arXiv subjects

P. L. Krapivsky

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

At least 199 records · Page 11Linked to original sources

arXiv2008

Smoothing a Rock by Chipping

We investigate an idealized model for the size reduction and smoothing of a polygonal rock due to repeated chipping at corners. Each chip is sufficiently small so that only a single corner and a fraction of its two adjacent sides are cut from the object in a single chipping event. After many chips have been cut away, the resulting shape of the rock is generally anisotropic, with facet lengths and corner angles distributed over a broad range. Although a well-defined shape is quickly reached for each realization, there are large fluctuations between realizations.

cond-mat.stat-mech↗

arXiv2008

Dynamics of Microtubule Growth and Catastrophe

We investigate a simple model of microtubule dynamics in which a microtubule evolves by: (i) attachment of guanosine triphosphate (GTP) to its end at rate lambda, (ii) GTP converting irreversibly to guanosine diphosphate (GDP) at rate 1, and (iii) detachment of GDP from the end of a microtubule at rate mu. As a function of these elemental rates, the microtubule can grow steadily or its length can fluctuate wildly. A master equation approach is developed to characterize these intriguing features. For mu=0, we find exact expressions for tubule and GTP cap length distributions, as well as a power-law length distributions of GTP and GDP islands. For mu=oo, we find the average time between catastrophes, where the microtubule shrinks to zero length, and extend this approach to also determine the size distribution of avalanches (sequence of consecutive GDP detachment events). We obtain the phase diagram for general rates and verify our predictions by numerical simulations.

q-bio.QM↗

arXiv2008

Phase Transition with Non-Thermodynamic States in Reversible Polymerization

We investigate a reversible polymerization process in which individual polymers aggregate and fragment at a rate proportional to their molecular weight. We find a nonequilibrium phase transition despite the fact that the dynamics are perfectly reversible. When the strength of the fragmentation process exceeds a critical threshold, the system reaches a thermodynamic steady state where the total number of polymers is proportional to the system size. The polymer length distribution has a sharp exponential tail in this case. When the strength of the fragmentation process falls below the critical threshold, the steady state becomes non-thermodynamic as the total number of polymers grows sub-linearly with the system size. Moreover, the length distribution has an algebraic tail and the characteristic exponent varies continuously with the fragmentation rate.

cond-mat.stat-mech↗

arXiv2007

Transition from small to large world in growing networks

We examine the global organization of growing networks in which a new vertex is attached to already existing ones with a probability depending on their age. We find that the network is infinite- or finite-dimensional depending on whether the attachment probability decays slower or faster than $(age)^{-1}$. The network becomes one-dimensional when the attachment probability decays faster than $(age)^{-2}$. We describe structural characteristics of these phases and transitions between them.

cond-mat.stat-mech↗

arXiv2007

Molecular Spiders in One Dimension

Molecular spiders are synthetic bio-molecular systems which have "legs" made of short single-stranded segments of DNA. Spiders move on a surface covered with single-stranded DNA segments complementary to legs. Different mappings are established between various models of spiders and simple exclusion processes. For spiders with simple gait and varying number of legs we compute the diffusion coefficient; when the hopping is biased we also compute their velocity.

cond-mat.stat-mech↗

arXiv2007

Molecular Spiders with Memory

Synthetic bio-molecular spiders with "legs" made of single-stranded segments of DNA can move on a surface which is also covered by single-stranded segments of DNA complementary to the leg DNA. In experimental realizations, when a leg detaches from a segment of the surface for the first time it alters that segment, and legs subsequently bound to these altered segments more weakly. Inspired by these experiments we investigate spiders moving along a one-dimensional substrate, whose legs leave newly visited sites at a slower rate than revisited sites. For a random walk (one-leg spider) the slowdown does not effect the long time behavior. For a bipedal spider, however, the slowdown generates an effective bias towards unvisited sites, and the spider behaves similarly to the excited walk. Surprisingly, the slowing down of the spider at new sites increases the diffusion coefficient and accelerates the growth of the number of visited sites.

cond-mat.stat-mech↗

arXiv2007

Addition-Deletion Networks

We study structural properties of growing networks where both addition and deletion of nodes are possible. Our model network evolves via two independent processes. With rate r, a node is added to the system and this node links to a randomly selected existing node. With rate 1, a randomly selected node is deleted, and its parent node inherits the links of its immediate descendants. We show that the in-component size distribution decays algebraically, c_k ~ k^{-beta}, as k-->infty. The exponent beta=2+1/(r-1) varies continuously with the addition rate r. Structural properties of the network including the height distribution, the diameter of the network, the average distance between two nodes, and the fraction of dangling nodes are also obtained analytically. Interestingly, the deletion process leads to a giant hub, a single node with a macroscopic degree whereas all other nodes have a microscopic degree.

cond-mat.stat-mech↗

arXiv2006

Probing Non-Integer Dimensions

We show that two-dimensional convection-diffusion problems with a radial sink or source at the origin may be recast as a pure diffusion problem in a fictitious space in which the spatial dimension is continuously-tunable with the Peclet number. This formulation allows us to probe various diffusion-controlled processes in non-integer dimensions.

physics.chem-ph↗

arXiv2006

Flows on Graphs with Random Capacities

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold that depends on the distribution of capacities. We then examine the maximal total flux from the root to the leaves. Our methods generalize to simple graphs with loops, e.g., to hierarchical lattices and to complete graphs.

