1,024 research outputs found
Pattern Selection and Super-patterns in the Bounded Confidence Model
We study pattern formation in the bounded confidence model of opinion
dynamics. In this random process, opinion is quantified by a single variable.
Two agents may interact and reach a fair compromise, but only if their
difference of opinion falls below a fixed threshold. Starting from a uniform
distribution of opinions with compact support, a traveling wave forms and it
propagates from the domain boundary into the unstable uniform state.
Consequently, the system reaches a steady state with isolated clusters that are
separated by distance larger than the interaction range. These clusters form a
quasi-periodic pattern where the sizes of the clusters and the separations
between them are nearly constant. We obtain analytically the average separation
between clusters L. Interestingly, there are also very small quasi-periodic
modulations in the size of the clusters. The spatial periods of these
modulations are a series of integers that follow from the continued fraction
representation of the irrational average separation L.Comment: 6 pages, 6 figure
Dynamics of Three Agent Games
We study the dynamics and resulting score distribution of three-agent games
where after each competition a single agent wins and scores a point. A single
competition is described by a triplet of numbers , and denoting the
probabilities that the team with the highest, middle or lowest accumulated
score wins. We study the full family of solutions in the regime, where the
number of agents and competitions is large, which can be regarded as a
hydrodynamic limit. Depending on the parameter values , we find six
qualitatively different asymptotic score distributions and we also provide a
qualitative understanding of these results. We checked our analytical results
against numerical simulations of the microscopic model and find these to be in
excellent agreement. The three agent game can be regarded as a social model
where a player can be favored or disfavored for advancement, based on his/her
accumulated score. It is also possible to decide the outcome of a three agent
game through a mini tournament of two-a gent competitions among the
participating players and it turns out that the resulting possible score
distributions are a subset of those obtained for the general three agent-games.
We discuss how one can add a steady and democratic decline rate to the model
and present a simple geometric construction that allows one to write down the
corresponding score evolution equations for -agent games
Self-Similarity in Random Collision Processes
Kinetics of collision processes with linear mixing rules are investigated
analytically. The velocity distribution becomes self-similar in the long time
limit and the similarity functions have algebraic or stretched exponential
tails. The characteristic exponents are roots of transcendental equations and
vary continuously with the mixing parameters. In the presence of conservation
laws, the velocity distributions become universal.Comment: 4 pages, 4 figure
Kinetics of Heterogeneous Single-Species Annihilation
We investigate the kinetics of diffusion-controlled heterogeneous
single-species annihilation, where the diffusivity of each particle may be
different. The concentration of the species with the smallest diffusion
coefficient has the same time dependence as in homogeneous single-species
annihilation, A+A-->0. However, the concentrations of more mobile species decay
as power laws in time, but with non-universal exponents that depend on the
ratios of the corresponding diffusivities to that of the least mobile species.
We determine these exponents both in a mean-field approximation, which should
be valid for spatial dimension d>2, and in a phenomenological Smoluchowski
theory which is applicable in d<2. Our theoretical predictions compare well
with both Monte Carlo simulations and with time series expansions.Comment: TeX, 18 page
Popularity-Driven Networking
We investigate the growth of connectivity in a network. In our model,
starting with a set of disjoint nodes, links are added sequentially. Each link
connects two nodes, and the connection rate governing this random process is
proportional to the degrees of the two nodes. Interestingly, this network
exhibits two abrupt transitions, both occurring at finite times. The first is a
percolation transition in which a giant component, containing a finite fraction
of all nodes, is born. The second is a condensation transition in which the
entire system condenses into a single, fully connected, component. We derive
the size distribution of connected components as well as the degree
distribution, which is purely exponential throughout the evolution.
Furthermore, we present a criterion for the emergence of sudden condensation
for general homogeneous connection rates.Comment: 5 pages, 2 figure
Universal statistical properties of poker tournaments
We present a simple model of Texas hold'em poker tournaments which retains
the two main aspects of the game: i. the minimal bet grows exponentially with
time; ii. players have a finite probability to bet all their money. The
distribution of the fortunes of players not yet eliminated is found to be
independent of time during most of the tournament, and reproduces accurately
data obtained from Internet tournaments and world championship events. This
model also makes the connection between poker and the persistence problem
widely studied in physics, as well as some recent physical models of biological
evolution, and extreme value statistics.Comment: Final longer version including data from Internet and WPT tournament
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.Comment: 4 pages, 3 figure
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.Comment: 6 pages, 4 figure
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