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P. L. Krapivsky

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

At least 235 records · Page 13Linked to original sources
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

Dynamics of Majority Rule

We introduce a 2-state opinion dynamics model where agents evolve by majority rule. In each update, a group of agents is specified whose members then all adopt the local majority state. In the mean-field limit, where a group consists of randomly-selected agents, consensus is reached in a time that scales ln N, where N is the number of agents. On finite-dimensional lattices, where a group is a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependent power of N. The upper critical dimension appears to be larger than 4. The final opinion always equals that of the initial majority except in one dimension.

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

Kinetic Anomalies in Addition-Aggregation Processes

We investigate irreversible aggregation in which monomer-monomer, monomer-cluster, and cluster-cluster reactions occur with constant but distinct rates K_{MM}, K_{MC}, and K_{CC}, respectively. The dynamics crucially depends on the ratio gamma=K_{CC}/K_{MC} and secondarily on epsilon=K_{MM}/K_{MC}. For epsilon=0 and gamma<2, there is conventional scaling in the long-time limit, with a single mass scale that grows linearly in time. For gamma >= 2, there is unusual behavior in which the concentration of clusters of mass k, c_k decays as a stretched exponential in time within a boundary layer k k*. When epsilon>0, analogous behaviors emerge for gamma<2 and gamma >= 2.

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

Freezing and Slow Evolution in a Constrained Opinion Dynamics Model

We study opinion formation in a population that consists of leftists, centrists, and rightist. In an interaction between neighboring agents, a centrist and a leftist can become both centrists or leftists (and similarly for a centrist and a rightist). In contrast, leftists and rightists do not affect each other. The initial density of centrists rho_0 controls the evolution. With probability rho_0 the system reaches a centrist consensus, while with probability 1-rho_0 a frozen population of leftists and rightists results. In one dimension, we determine this frozen state and the opinion dynamics by mapping the system onto a spin-1 Ising model with zero-temperature Glauber kinetics. In the frozen state, the length distribution of single-opinion domains has an algebraic small-size tail x^{-2(1-psi)} and the average domain size grows as L^{2*psi}, where L is the system length. The approach to this frozen state is governed by a t^{-psi} long-time tail with psi-->2*rho_0/pi as rho_0-->0.

cond-mat.stat-mech↗
arXiv2002

Ordering of Random Walks: The Leader and the Laggard

We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\cal L}_N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability {\cal R}_N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N-1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for {\cal L}_N(t) for N=4, the first case that is not exactly soluble: {\cal L}_4(t) ~ t^{-β_4}, with β_4=0.91342(8). The probability of being the laggard also decays algebraically, {\cal R}_N(t) ~ t^{-γ_N}; we derive γ_2=1/2, γ_3=3/8, and argue that γ_N--> ln N/N$ as N-->oo.

cond-mat.stat-mech↗
arXiv2002

Statistics of Lead Changes in Popularity-Driven Systems

We study statistical properties of the highest degree, or most popular, nodes in growing networks. We show that the number of lead changes increases logarithmically with network size N, independent of the details of the growth mechanism. The probability that the first node retains the lead approaches a finite constant for popularity-driven growth, and decays as N^{-phi}(ln N)^{-1/2}, with phi=0.08607..., for growth with no popularity bias.

cond-mat.stat-mech↗
arXiv2002

Infinite-Order Percolation and Giant Fluctuations in a Protein Interaction Network

We investigate a model protein interaction network whose links represent interactions between individual proteins. This network evolves by the functional duplication of proteins, supplemented by random link addition to account for mutations. When link addition is dominant, an infinite-order percolation transition arises as a function of the addition rate. In the opposite limit of high duplication rate, the network exhibits giant structural fluctuations in different realizations. For biologically-relevant growth rates, the node degree distribution has an algebraic tail with a peculiar rate dependence for the associated exponent.

cond-mat.stat-mech↗
arXiv2002

Finiteness and Fluctuations in Growing Networks

We study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks. We first determine the exact degree distribution of finite networks by generating function approaches. The resulting distributions exhibit an unusual finite-size scaling behavior and they are also sensitive to the initial conditions. We argue that fluctuations in the number of nodes of degree k become Gaussian for fixed degree as the size of the network diverges. We also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution.

cond-mat.stat-mech↗
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↗
arXiv2002

Impurity in a Granular Fluid

We investigate velocity statistics of an impurity immersed in a uniform granular fluid. An analytic solution of the inelastic Maxwell model shows that a phase transition occurs in the cooling case. Light impurities have similar velocity statistics as the fluid background, although their temperature is generally different. Asymptotically, the temperature ratio increases with the impurity mass, and it diverges at some critical mass. Impurities heavier than this critical mass essentially scatter of a static fluid background. The impurity velocity distribution is generally characterized by multiscaling asymptotic behavior of its moments. Additionally, in the heavy impurity phase, it consists of modified replicas of the initial distribution.

cond-mat.stat-mech↗
arXiv2002

A Statistical Physics Perspective on Web Growth

Approaches from statistical physics are applied to investigate the structure of network models whose growth rules mimic aspects of the evolution of the world-wide web. We first determine the degree distribution of a growing network in which nodes are introduced one at a time and attach to an earlier node of degree k with rate A_ksim k^gamma. Very different behaviors arise for gamma<1, gamma=1, and gamma>1. We also analyze the degree distribution of a heterogeneous network, the joint age-degree distribution, the correlation between degrees of neighboring nodes, as well as global network properties. An extension to directed networks is then presented. By tuning model parameters to reasonable values, we obtain distinct power-law forms for the in-degree and out-degree distributions with exponents that are in good agreement with current data for the web. Finally, a general growth process with independent introduction of nodes and links is investigated. This leads to independently growing sub-networks that may coalesce with other sub-networks. General results for both the size distribution of sub-networks and the degree distribution are obtained.

cs.NI↗
arXiv2002

Kinetic Theory of Traffic Flows

We describe traffic flows in one lane roadways using kinetic theory, with special emphasis on the role of quenched randomness in the velocity distributions. When passing is forbidden, growing clusters are formed behind slow cars and the cluster velocity distribution is governed by an exact Boltzmann equation which is linear and has an infinite memory. The distributions of the cluster size and the cluster velocity exhibit scaling behaviors, with exponents dominated solely by extremal characteristics of the intrinsic velocity distribution. When passing is allowed, the system approaches a steady state, whose nature is determined by a single dimensionless number, the ratio of the passing time to the collision time, the two time scales in the problem. The flow exhibits two regimes, a laminar flow regime, and a congested regime where large slow clusters dominate the flow. A phase transition separates these two regimes when only the next-to-leading car can pass.

cond-mat.stat-mech↗
arXiv2002

Synchronization and Coarsening (without SOC) in a Forest-Fire Model

We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length grows linearly with time as t-->oo. The number density of patches of length L, N(L,t), scales as ^{-2}M(L/ ), and within a mean-field rate equation description we find that this scaling function decays as e^{-1/x} for x-->0, and as e^{-x} for x-->oo. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.

cond-mat.stat-mech↗