), α=1/2 when μ>0, and α=(μ+1)/(μ+2) when μ<0.

cond-mat.stat-mech↗

arXiv1997

Wealth Distributions in Models of Capital Exchange

A dynamical model of capital exchange is introduced in which a specified amount of capital is exchanged between two individuals when they meet. The resulting time dependent wealth distributions are determined for a variety of exchange rules. For ``greedy'' exchange, an interaction between a rich and a poor individual results in the rich taking a specified amount of capital from the poor. When this amount is independent of the capitals of the two traders, a mean-field analysis yields a Fermi-like scaled wealth distribution in the long-time limit. This same distribution also arises in greedier exchange processes, where the interaction rate is an increasing function of the capital difference of the two traders. The wealth distribution in multiplicative processes, where the amount of capital exchanged is a finite fraction of the capital of one of the traders, are also discussed. For random multiplicative exchange, a steady state wealth distribution is reached, while in greedy multiplicative exchange a non-steady power law wealth distribution arises, in which the support of the distribution continuously increases. Finally, extensions of our results to arbitrary spatial dimension and to growth processes, where capital is created in an interaction, are presented.

cond-mat.stat-mech↗

arXiv1997

Ballistic Coalescence Model

We study statistical properties of a one dimensional infinite system of coalescing particles. Each particle moves with constant velocity $\pm v$ towards its closest neighbor and merges with it upon collision. We propose a mean-field theory that confirms a $t^{-1}$ concentration decay obtained in simulations and provides qualitative description for the densities of growing, constant, and shrinking inter-particle gaps.

cond-mat.stat-mech↗

arXiv1997

Domain Statistics in Coarsening Systems

We study the domain number and size distributions in the one-dimensional Ising and $q$-state Potts models subject to zero-temperature Glauber dynamics. The survival probability of a domain, $S(t)\sim t^{-ψ}$, and an unreacted domain, $Q_1(t)\sim t^{-δ}$, are characterized by two independent nonrtrivial exponents. For the Ising case, we find $ψ=0.126$ and $δ=1.27$ using numerical simulations. We develop an independent interval approximation (IIA) that predicts the qualitative behavior of the domain distribution and provides good estimates for the exponents. Exact results for the domain distribution are also obtained in several solvable cases.

cond-mat.stat-mech↗

arXiv1996

Slowly Divergent Drift in the Field-Driven Lorentz Gas

The dynamics of a point charged particle which is driven by a uniform external electric field and moves in a medium of elastic scatterers is investigated. Using rudimentary approaches, we reproduce, in one dimension, the known results that the typical speed grows with time as t^{1/3} and that the leading behavior of the velocity distribution is exp(-|v|^3/t). In spatial dimension d>1, we develop an effective medium theory which provides a simple and comprehensive description for the motion of a test particle. This approach predicts that the typical speed grows as t^{1/3} for all d, while the speed distribution is given by the scaling form P(u,t)= ^{-1}f(u/ ), where u=|v|^{3/2}, ~t^{1/2}, and f(z) is proportional to z^{(d-1)/3}exp(-z^2/2). For a periodic Lorentz gas with an infinite horizon, e. g., for a hypercubic lattice of scatters, a logarithmic correction to the effective medium result is predicted; in particular, the typical speed grows as (t ln t)^{1/3}.

cond-mat.stat-mech↗

arXiv1996

Aging and its Distribution in Coarsening Processes

We investigate the age distribution function P(tau,t) in prototypical one-dimensional coarsening processes. Here P(tau,t) is the probability density that in a time interval (0,t) a given site was last crossed by an interface in the coarsening process at time tau. We determine P(tau,t) analytically for two cases, the (deterministic) two-velocity ballistic annihilation process, and the (stochastic) infinite-state Potts model with zero temperature Glauber dynamics. Surprisingly, we find that in the scaling limit, P(tau,t) is identical for these two models. We also show that the average age, i. e., the average time since a site was last visited by an interface, grows linearly with the observation time t. This latter property is also found in the one-dimensional Ising model with zero temperature Glauber dynamics. We also discuss briefly the age distribution in dimension d greater than or equal to 2.

cond-mat↗

arXiv1996

Deterministic Soluble Model of Coarsening

We investigate a 3-phase deterministic one-dimensional phase ordering model in which interfaces move ballistically and annihilate upon colliding. We determine analytically the autocorrelation function A(t). This is done by computing generalized first-passage type probabilities P_n(t) which measure the fraction of space crossed by exactly n interfaces during the time interval (0,t), and then expressing the autocorrelation function via P_n's. We further reveal the spatial structure of the system by analyzing the domain size distribution.

