2,515 research outputs found
A non-local problem for the Fokker-Planck equation related to the Becker-D\"{o}ring model
This paper concerns a Fokker-Planck equation on the positive real line
modeling nucleation and growth of clusters. The main feature of the equation is
the dependence of the driving vector field and boundary condition on a
non-local order parameter related to the excess mass of the system.
The first main result concerns the well-posedness and regularity of the
Cauchy problem. The well-posedness is based on a fixed point argument, and the
regularity on Schauder estimates. The first a priori estimates yield H\"older
regularity of the non-local order parameter, which is improved by an iteration
argument.
The asymptotic behavior of solutions depends on some order parameter
depending on the initial data. The system shows different behavior depending on
a value , determined from the potentials and diffusion coefficient.
For , there exists an equilibrium solution
. If the solution converges strongly to
, while if the solution converges
weakly to . The excess gets lost due
to the formation of larger and larger clusters. In this regard, the model
behaves similarly to the classical Becker-D\"oring equation.
The system possesses a free energy, strictly decreasing along the evolution,
which establishes the long time behavior. In the subcritical case
the entropy method, based on suitable weighted logarithmic Sobolev inequalities
and interpolation estimates, is used to obtain explicit convergence rates to
the equilibrium solution.
The close connection of the presented model and the Becker-D\"oring model is
outlined by a family of discrete Fokker-Planck type equations interpolating
between both of them. This family of models possesses a gradient flow
structure, emphasizing their commonality.Comment: Minor revised version accepted for publication in Discrete &
Continuous Dynamical Systems -
Strong Convergence to the Homogenized Limit of Parabolic Equations with Random Coefficients
This paper is concerned with the study of solutions to discrete parabolic
equations in divergence form with random coefficients, and their convergence to
solutions of a homogenized equation. It has previously been shown that if the
random environment is translational invariant and ergodic, then solutions of
the random equation converge under diffusive scaling to solutions of a
homogenized parabolic PDE. In this paper point-wise estimates are obtained on
the difference between the averaged solution to the random equation and the
solution to the homogenized equation for certain random environments which are
strongly mixing.Comment: 46 page
Young stars at large distances from the galactic plane: mechanisms of formation
We have collected from the literature a list of early-type stars, situated at
large distances from the galactic plane, for which evidence of youth seems
convincing. We discuss two possible formation mechanisms for these stars:
ejection from the plane by dynamical interactions within small clusters, and
formation away from the plane, via induced shocks created by spiral density
waves. We identify the stars that could be explained by each mechanism. We
conclude that the ejection mechanism can account for about two thirds of the
stars, while a combination of star formation at z = 500-800 pc from the plane
and ejection, can account for 90 percent of these stars. Neither mechanism, nor
both together, can explain the most extreme examples.Comment: 6 pages, No figures. Sixth Pacific Rim Conference on Stellar
Astrophysics - A tribute to Helmut Abt, (Kluwer
Towards Realistic String Vacua From Branes At Singularities
We report on progress towards constructing string models incorporating both
realistic D-brane matter content and moduli stabilisation with dynamical
low-scale supersymmetry breaking. The general framework is that of local
D-brane models embedded into the LARGE volume approach to moduli stabilisation.
We review quiver theories on del Pezzo () singularities including
both D3 and D7 branes. We provide supersymmetric examples with three
quark/lepton families and the gauge symmetries of the Standard, Left-Right
Symmetric, Pati-Salam and Trinification models, without unwanted chiral
exotics. We describe how the singularity structure leads to family symmetries
governing the Yukawa couplings which may give mass hierarchies among the
different generations. We outline how these models can be embedded into compact
Calabi-Yau compactifications with LARGE volume moduli stabilisation, and state
the minimal conditions for this to be possible. We study the general structure
of soft supersymmetry breaking. At the singularity all leading order
contributions to the soft terms (both gravity- and anomaly-mediation) vanish.
We enumerate subleading contributions and estimate their magnitude. We also
describe model-independent physical implications of this scenario. These
include the masses of anomalous and non-anomalous U(1)'s and the generic
existence of a new hyperweak force under which leptons and/or quarks could be
charged. We propose that such a gauge boson could be responsible for the ghost
muon anomaly recently found at the Tevatron's CDF detector.Comment: 40 pages, 10 figure
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