105 research outputs found
Random equations in aerodynamics
Literature was reviewed to identify aerodynamic models which might be treated by probablistic methods. The numerical solution of some integral equations that arise in aerodynamical problems were investigated. On the basis of the numerical studies a qualitative theory of random integral equations was developed to provide information on the behavior of the solutions of these equations (in particular, boundary and asymptotic behavior, and stability) and their statistical properties without actually obtaining explicit solutions of the equations
QED cascades induced by circularly polarized laser fields
The results of Monte-Carlo simulations of electron-positron-photon cascades
initiated by slow electrons in circularly polarized fields of ultra-high
strength are presented and discussed. Our results confirm previous qualitative
estimations [A.M. Fedotov, et al., PRL 105, 080402 (2010)] of the formation of
cascades. This sort of cascades has revealed the new property of the
restoration of energy and dynamical quantum parameter due to the acceleration
of electrons and positrons by the field and may become a dominating feature of
laser-matter interactions at ultra-high intensities. Our approach incorporates
radiation friction acting on individual electrons and positrons.Comment: 13 pages, 10 figure
Correlations between zeros of a random polynomial
We obtain exact analytical expressions for correlations between real zeros of
the Kac random polynomial. We show that the zeros in the interval are
asymptotically independent of the zeros outside of this interval, and that the
straightened zeros have the same limit translation invariant correlations. Then
we calculate the correlations between the straightened zeros of the SO(2)
random polynomial.Comment: 31 pages, 2 figures; a revised version of the J. Stat. Phys. pape
Multifractal Multiplicity Distribution in Bunching-Parameter Analysis
A new multiplicity distribution with multifractal properties which can be
used in high-energy physics and quantum optics is proposed. It may be
considered as a generalization of the negative-binomial distribution. We find
the structure of the generating function for such distribution and discuss its
properties.Comment: LaTex, 12 pages, cite.st
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Finite-size scaling of directed percolation in the steady state
Recently, considerable progress has been made in understanding finite-size
scaling in equilibrium systems. Here, we study finite-size scaling in
non-equilibrium systems at the instance of directed percolation (DP), which has
become the paradigm of non-equilibrium phase transitions into absorbing states,
above, at and below the upper critical dimension. We investigate the
finite-size scaling behavior of DP analytically and numerically by considering
its steady state generated by a homogeneous constant external source on a
d-dimensional hypercube of finite edge length L with periodic boundary
conditions near the bulk critical point. In particular, we study the order
parameter and its higher moments using renormalized field theory. We derive
finite-size scaling forms of the moments in a one-loop calculation. Moreover,
we introduce and calculate a ratio of the order parameter moments that plays a
similar role in the analysis of finite size scaling in absorbing nonequilibrium
processes as the famous Binder cumulant in equilibrium systems and that, in
particular, provides a new signature of the DP universality class. To
complement our analytical work, we perform Monte Carlo simulations which
confirm our analytical results.Comment: 21 pages, 6 figure
Natural boundaries for the Smoluchowski equation and affiliated diffusion processes
The Schr\"{o}dinger problem of deducing the microscopic dynamics from the
input-output statistics data is known to admit a solution in terms of Markov
diffusions. The uniqueness of solution is found linked to the natural
boundaries respected by the underlying random motion. By choosing a reference
Smoluchowski diffusion process, we automatically fix the Feynman-Kac potential
and the field of local accelerations it induces. We generate the family of
affiliated diffusions with the same local dynamics, but different inaccessible
boundaries on finite, semi-infinite and infinite domains. For each diffusion
process a unique Feynman-Kac kernel is obtained by the constrained (Dirichlet
boundary data) Wiener path integration.As a by-product of the discussion, we
give an overview of the problem of inaccessible boundaries for the diffusion
and bring together (sometimes viewed from unexpected angles) results which are
little known, and dispersed in publications from scarcely communicating areas
of mathematics and physics.Comment: Latex file, Phys. Rev. E 49, 3815-3824, (1994
Boundary-layer turbulence as a kangaroo process
A nonlocal mixing-length theory of turbulence transport by finite size eddies is developed by means of a novel evaluation of the Reynolds stress. The analysis involves the contruct of a sample path space and a stochastic closure hypothesis. The simplifying property of exhange (strong eddies) is satisfied by an analytical sampling rate model. A nonlinear scaling relation maps the path space onto the semi-infinite boundary layer. The underlying near-wall behavior of fluctuating velocities perfectly agrees with recent direct numerical simulations. The resulting integro-differential equation for the mixing of scalar densities represents fully developed boundary-layer turbulence as a nondiffusive (Kubo-Anderson or kangaroo) type of stochastic process. The model involves a scaling exponent (with → in the diffusion limit). For the (partly analytical) solution for the mean velocity profile, excellent agreement with the experimental data yields 0.58. © 1995 The American Physical Society
On the expected number of internal equilibria in random evolutionary games with correlated payoff matrix
The analysis of equilibrium points in random games has been of great interest
in evolutionary game theory, with important implications for understanding of
complexity in a dynamical system, such as its behavioural, cultural or
biological diversity. The analysis so far has focused on random games of
independent payoff entries. In this paper, we overcome this restrictive
assumption by considering multi-player two-strategy evolutionary games where
the payoff matrix entries are correlated random variables. Using techniques
from the random polynomial theory we establish a closed formula for the mean
numbers of internal (stable) equilibria. We then characterise the asymptotic
behaviour of this important quantity for large group sizes and study the effect
of the correlation. Our results show that decreasing the correlation among
payoffs (namely, of a strategist for different group compositions) leads to
larger mean numbers of (stable) equilibrium points, suggesting that the system
or population behavioural diversity can be promoted by increasing independence
of the payoff entries. Numerical results are provided to support the obtained
analytical results.Comment: Revision from the previous version; 27 page
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