660 research outputs found
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
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
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.Comment: 8 pages, 8 figure
Statistics of Partial Minima
Motivated by multi-objective optimization, we study extrema of a set of N
points independently distributed inside the d-dimensional hypercube. A point in
this set is k-dominated by another point when at least k of its coordinates are
larger, and is a k-minimum if it is not k-dominated by any other point. We
obtain statistical properties of these partial minima using exact probabilistic
methods and heuristic scaling techniques. The average number of partial minima,
A, decays algebraically with the total number of points, A ~ N^{-(d-k)/k}, when
1<=k<d. Interestingly, there are k-1 distinct scaling laws characterizing the
largest coordinates as the distribution P(y_j) of the jth largest coordinate,
y_j, decays algebraically, P(y_j) ~ (y_j)^{-alpha_j-1}, with
alpha_j=j(d-k)/(k-j) for 1<=j<=k-1. The average number of partial minima grows
logarithmically, A ~ [1/(d-1)!](ln N)^{d-1}, when k=d. The full distribution of
the number of minima is obtained in closed form in two-dimensions.Comment: 6 pages, 1 figur
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
Discrete Analog of the Burgers Equation
We propose the set of coupled ordinary differential equations
dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers
equation. We focus on traveling waves and triangular waves, and find that these
special solutions of the discrete system capture major features of their
continuous counterpart. In particular, the propagation velocity of a traveling
wave and the shape of a triangular wave match the continuous behavior. However,
there are some subtle differences. For traveling waves, the propagating front
can be extremely sharp as it exhibits double exponential decay. For triangular
waves, there is an unexpected logarithmic shift in the location of the front.
We establish these results using asymptotic analysis, heuristic arguments, and
direct numerical integration.Comment: 6 pages, 5 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
Entropic Tightening of Vibrated Chains
We investigate experimentally the distribution of configurations of a ring
with an elementary topological constraint, a ``figure-8'' twist. Using vibrated
granular chains, which permit controlled preparation and direct observation of
such a constraint, we show that configurations where one of the loops is tight
and the second is large are strongly preferred. This agrees with recent
predictions for equilibrium properties of topologically-constrained polymers.
However, the dynamics of the tightening process weakly violate detailed
balance, a signature of the nonequilibrium nature of this system.Comment: 4 pages, 4 figure
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.Comment: 6 pages, 7 figure
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.Comment: 11 pages, 9 figure
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