cond-mat↗

arXiv1996

Spatial organization in cyclic Lotka-Volterra systems

We study the evolution of a system of $N$ interacting species which mimics the dynamics of a cyclic food chain. On a one-dimensional lattice with N<5 species, spatial inhomogeneities develop spontaneously in initially homogeneous systems. The arising spatial patterns form a mosaic of single-species domains with algebraically growing size, $\ell(t)\sim t^α$, where $α=3/4$ (1/2) and 1/3 for N=3 with sequential (parallel) dynamics and N=4, respectively. The domain distribution also exhibits a self-similar spatial structure which is characterized by an additional length scale, ${\cal L}(t)\sim t^β$, with $β=1$ and 2/3 for N=3 and 4, respectively. For $N\geq 5$, the system quickly reaches a frozen state with non interacting neighboring species. We investigate the time distribution of the number of mutations of a site using scaling arguments as well as an exact solution for N=3. Some possible extensions of the system are analyzed.

cond-mat↗

arXiv1996

Fractal formation and ordering in random sequential adsorption

We reveal the fractal nature of patterns arising in random sequential adsorption of particles with continuum power-law size distribution, $P(R)\sim R^{α-1}$, $R \le R_{\rm max}$. We find that the patterns become more and more ordered as $α$ increases, and that the Apollonian packing is obtained at $α\to \infty $ limit. We introduce the entropy production rate as a quantitative criteria of regularity and observe a transition from an irregular regime of the pattern formation to a regular one. We develop a scaling theory that relates kinetic and structural properties of the system.

cond-mat↗

arXiv1996

Two-Scale Annihilation

The kinetics of single-species annihilation, $A+A\to 0$, is investigated in which each particle has a fixed velocity which may be either $\pm v$ with equal probability, and a finite diffusivity. In one dimension, the interplay between convection and diffusion leads to a decay of the density which is proportional to $t^{-3/4}$. At long times, the reactants organize into domains of right- and left-moving particles, with the typical distance between particles in a single domain growing as $t^{3/4}$, and the distance between domains growing as $t$. The probability that an arbitrary particle reacts with its $n^{\rm th}$ neighbor is found to decay as $n^{-5/2}$ for same-velocity pairs and as $n^{-7/4}$ for $+-$ pairs. These kinetic and spatial exponents and their interrelations are obtained by scaling arguments. Our predictions are in excellent agreement with numerical simulations.

cond-mat↗

arXiv1996

Kinetics of a Diffusive Capture Process: Lamb Besieged by a Pride of Lions

The survival probability, S_N(t), of a diffusing prey (``lamb'') in the proximity of N diffusing predators (a ``pride of lions'') in one dimension is investigated. When the lions are all to one side of the lamb, the survival probability decays as a non-universal power law, S_N(t) is proportional to t^{-beta_N}, with the decay exponent beta_N proportional to ln N. The crossover behavior as a function of the relative diffusivities of the lions and the lamb is also discussed. When N--->oo, the lamb survival probability exhibits a log-normal decay, exp(-ln^2 t).

cond-mat↗

arXiv1996

Exact results for nucleation-and-growth in one dimension

We study statistical properties of the Kolmogorov-Avrami-Johnson-Mehl nucleation-and-growth model in one dimension. We obtain exact results for the gap density as well as the island distribution. When all nucleation events occur simultaneously, the island distribution has discontinuous derivatives on the rays x_n(t)=nt, n=1,2,3... We introduce an accelerated growth mechanism where the velocity increases linearly with the island size. We solve for the inter-island gap density and show that the system reaches complete coverage in a finite time and that the near-critical behavior of the system is robust, i.e., it is insensitive to details such as the nucleation mechanism.

