arXiv subjects

E. Ben-Naim

Publications and source records attributed to E. Ben-Naim.

At least 55 records · Page 3Linked to original sources

arXiv2007

Singular Energy Distributions in Granular Media

We study the kinetic theory of driven granular gases, taking into account both translational and rotational degrees of freedom. We obtain the high-energy tail of the stationary bivariate energy distribution, depending on the total energy E and the ratio x=sqrt{E_w/E} of rotational energy E_w to total energy. Extremely energetic particles have a unique and well-defined distribution f(x) which has several remarkable features: x is not uniformly distributed as in molecular gases; f(x) is not smooth but has multiple singularities. The latter behavior is sensitive to material properties such as the collision parameters, the moment of inertia and the collision rate. Interestingly, there are preferred ratios of rotational-to-total energy. In general, f(x) is strongly correlated with energy and the deviations from a uniform distribution grow with energy. We also solve for the energy distribution of freely cooling Maxwell Molecules and find qualitatively similar behavior.

cond-mat.soft↗

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

How to Choose a Champion

League competition is investigated using random processes and scaling techniques. In our model, a weak team can upset a strong team with a fixed probability. Teams play an equal number of head-to-head matches and the team with the largest number of wins is declared to be the champion. The total number of games needed for the best team to win the championship with high certainty, T, grows as the cube of the number of teams, N, i.e., T ~ N^3. This number can be substantially reduced using preliminary rounds where teams play a small number of games and subsequently, only the top teams advance to the next round. When there are k rounds, the total number of games needed for the best team to emerge as champion, T_k, scales as follows, T_k ~N^(γ_k) with gamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 for k=1,2,3. These results suggest an algorithm for how to infer the best team using a schedule that is linear in N. We conclude that league format is an ineffective method of determining the best team, and that sequential elimination from the bottom up is fair and efficient.

physics.soc-ph↗

arXiv2006

Scaling in Tournaments

We study a stochastic process that mimics single-game elimination tournaments. In our model, the outcome of each match is stochastic: the weaker player wins with upset probability q<=1/2, and the stronger player wins with probability 1-q. The loser is eliminated. Extremal statistics of the initial distribution of player strengths governs the tournament outcome. For a uniform initial distribution of strengths, the rank of the winner, x_*, decays algebraically with the number of players, N, as x_* ~ N^(-beta). Different decay exponents are found analytically for sequential dynamics, beta_seq=1-2q, and parallel dynamics, beta_par=1+[ln (1-q)]/[ln 2]. The distribution of player strengths becomes self-similar in the long time limit with an algebraic tail. Our theory successfully describes statistics of the US college basketball national championship tournament.

cond-mat.stat-mech↗

arXiv2006

Nonlinear Integral-Equation Formulation of Orthogonal Polynomials

The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure x w(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed.

math-ph↗

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

Parity and Predictability of Competitions

We present an extensive statistical analysis of the results of all sports competitions in five major sports leagues in England and the United States. We characterize the parity among teams by the variance in the winning fraction from season-end standings data and quantify the predictability of games by the frequency of upsets from game results data. We introduce a novel mathematical model in which the underdog team wins with a fixed upset probability. This model quantitatively relates the parity among teams with the predictability of the games, and it can be used to estimate the upset frequency from standings data.

physics.soc-ph↗

arXiv2006

Dynamics of Multi-Player Games

We analyze the dynamics of competitions with a large number of players. In our model, n players compete against each other and the winner is decided based on the standings: in each competition, the mth ranked player wins. We solve for the long time limit of the distribution of the number of wins for all n and m and find three different scenarios. When the best player wins, the standings are most competitive as there is one-tier with a clear differentiation between strong and weak players. When an intermediate player wins, the standings are two-tier with equally-strong players in the top tier and clearly-separated players in the lower tier. When the worst player wins, the standings are least competitive as there is one tier in which all of the players are equal. This behavior is understood via scaling analysis of the nonlinear evolution equations.

physics.soc-ph↗

arXiv2006

Experimental characterization of vibrated granular rings

We report an experimental study of the statistical properties of vibrated granular rings. In this system, a linked rod and bead metallic chain in the form of a ring is collisionally excited by a vertically oscillating plate. The dynamics are driven primarily by inelastic bead-plate collisions and are simultaneously constrained by the rings' physical connectedness. By imaging many instances of the ring configurations, we measure the ensemble averages and distributions of several physical characteristics on the scale of individual beads and composite ring. We study local properties such as inter-bead seperation and inter-bonds angles, and global properties such as the radius of gyration and center of mass motion. We characterize scaling with respect to the size of the chain.

cond-mat.soft↗

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

What is the most competitive sport?

We present an extensive statistical analysis of the results of all sports competitions in five major sports leagues in England and the United States. We characterize the parity among teams by the variance in the winning fraction from season-end standings data and quantify the predictability of games by the frequency of upsets from game results data. We introduce a mathematical model in which the underdog team wins with a fixed upset probability. This model quantitatively relates the parity among teams with the predictability of the games, and it can be used to estimate the upset frequency from standings data. We propose the likelihood of upsets as a measure of competitiveness.

physics.data-an↗

arXiv2005

On The Structure of Competitive Societies

We model the dynamics of social structure by a simple interacting particle system. The social standing of an individual agent is represented by an integer-valued fitness that changes via two offsetting processes. When two agents interact one advances: the fitter with probability p and the less fit with probability 1-p. The fitness of an agent may also decline with rate r. From a scaling analysis of the underlying master equations for the fitness distribution of the population, we find four distinct social structures as a function of the governing parameters p and r. These include: (i) a static lower-class society where all agents have finite fitness; (ii) an upwardly-mobile middle-class society; (iii) a hierarchical society where a finite fraction of the population belongs to a middle class and a complementary fraction to the lower class; (iv) an egalitarian society where all agents are upwardly mobile and have nearly the same fitness. We determine the basic features of the fitness distributions in these four phases.

physics.soc-ph↗

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

Velocity Distributions of Granular Gases with Drag and with Long-Range Interactions

We study velocity statistics of electrostatically driven granular gases. For two different experiments: (i) non-magnetic particles in a viscous fluid and (ii) magnetic particles in air, the velocity distribution is non-Maxwellian, and its high-energy tail is exponential, P(v) ~ exp(-|v|). This behavior is consistent with kinetic theory of driven dissipative particles. For particles immersed in a fluid, viscous damping is responsible for the exponential tail, while for magnetic particles, long-range interactions cause the exponential tail. We conclude that velocity statistics of dissipative gases are sensitive to the fluid environment and to the form of the particle interaction.

cond-mat.soft↗

arXiv2005

Power-law velocity distributions in granular gases

We report a general class of steady and transient states of granular gases. We find that the kinetic theory of inelastic gases admits stationary solutions with a power-law velocity distribution, f(v) ~ v^(-sigma). The exponent sigma is found analytically and depends on the spatial dimension, the degree of inelasticity, and the homogeneity degree of the collision rate. Driven steady-states, with the same power-law tail and a cut-off can be maintained by injecting energy at a large velocity scale, which then cascades to smaller velocities where it is dissipated. Associated with these steady-states are freely cooling time-dependent states for which the cut-off decreases and the velocity distribution is self-similar.

cond-mat.soft↗

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↗