401 research outputs found
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
Self-gravitating Brownian particles in two dimensions: the case of N=2 particles
We study the motion of N=2 overdamped Brownian particles in gravitational
interaction in a space of dimension d=2. This is equivalent to the simplified
motion of two biological entities interacting via chemotaxis when time delay
and degradation of the chemical are ignored. This problem also bears some
similarities with the stochastic motion of two point vortices in viscous
hydrodynamics [Agullo & Verga, Phys. Rev. E, 63, 056304 (2001)]. We
analytically obtain the density probability of finding the particles at a
distance r from each other at time t. We also determine the probability that
the particles have coalesced and formed a Dirac peak at time t (i.e. the
probability that the reduced particle has reached r=0 at time t). Finally, we
investigate the variance of the distribution and discuss the proper form
of the virial theorem for this system. The reduced particle has a normal
diffusion behaviour for small times with a gravity-modified diffusion
coefficient =r_0^2+(4k_B/\xi\mu)(T-T_*)t, where k_BT_{*}=Gm_1m_2/2 is a
critical temperature, and an anomalous diffusion for large times
~t^(1-T_*/T). As a by-product, our solution also describes the growth of
the Dirac peak (condensate) that forms in the post-collapse regime of the
Smoluchowski-Poisson system (or Keller-Segel model) for T<T_c=GMm/(4k_B). We
find that the saturation of the mass of the condensate to the total mass is
algebraic in an infinite domain and exponential in a bounded domain.Comment: Revised version (20/5/2010) accepted for publication in EPJ
Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion
For a specific choice of the diffusion, the parabolic-elliptic
Patlak-Keller-Segel system with non-linear diffusion (also referred to as the
quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold
phenomenon: there is a critical mass such that all the solutions with
initial data of mass smaller or equal to exist globally while the
solution blows up in finite time for a large class of initial data with mass
greater than . Unlike in space dimension 2, finite mass self-similar
blowing-up solutions are shown to exist in space dimension
The spatial correlations in the velocities arising from a random distribution of point vortices
This paper is devoted to a statistical analysis of the velocity fluctuations
arising from a random distribution of point vortices in two-dimensional
turbulence. Exact results are derived for the correlations in the velocities
occurring at two points separated by an arbitrary distance. We find that the
spatial correlation function decays extremely slowly with the distance. We
discuss the analogy with the statistics of the gravitational field in stellar
systems.Comment: 37 pages in RevTeX format (no figure); submitted to Physics of Fluid
Evolutionary dynamics of the most populated genotype on rugged fitness landscapes
We consider an asexual population evolving on rugged fitness landscapes which
are defined on the multi-dimensional genotypic space and have many local
optima. We track the most populated genotype as it changes when the population
jumps from a fitness peak to a better one during the process of adaptation.
This is done using the dynamics of the shell model which is a simplified
version of the quasispecies model for infinite populations and standard
Wright-Fisher dynamics for large finite populations. We show that the
population fraction of a genotype obtained within the quasispecies model and
the shell model match for fit genotypes and at short times, but the dynamics of
the two models are identical for questions related to the most populated
genotype. We calculate exactly several properties of the jumps in infinite
populations some of which were obtained numerically in previous works. We also
present our preliminary simulation results for finite populations. In
particular, we measure the jump distribution in time and find that it decays as
as in the quasispecies problem.Comment: Minor changes. To appear in Phys Rev
Data-driven discovery of stochastic dynamical equations of collective motion
Coarse-grained descriptions of collective motion of flocking systems are
often derived for the macroscopic or the thermodynamic limit. However, many
real flocks are small sized (10 to 100 individuals), called the mesoscopic
scales, where stochasticity arising from the finite flock sizes is important.
