arXiv subjects

P. L. Krapivsky

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

At least 253 records · Page 14Linked to original sources

arXiv2002

Scaling, Multiscaling, and Nontrivial Exponents in Inelastic Collision Processes

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large velocity tail, P(v,t) ~ v^{-sigma}. The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension, d, and the dissipation coefficient, epsilon. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, A(t)= ~ t^{-alpha}, with a non-universal dissipation-dependent exponent alpha=1/epsilon. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

cond-mat.stat-mech↗

arXiv2001

Nontrivial Velocity Distributions in Inelastic Gases

We study freely evolving and forced inelastic gases using the Boltzmann equation. We consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient epsilon. In the freely evolving case, we find that the velocity distribution decays algebraically, P(v,t) ~ v^{-sigma} for sufficiently large velocities. We derive the exponent sigma(d,epsilon), which exhibits nontrivial dependence on both d and epsilon, exactly. In the forced case, the velocity distribution approaches a steady-state with a Gaussian large velocity tail.

cond-mat.stat-mech↗

arXiv2001

Extreme Value Statistics and Traveling Fronts: An Application to Computer Science

We study the statistics of height and balanced height in the binary search tree problem in computer science. The search tree problem is first mapped to a fragmentation problem which is then further mapped to a modified directed polymer problem on a Cayley tree. We employ the techniques of traveling fronts to solve the polymer problem and translate back to derive exact asymptotic properties in the original search tree problem. The second mapping allows us not only to re-derive the already known results for random binary trees but to obtain new exact results for search trees where the entries arrive according to an arbitrary distribution, not necessarily randomly. Besides it allows us to derive the asymptotic shape of the full probability distribution of height and not just its moments. Our results are then generalized to $m$-ary search trees with arbitrary distribution. An attempt has been made to make the article accessible to both physicists and computer scientists.

cond-mat.stat-mech↗

arXiv2001

Stable Distributions in Stochastic Fragmentation

We investigate a class of stochastic fragmentation processes involving stable and unstable fragments. We solve analytically for the fragment length density and find that a generic algebraic divergence characterizes its small-size tail. Furthermore, the entire range of acceptable values of decay exponent consistent with the length conservation can be realized. We show that the stochastic fragmentation process is non-self-averaging as moments exhibit significant sample-to-sample fluctuations. Additionally, we find that the distributions of the moments and of extremal characteristics possess an infinite set of progressively weaker singularities.

cond-mat.stat-mech↗

arXiv2001

Parity and Ruin in a Stochastic Game

We study an elementary two-player card game where in each round players compare cards and the holder of the smallest card wins. Using the rate equations approach, we treat the stochastic version of the game in which cards are drawn randomly. We obtain an exact solution for arbitrary initial conditions. In general, the game approaches a steady state where the card densities of the two players are proportional to each other. The leading small size behavior of the initial card densities determines the corresponding proportionality constant, while the next correction governs the asymptotic time dependence. The relaxation towards the steady state exhibits a rich behavior, e.g., it may be algebraically slow or exponentially fast. Moreover, in ruin situations where one player eventually wins all cards, the game may even end in a finite time.

nlin.AO↗

arXiv2001

Organization of Growing Random Networks

The organizational development of growing random networks is investigated. These growing networks are built by adding nodes successively and linking each to an earlier node of degree k with attachment probability A_k. When A_k grows slower than linearly with k, the number of nodes with k links, N_k(t), decays faster than a power-law in k, while for A_k growing faster than linearly in k, a single node emerges which connects to nearly all other nodes. When A_k is asymptotically linear, N_k(t) tk^{-nu}, with nu dependent on details of the attachment probability, but in the range 2<nu<infty. The combined age and degree distribution of nodes shows that old nodes typically have a large degree. There is also a significant correlation in the degrees of neighboring nodes, so that nodes of similar degree are more likely to be connected. The size distributions of the in-components and out-components of the network with respect to a given node -- namely, its "descendants" and "ancestors" -- are also determined. The in-component exhibits a robust s^{-2} power-law tail, where s is the component size. The out component has a typical size of order ln t and it provides basic insights about the genealogy of the network.