cond-mat↗

arXiv1996

On War: The Dynamics of Vicious Civilizations

The dynamics of ``vicious'', continuously growing civilizations (domains), which engage in ``war'' whenever two domains meet, is investigated. In the war event, the smaller domain is annihilated, while the larger domain is reduced in size by a fraction $\e$ of the casualties of the loser. Here $\e$ quantifies the fairness of the war, with $\e=1$ corresponding to a fair war with equal casualties on both side, and $\e=0$ corresponding to a completely unfair war where the winner suffers no casualties. In the heterogeneous version of the model, evolution begins from a specified initial distribution of domains, while in the homogeneous system, there is a continuous and spatially uniform input of point domains, in addition to the growth and warfare. For the heterogeneous case, the rate equations are derived and solved, and comparisons with numerical simulations are made. An exact solution is also derived for the case of equal size domains in one dimension. The heterogeneous system is found to coarsen, with the typical cluster size growing linearly in time $t$ and the number density of domains decreases as $1/t$. For the homogeneous system, two different long-time behaviors arise as a function of $\e$. When $1/2<\e\leq 1$ (relatively fair wars), a steady state arises which is characterized by egalitarian competition between domains of comparable size. In the limiting case of $\e=1$, rate equations which simultaneously account for the distribution of domains and that of the intervening gaps are derived and solved. The steady state is characterized by domains whose age is typically much larger than their size. When $0\leq\e<1/2$ (unfair wars), a few ``superpowers'' ultimately dominate. Simulations indicate that this coarsening process is characterized by power-law temporal behavior, with non-universal

cond-mat↗

arXiv1995

Coarsening and persistence in the voter model

We investigate coarsening and persistence in the voter model by introducing the quantity $P_n(t)$, defined as the fraction of voters who changed their opinion n times up to time t. We show that $P_n(t)$ exhibits scaling behavior that strongly depends on the dimension as well as on the initial opinion concentrations. Exact results are obtained for the average number of opinion changes, , and the autocorrelation function, $A(t)\equiv \sum (-1)^n P_n\sim t^{-d/2}$ in arbitrary dimension d. These exact results are complemented by a mean-field theory, heuristic arguments and numerical simulations. For dimensions d>2, the system does not coarsen, and the opinion changes follow a nearly Poissonian distribution, in agreement with mean-field theory. For dimensions d<=2, the distribution is given by a different scaling form, which is characterized by nontrivial scaling exponents. For unequal opinion concentrations, an unusual situation occurs where different scaling functions correspond to the majority and the minority, as well as for even and odd n.

cond-mat↗

arXiv1995

Space covering by growing rays

We study kinetic and jamming properties of a space covering process in one dimension. The stochastic process is defined as follows: Seeds are nucleated randomly in space and produce rays which grow with a constant velocity. The growth stops upon collision with another ray. For arbitrary distributions of the growth velocity, the exact coverage, velocity and size distributions are evaluated for both simultaneous and continuous nucleation. In general, simultaneous nucleation exhibits a stronger dependence on the details of the growth velocity distribution in the asymptotic time regime. The coverage in the continuous case exhibits a universal $1/t$ approach to the jammed state, while an inhomogeneous version of the process leads to nonuniversal $t^{-p_+}$ decay, with $0\le p_+\le 1$ the fraction of right growing rays.

cond-mat↗

arXiv1995

Exact Results for Kinetics of Catalytic Reactions

The kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined. In the limit of infinitesimal reaction rate (reaction-controlled limit), we solve the dimer-dimer surface reaction model (or voter model) exactly in arbitrary dimension $D$. The density of reactive interfaces is found to exhibit a power law decay for $D<2$ and a slow logarithmic decay in two dimensions. We discuss the relevance of these results for the monomer-monomer surface reaction model.

cond-mat↗

arXiv1995

Kinetics of Aggregation-Annihilation Processes

We investigate the kinetics of many-species systems with aggregation of similar species clusters and annihilation of opposite species clusters. We find that the interplay between aggregation and annihilation leads to rich kinetic behaviors and unusual conservation laws. On the mean-field level, an exact solution for the cluster-mass distribution is obtained. Asymptotically, this solution exhibits a novel scaling form if the initial species densities are the same while in the general case of unequal densities the process approaches single species aggregation. The theoretical predictions are compared with numerical simulations in 1D, 2D, and 3D. Nontrivial growth exponents characterize the mass distribution in one dimension.

cond-mat↗

arXiv1995

Extremal Properties of Random Systems

We find that the probability distribution for the largest intervals $p(l)$ exhibits universal properties for different systems including random walk and random cutting models. In particular, $p(l)$ has an infinite set of singularities at $l=1/k$ with $k=2,3,\ldots$ which become weaker and weaker as $k \to \infty$; additionally, $p(l)$ has an essential singularity at $l=0$. These properties are found also in many dimensional situation.

cond-mat↗