Developing mesoscopic scale equations, typically in the form of stochastic
differential equations, can be challenging even for the simplest of the
collective motion models. Here, we take a novel data-driven equation learning
approach to construct the stochastic mesoscopic descriptions of a simple
self-propelled particle (SPP) model of collective motion. In our SPP model, a
focal individual can interact with k randomly chosen neighbours within an
interaction radius. We consider k = 1 (called stochastic pairwise
interactions), k = 2 (stochastic ternary interactions), and k equalling all
available neighbours within the interaction radius (equivalent to Vicsek-like
local averaging). The data-driven mesoscopic equations reveal that the
stochastic pairwise interaction model produces a novel form of collective
motion driven by a multiplicative noise term (hence termed, noise-induced
flocking). In contrast, for higher order interactions (k > 1), including
Vicsek-like averaging interactions, yield collective motion driven primarily by
the deterministic forces. We find that the relation between the parameters of
the mesoscopic equations describing the dynamics and the population size are
sensitive to the density and to the interaction radius, exhibiting deviations
from mean-field theoretical expectations. We provide semi-analytic arguments
potentially explaining these observed deviations. In summary, our study
emphasizes the importance of mesoscopic descriptions of flocking systems and
demonstrates the potential of the data-driven equation discovery methods for
complex systems studies
On War: The Dynamics of Vicious Civilizations
The dynamics of ``vicious'', continuously growing civilizations (domains),
which engage in ``war'' whenever two domains meet, is investigated. In the war
event, the smaller domain is annihilated, while the larger domain is reduced in
size by a fraction \e of the casualties of the loser. Here \e quantifies
the fairness of the war, with \e=1 corresponding to a fair war with equal
casualties on both side, and \e=0 corresponding to a completely unfair war
where the winner suffers no casualties. In the heterogeneous version of the
model, evolution begins from a specified initial distribution of domains, while
in the homogeneous system, there is a continuous and spatially uniform input of
point domains, in addition to the growth and warfare. For the heterogeneous
case, the rate equations are derived and solved, and comparisons with numerical
simulations are made. An exact solution is also derived for the case of equal
size domains in one dimension. The heterogeneous system is found to coarsen,
with the typical cluster size growing linearly in time and the number
density of domains decreases as . For the homogeneous system, two
different long-time behaviors arise as a function of \e. When 1/2<\e\leq 1
(relatively fair wars), a steady state arises which is characterized by
egalitarian competition between domains of comparable size. In the limiting
case of \e=1, rate equations which simultaneously account for the
distribution of domains and that of the intervening gaps are derived and
solved. The steady state is characterized by domains whose age is typically
much larger than their size. When 0\leq\e<1/2 (unfair wars), a few
``superpowers'' ultimately dominate. Simulations indicate that this coarsening
process is characterized by power-law temporal behavior, with non-universalComment: 43 pages, plain TeX, 12 figures included, gzipped and uuencode
Exponents appearing in heterogeneous reaction-diffusion models in one dimension
We study the following 1D two-species reaction diffusion model : there is a
small concentration of B-particles with diffusion constant in an
homogenous background of W-particles with diffusion constant ; two
W-particles of the majority species either coagulate ()
or annihilate () with the respective
probabilities and ; a B-particle and a
W-particle annihilate () with probability 1. The
exponent describing the asymptotic time decay of
the minority B-species concentration can be viewed as a generalization of the
exponent of persistent spins in the zero-temperature Glauber dynamics of the 1D
-state Potts model starting from a random initial condition : the
W-particles represent domain walls, and the exponent
characterizes the time decay of the probability that a diffusive "spectator"
does not meet a domain wall up to time . We extend the methods introduced by
Derrida, Hakim and Pasquier ({\em Phys. Rev. Lett.} {\bf 75} 751 (1995); Saclay
preprint T96/013, to appear in {\em J. Stat. Phys.} (1996)) for the problem of
persistent spins, to compute the exponent in perturbation
at first order in for arbitrary and at first order in
for arbitrary .Comment: 29 pages. The three figures are not included, but are available upon
reques
Topological correlations in soap froths
Correlation in two-dimensional soap froth is analysed with an effective
potential for the first time. Cells with equal number of sides repel (with
linear correlation) while cells with different number of sides attract (with
NON-bilinear) for nearest neighbours, which cannot be explained by the maximum
entropy argument. Also, the analysis indicates that froth is correlated up to
the third shell neighbours at least, contradicting the conventional ideas that
froth is not strongly correlated.Comment: 10 Pages LaTeX, 6 Postscript figure
Analytical results for random walk persistence
In this paper, we present the detailed calculation of the persistence
exponent for a nearly-Markovian Gaussian process , a problem
initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the
probability that the walker never crosses the origin. New resummed perturbative
and non-perturbative expressions for are obtained, which suggest a
connection with the result of the alternative independent interval
approximation (IIA). The perturbation theory is extended to the calculation of
for non-Gaussian processes, by making a strong connection between the
problem of persistence and the calculation of the energy eigenfunctions of a
quantum mechanical problem. Finally, we give perturbative and non-perturbative
expressions for the persistence exponent , describing the
probability that the process remains bigger than .Comment: 23 pages; accepted for publication to Phys. Rev. E (Dec. 98
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