cond-mat.stat-mech↗

arXiv2001

Degree Distributions of Growing Networks

The in-degree and out-degree distributions of a growing network model are determined. The in-degree is the number of incoming links to a given node (and vice versa for out-degree. The network is built by (i) creation of new nodes which each immediately attach to a pre-existing node, and (ii) creation of new links between pre-existing nodes. This process naturally generates correlated in- and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained.

cond-mat.stat-mech↗

arXiv2001

Steady State of an Inhibitory Neural Network

We investigate the dynamics of a neural network where each neuron evolves according to the combined effects of deterministic integrate-and-fire dynamics and purely inhibitory coupling with K randomly-chosen "neighbors". The inhibition reduces the voltage of a given neuron by an amount Delta when one of its neighbors fires. The interplay between the integration and inhibition leads to a steady state which is determined by solving the rate equations for the neuronal voltage distribution. We also study the evolution of a single neuron and find that the mean lifetime between firing events equals 1+K*Delta and that the probability that a neuron has not yet fired decays exponentially with time.

nlin.AO↗

arXiv2001

Extremal properties of random trees

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights approaches a traveling wave form. The wave front in the minimal case is governed by the small-extremal-height tail of the distribution, and conversely, the front in the maximal case is governed by the large-extremal-height tail of the distribution. We determine several statistical characteristics of the extremal height distribution analytically. In particular, the expected minimal and maximal heights grow logarithmically with the tree size, N, hmin ~ vmin ln N, and hmax ~ vmax ln N, with vmin=0.373365 and vmax=4.31107, respectively. Corrections to this asymptotic behavior are of order O(ln ln N).

cond-mat.stat-mech↗

arXiv2001

Freezing in Ising Ferromagnets

We investigate the final state of zero-temperature Ising ferromagnets which are endowed with single-spin flip Glauber dynamics. Surprisingly, the ground state is generally not reached for zero initial magnetization. In two dimensions, the system either reaches a frozen stripe state with probability approximately equal to 1/3 or the ground state with probability approximately equal to 2/3. In greater than two dimensions, the probability to reach the ground state or a frozen state rapidly vanishes as the system size increases and the system wanders forever in an iso-energy set of metastable states. An external magnetic field changes the situation drastically -- in two dimensions the favorable ground state is always reached, while in three dimensions the field must exceed a threshold value to reach the ground state. For small but non-zero temperature, relaxation to the final state first proceeds by the formation of very long-lived metastable states, similar to the zero-temperature case, before equilibrium is reached.

cond-mat.stat-mech↗

arXiv2000

The Dynamics of Efficiency: A Simple Model

We propose a simple model that describes the dynamics of efficiencies of competing agents. Agents communicate leading to increase of efficiencies of underachievers, and an efficiency of each agent can increase or decrease irrespectively of other agents. When the rate of deleterious changes exceeds a certain threshold, the economy falls into a stagnant phase. In the opposite situation, the economy improves with asymptotically constant rate and the efficiency distribution has a finite width. The leading algebraic corrections to the asymptotic growth rate are also computed.

cond-mat.stat-mech↗

arXiv2000

Connectivity of Growing Random Networks

A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites which link to earlier sites with a probability A_k which depends on the number of pre-existing links k to that site. For homogeneous connection kernels, A_k ~ k^gamma, different behaviors arise for gamma<1, gamma>1, and gamma=1. For gamma<1, the number of sites with k links, N_k, varies as stretched exponential. For gamma>1, a single site connects to nearly all other sites. In the borderline case A_k ~ k, the power law N_k ~k^{-nu} is found, where the exponent nu can be tuned to any value in the range 2<nu<infinity.

