3,325 research outputs found
Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition
We consider the spatially homogeneous Boltzmann equation for (true) hard
and moderately soft potentials. We assume that the initial condition is a
probability measure with finite energy and is not a Dirac mass. For hard
potentials, we prove that any reasonable weak solution immediately belongs to
some Besov space. For moderately soft potentials, we assume additionally that
the initial condition has a moment of sufficiently high order ( is enough)
and prove the existence of a solution that immediately belongs to some Besov
space. The considered solutions thus instantaneously become functions with a
finite entropy. We also prove that in any case, any weak solution is
immediately supported by .Comment: Published in at http://dx.doi.org/10.1214/14-AAP1012 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques
We review phase space techniques based on the Wigner representation that
provide an approximate description of dilute ultra-cold Bose gases. In this
approach the quantum field evolution can be represented using equations of
motion of a similar form to the Gross-Pitaevskii equation but with stochastic
modifications that include quantum effects in a controlled degree of
approximation. These techniques provide a practical quantitative description of
both equilibrium and dynamical properties of Bose gas systems. We develop
versions of the formalism appropriate at zero temperature, where quantum
fluctuations can be important, and at finite temperature where thermal
fluctuations dominate. The numerical techniques necessary for implementing the
formalism are discussed in detail, together with methods for extracting
observables of interest. Numerous applications to a wide range of phenomena are
presented.Comment: 110 pages, 32 figures. Updated to address referee comments. To appear
in Advances in Physic
Kac's chaos and Kac's program
In this note I present the main results about the quantitative and
qualitative propagation of chaos for the Boltzmann-Kac system obtained in
collaboration with C. Mouhot in \cite{MMinvent} which gives a possible answer
to some questions formulated by Kac in \cite{Kac1956}. We also present some
related recent results about Kac's chaos and Kac's program obtained in
\cite{MMWchaos,HaurayMischler,KleberSphere} by K. Carrapatoso, M. Hauray, C.
Mouhot, B. Wennberg and myself
Kinetics of the chiral phase transition in a linear model
We study the dynamics of the chiral phase transition in a linear quark-meson
model using a novel approach based on semiclassical wave-particle
duality. The quarks are treated as test particles in a Monte-Carlo simulation
of elastic collisions and the coupling to the meson, which is treated
as a classical field. The exchange of energy and momentum between particles and
fields is described in terms of appropriate Gaussian wave packets. It has been
demonstrated that energy-momentum conservation and the principle of detailed
balance are fulfilled, and that the dynamics leads to the correct equilibrium
limit. First schematic studies of the dynamics of matter produced in heavy-ion
collisions are presented.Comment: 15 pages, 12 figures, accepted by EPJA, dedicated to memory of Walter
Greiner; v2: corrected typos, added references and an acknowledgmen
Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade
We develop a new 3+1 dimensional Monte Carlo cascade solving the kinetic
on-shell Boltzmann equations for partons including the inelastic gg ggg
pQCD processes. The back reaction channel is treated -- for the first time --
fully consistently within this scheme. An extended stochastic method is used to
solve the collision integral. The frame dependence and convergency are studied
for a fixed tube with thermal initial conditions. The detailed numerical
analysis shows that the stochastic method is fully covariant and that
convergency is achieved more efficiently than within a standard geometrical
formulation of the collision term, especially for high gluon interaction rates.
The cascade is then applied to simulate parton evolution and to investigate
thermalization of gluons for a central Au+Au collision at RHIC energy. For this
study the initial conditions are assumed to be generated by independent
minijets with p_T > p_0=2 GeV. With that choice it is demonstrated that overall
kinetic equilibration is driven mainly by the inelastic processes and is
achieved on a scale of 1 fm/c. The further evolution of the expanding gluonic
matter in the central region then shows almost an ideal hydrodynamical
behavior. In addition, full chemical equilibration of the gluons follows on a
longer timescale of about 3 fm/c.Comment: 121 pages with 55 figures, revised version. Two eps-figures and
comments are added. Formula (54) which has typo in journal version is given
correctl
Quantitative uniform in time chaos propagation for Boltzmann collision processes
This paper is devoted to the study of mean-field limit for systems of
indistinguables particles undergoing collision processes. As formulated by Kac
\cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1)
prove and quantify this property for Boltzmann collision processes with
unbounded collision rates (hard spheres or long-range interactions), (2) prove
and quantify this property \emph{uniformly in time}. This yields the first
chaos propagation result for the spatially homogeneous Boltzmann equation for
true (without cut-off) Maxwell molecules whose "Master equation" shares
similarities with the one of a L\'evy process and the first {\em quantitative}
chaos propagation result for the spatially homogeneous Boltzmann equation for
hard spheres (improvement of the %non-contructive convergence result of
Sznitman \cite{S1}). Moreover our chaos propagation results are the first
uniform in time ones for Boltzmann collision processes (to our knowledge),
which partly answers the important question raised by Kac of relating the
long-time behavior of a particle system with the one of its mean-field limit,
and we provide as a surprising application a new proof of the well-known result
of gaussian limit of rescaled marginals of uniform measure on the
-dimensional sphere as goes to infinity (more applications will be
provided in a forthcoming work). Our results are based on a new method which
reduces the question of chaos propagation to the one of proving a purely
functional estimate on some generator operators ({\em consistency estimate})
together with fine stability estimates on the flow of the limiting non-linear
equation ({\em stability estimates})
Loewner Chains
These lecture notes on 2D growth processes are divided in two parts. The
first part is a non-technical introduction to stochastic Loewner evolutions
(SLEs). Their relationship with 2D critical interfaces is illustrated using
numerical simulations. Schramm's argument mapping conformally invariant
interfaces to SLEs is explained. The second part is a more detailed
introduction to the mathematically challenging problems of 2D growth processes
such as Laplacian growth, diffusion limited aggregation (DLA), etc. Their
description in terms of dynamical conformal maps, with discrete or continuous
time evolution, is recalled. We end with a conjecture based on possible
dendritic anomalies which, if true, would imply that the Hele-Shaw problem and
DLA are in different universality classes.Comment: 46 pages, 21 figure
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