293 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
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
Soccer: is scoring goals a predictable Poissonian process?
The non-scientific event of a soccer match is analysed on a strictly
scientific level. The analysis is based on the recently introduced concept of a
team fitness (Eur. Phys. J. B 67, 445, 2009) and requires the use of
finite-size scaling. A uniquely defined function is derived which
quantitatively predicts the expected average outcome of a soccer match in terms
of the fitness of both teams. It is checked whether temporary fitness
fluctuations of a team hamper the predictability of a soccer match.
To a very good approximation scoring goals during a match can be
characterized as independent Poissonian processes with pre-determined
expectation values. Minor correlations give rise to an increase of the number
of draws. The non-Poissonian overall goal distribution is just a consequence of
the fitness distribution among different teams. The limits of predictability of
soccer matches are quantified. Our model-free classification of the underlying
ingredients determining the outcome of soccer matches can be generalized to
different types of sports events
Ballistic Annihilation
Ballistic annihilation with continuous initial velocity distributions is
investigated in the framework of Boltzmann equation. The particle density and
the rms velocity decay as and , with the
exponents depending on the initial velocity distribution and the spatial
dimension. For instance, in one dimension for the uniform initial velocity
distribution we find . We also solve the Boltzmann equation
for Maxwell particles and very hard particles in arbitrary spatial dimension.
These solvable cases provide bounds for the decay exponents of the hard sphere
gas.Comment: 4 RevTeX pages and 1 Eps figure; submitted to Phys. Rev. Let
Kinetics of Clustering in Traffic Flows
We study a simple aggregation model that mimics the clustering of traffic on
a one-lane roadway. In this model, each ``car'' moves ballistically at its
initial velocity until it overtakes the preceding car or cluster. After this
encounter, the incident car assumes the velocity of the cluster which it has
just joined. The properties of the initial distribution of velocities in the
small velocity limit control the long-time properties of the aggregation
process. For an initial velocity distribution with a power-law tail at small
velocities, \pvim as , a simple scaling argument shows that the
average cluster size grows as n \sim t^{\va} and that the average velocity
decays as v \sim t^{-\vb} as . We derive an analytical solution
for the survival probability of a single car and an asymptotically exact
expression for the joint mass-velocity distribution function. We also consider
the properties of spatially heterogeneous traffic and the kinetics of traffic
clustering in the presence of an input of cars.Comment: 18 pages, Plain TeX, 2 postscript figure
Maxwell Model of Traffic Flows
We investigate traffic flows using the kinetic Boltzmann equations with a
Maxwell collision integral. This approach allows analytical determination of
the transient behavior and the size distributions. The relaxation of the car
and cluster velocity distributions towards steady state is characterized by a
wide range of velocity dependent relaxation scales, , with
the ratio of the passing and the collision rates. Furthermore, these
relaxation time scales decrease with the velocity, with the smallest scale
corresponding to the decay of the overall density. The steady state cluster
size distribution follows an unusual scaling form . This distribution is primarily algebraic, , for , and is exponential otherwise.Comment: revtex, 10 page
Inelastically scattering particles and wealth distribution in an open economy
Using the analogy with inelastic granular gasses we introduce a model for
wealth exchange in society. The dynamics is governed by a kinetic equation,
which allows for self-similar solutions. The scaling function has a power-law
tail, the exponent being given by a transcendental equation. In the limit of
continuous trading, closed form of the wealth distribution is calculated
analytically.Comment: 8 pages 5 figure
Persistence with Partial Survival
We introduce a parameter , called partial survival, in the persistence of
stochastic processes and show that for smooth processes the persistence
exponent changes continuously with , being the usual
persistence exponent. We compute exactly for a one-dimensional
deterministic coarsening model, and approximately for the diffusion equation.
Finally we develop an exact, systematic series expansion for , in
powers of , for a general Gaussian process with finite density of
zero crossings.Comment: 5 pages, 2 figures, references added, to appear in Phys.Rev.Let
The different paths to entropy
In order to undestand how the complex concept of entropy emerged,we propose a
trip towards the past reviewing the works of Clausius, Boltzmann, Gibbs and
Planck. In particular, since the Gibbs's work is not very well known, we
present a detailed analysis, recalling the three definitions of the entropy
that Gibbs gives. May be one of the most important aspect of the entropy is to
see it as a thermodynamic potential like the other thermodynamic potentials as
proposed by Callen. We close with some remarks on entropy and irreversibility.Comment: 32 page
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