cond-mat.stat-mech↗

arXiv2000

Fate of Zero-Temperature Ising Ferromagnets

We investigate the relaxation of homogeneous Ising ferromagnets on finite lattices with zero-temperature spin-flip dynamics. On the square lattice, a frozen two-stripe state is apparently reached approximately 1/4 of the time, while the ground state is reached otherwise. The asymptotic relaxation is characterized by two distinct time scales, with the longer stemming from the influence of a long-lived diagonal stripe ``defect''. In greater than two dimensions, the probability to reach the ground state rapidly vanishes as the size increases and the system typically ends up wandering forever within an iso-energy set of stochastically ``blinking'' metastable states.

cond-mat.stat-mech↗

arXiv2000

Fragmentation with a Steady Source

We investigate fragmentation processes with a steady input of fragments. We find that the size distribution approaches a stationary form which exhibits a power law divergence in the small size limit, P(x) ~ x^{-3}. This algebraic behavior is robust as it is independent of the details of the input as well as the spatial dimension. The full time dependent behavior is obtained analytically for arbitrary inputs, and is found to exhibit a universal scaling behavior.

cond-mat.stat-mech↗

arXiv2000

Extremal Paths on a Random Cayley Tree

We investigate the statistics of extremal path(s) (both the shortest and the longest) from the root to the bottom of a Cayley tree. The lengths of the edges are assumed to be independent identically distributed random variables drawn from a distribution ρ(l). Besides, the number of branches from any node is also random. Exact results are derived for arbitrary distribution ρ(l). In particular, for the binary {0,1} distribution ρ(l)=pδ_{l,1}+(1-p)δ_{l,0}, we show that as p increases, the minimal length undergoes an unbinding transition from a `localized' phase to a `moving' phase at the critical value, p=p_c=1-b^{-1}, where b is the average branch number of the tree. As the height n of the tree increases, the minimal length saturates to a finite constant in the localized phase (p p_c) where the velocity v_{min}(p) is determined via a front selection mechanism. At p=p_c, the minimal length grows with n in an extremely slow double logarithmic fashion. The length of the maximal path, on the other hand, increases linearly as v_{max}(p)n for all p. The maximal and minimal velocities satisfy a general duality relation, v_{min}(p)+v_{max}(1-p)=1, which is also valid for directed paths on finite-dimensional lattices.

cond-mat.stat-mech↗

arXiv2000

Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem

We study a random bisection problem where an initial interval of length x is cut into two random fragments at the first stage, then each of these two fragments is cut further, etc. We compute the probability P_n(x) that at the n-th stage, each of the 2^n fragments is shorter than 1. We show that P_n(x) approaches a traveling wave form, and the front position x_n increases as x_n\sim n^βρ^n for large n. We compute exactly the exponents ρ=1.261076... and β=0.453025.... as roots of transcendental equations. We also solve the m-section problem where each interval is broken into m fragments. In particular, the generalized exponents grow as ρ_m\approx m/(\ln m) and β_m\approx 3/(2\ln m) in the large m limit. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.

cond-mat.stat-mech↗

arXiv2000

Stochastic Aggregation: Rate Equations Approach

We investigate a class of stochastic aggregation processes involving two types of clusters: active and passive. The mass distribution is obtained analytically for several aggregation rates. When the aggregation rate is constant, we find that the mass distribution of passive clusters decays algebraically. Furthermore, the entire range of acceptable decay exponents is possible. For aggregation rates proportional to the cluster masses, we find that gelation is suppressed. In this case, the tail of the mass distribution decays exponentially for large masses, and as a power law over an intermediate size range.

cond-mat.stat-mech↗

arXiv2000

Stochastic Aggregation: Scaling Properties

We study scaling properties of stochastic aggregation processes in one dimension. Numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent nontrivial exponents corresponding to the survival probability of particles and monomers. The overall behavior agrees qualitatively with the mean-field theory. This theory also provides a useful approximation for the decay exponents, as well as the limiting mass distribution.

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