893 research outputs found
Stationary states and energy cascades in inelastic gases
We find a general class of nontrivial stationary states in inelastic gases
where, due to dissipation, energy is transfered from large velocity scales to
small velocity scales. These steady-states exist for arbitrary collision rules
and arbitrary dimension. Their signature is a stationary velocity distribution
f(v) with an algebraic high-energy tail, f(v) ~ v^{-sigma}. The exponent sigma
is obtained analytically and it varies continuously with the spatial dimension,
the homogeneity index characterizing the collision rate, and the restitution
coefficient. We observe these stationary states in numerical simulations in
which energy is injected into the system by infrequently boosting particles to
high velocities. We propose that these states may be realized experimentally in
driven granular systems.Comment: 4 pages, 4 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
Maxima of Two Random Walks: Universal Statistics of Lead Changes
We investigate statistics of lead changes of the maxima of two discrete-time
random walks in one dimension. We show that the average number of lead changes
grows as in the long-time limit. We present theoretical and
numerical evidence that this asymptotic behavior is universal. Specifically,
this behavior is independent of the jump distribution: the same asymptotic
underlies standard Brownian motion and symmetric Levy flights. We also show
that the probability to have at most n lead changes behaves as for Brownian motion and as for symmetric Levy
flights with index . The decay exponent varies continuously
with the Levy index when .Comment: 7 pages, 6 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
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
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.Comment: 4 pages, 2 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
Percolation with Multiple Giant Clusters
We study the evolution of percolation with freezing. Specifically, we
consider cluster formation via two competing processes: irreversible
aggregation and freezing. We find that when the freezing rate exceeds a certain
threshold, the percolation transition is suppressed. Below this threshold, the
system undergoes a series of percolation transitions with multiple giant
clusters ("gels") formed. Giant clusters are not self-averaging as their total
number and their sizes fluctuate from realization to realization. The size
distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~
k^{-3}. We propose freezing as a practical mechanism for controlling the gel
size.Comment: 4 pages, 3 figure
Random Geometric Series
Integer sequences where each element is determined by a previous randomly
chosen element are investigated analytically. In particular, the random
geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments
grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical
behavior is x_n n^ln 2. The probability distribution is obtained explicitly in
terms of the Stirling numbers of the first kind and it approaches a log-normal
distribution asymptotically.Comment: 6 pages, 2 figure
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