cond-mat.stat-mech↗

arXiv2006

Condensates in Driven Aggregation Processes

We investigate aggregation driven by mass injection. In this stochastic process, mass is added with constant rate r and clusters merge at a constant total rate 1, so that both the total number of clusters and the total mass steadily grow with time. Analytic results are presented for the three classic aggregation rates K_{i,j} between clusters of size i and j. When K_{i,j}=const, the cluster size distribution decays exponentially. When K_{i,j} (i+j) or K_{i,j} (ij), there are two phases: (i) a condensate phase with a condensate containing a finite fraction of the mass in the system as well as finite clusters, and (ii) a cluster phase with finite clusters only. For K_{i,j} (i+j), the cluster size distribution, c_k, has a power-law tail, c_k~k^{-gamma} in either phase. The exponent is a non-monotonic function of the injection rate: gamma=r/(r-1) in the condensate phase, r<2, and γ=r in the cluster phase, r>2.

cond-mat.stat-mech↗

arXiv2006

Social Balance on Networks: The Dynamics of Friendship and Enmity

How do social networks evolve when both friendly and unfriendly relations exist? Here we propose a simple dynamics for social networks in which the sense of a relationship can change so as to eliminate imbalanced triads--relationship triangles that contains 1 or 3 unfriendly links. In this dynamics, a friendly link changes to unfriendly or vice versa in an imbalanced triad to make the triad balanced. Such networks undergo a dynamic phase transition from a steady state to "utopia"--all friendly links--as the amount of network friendliness is changed. Basic features of the long-time dynamics and the phase transition are discussed.

physics.gen-ph↗

arXiv2006

Weak Disorder in Fibonacci Sequences

We study how weak disorder affects the growth of the Fibonacci series. We introduce a family of stochastic sequences that grow by the normal Fibonacci recursion with probability 1-epsilon, but follow a different recursion rule with a small probability epsilon. We focus on the weak disorder limit and obtain the Lyapunov exponent, that characterizes the typical growth of the sequence elements, using perturbation theory. The limiting distribution for the ratio of consecutive sequence elements is obtained as well. A number of variations to the basic Fibonacci recursion including shift, doubling, and copying are considered.

cond-mat.stat-mech↗

arXiv2005

Alignment of Rods and Partition of Integers

We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.

cond-mat.stat-mech↗

arXiv2005

Rank Statistics in Biological Evolution

We present a statistical analysis of biological evolution processes. Specifically, we study the stochastic replication-mutation-death model where the population of a species may grow or shrink by birth or death, respectively, and additionally, mutations lead to the creation of new species. We rank the various species by the chronological order by which they originate. The average population N_k of the kth species decays algebraically with rank, N_k ~ M^{mu} k^{-mu}, where M is the average total population. The characteristic exponent mu=(alpha-gamma)/(alpha+beta-gamma)$ depends on alpha, beta, and gamma, the replication, mutation, and death rates. Furthermore, the average population P_k of all descendants of the kth species has a universal algebraic behavior, P_k ~ M/k.

q-bio.PE↗

arXiv2005

Polymerization with Freezing

Irreversible aggregation processes involving reactive and frozen clusters are investigated using the rate equation approach. In aggregation events, two clusters join irreversibly to form a larger cluster, and additionally, reactive clusters may spontaneously freeze. Frozen clusters do not participate in merger events. Generally, freezing controls the nature of the aggregation process, as demonstrated by the final distribution of frozen clusters. The cluster mass distribution has a power-law tail, F_k ~ k^{-gamma}, when the freezing process is sufficiently slow. Different exponents, gamma=1, 3 are found for the constant and the product aggregation rates, respectively. For the latter case, the standard polymerization model, either no gels, or a single gel, or even multiple gels may be produced.

cond-mat.stat-mech↗

arXiv2005

Dynamics of Social Balance on Networks

We study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes. The network is endowed with dynamics in which the sense of a link in an imbalanced triad--a triangular loop with 1 or 3 unfriendly links--is reversed to make the triad balanced. With this dynamics, an infinite network undergoes a dynamic phase transition from a steady state to "paradise"--all links are friendly--as the propensity p for friendly links in an update event passes through 1/2. A finite network always falls into a socially-balanced absorbing state where no imbalanced triads remain. If the additional constraint that the number of imbalanced triads in the network does not increase in an update is imposed, then the network quickly reaches a balanced final state.

cond-mat.stat-mech↗

arXiv2005

A "Burnt Bridge'' Brownian Ratchet

Motivated by a biased diffusion of molecular motors with the bias dependent on the state of the substrate, we investigate a random walk on a one-dimensional lattice that contains weak links (called "bridges'') which are affected by the walker. Namely, a bridge is destroyed with probability p when the walker crosses it; the walker is not allowed to cross it again and this leads to a directed motion. The velocity of the walker is determined analytically for equidistant bridges. The special case of p=1 is more tractable -- both the velocity and the diffusion constant are calculated for uncorrelated locations of bridges, including periodic and random distributions.

cond-mat.stat-mech↗

arXiv2005

Kinetic Theory of Random Graphs

Statistical properties of evolving random graphs are analyzed using kinetic theory. Treating the linking process dynamically, structural characteristics such as links, paths, cycles, and components are obtained analytically using the rate equation approach. Scaling laws for finite systems are derived using extreme statistics and scaling arguments.

cond-mat.stat-mech↗