100 research outputs found
Chaos and stability in a two-parameter family of convex billiard tables
We study, by numerical simulations and semi-rigorous arguments, a
two-parameter family of convex, two-dimensional billiard tables, generalizing
the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A
17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard
tables are continuously deformed from the integrable circular billiard to
different versions of completely-chaotic stadia. In particular, we conjecture
that a new class of ergodic billiard tables is obtained in certain regions of
the two-dimensional parameter space, when the billiards are close to skewed
stadia. We provide heuristic arguments supporting this conjecture, and give
numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video
available at http://sistemas.fciencias.unam.mx/~dsanders
Asymmetric Primitive-Model Electrolytes: Debye-Huckel Theory, Criticality and Energy Bounds
Debye-Huckel (DH) theory is extended to treat two-component size- and
charge-asymmetric primitive models, focussing primarily on the 1:1 additive
hard-sphere electrolyte with, say, negative ion diameters, a--, larger than the
positive ion diameters, a++. The treatment highlights the crucial importance of
the charge-unbalanced ``border zones'' around each ion into which other ions of
only one species may penetrate. Extensions of the DH approach which describe
the border zones in a physically reasonable way are exact at high and low
density, , and, furthermore, are also in substantial agreement with
recent simulation predictions for \emph{trends} in the critical parameters,
and , with increasing size asymmetry. Conversely, the simplest
linear asymmetric DH description, which fails to account for physically
expected behavior in the border zones at low , can violate a new lower bound
on the energy (which applies generally to models asymmetric in both charge and
size). Other recent theories, including those based on the mean spherical
approximation, have predicted trends in the critical parameters quite opposite
to those established by the simulations.Comment: to appear in Physical Review
Density Fluctuations in an Electrolyte from Generalized Debye-Hueckel Theory
Near-critical thermodynamics in the hard-sphere (1,1) electrolyte is well
described, at a classical level, by Debye-Hueckel (DH) theory with (+,-) ion
pairing and dipolar-pair-ionic-fluid coupling. But DH-based theories do not
address density fluctuations. Here density correlations are obtained by
functional differentiation of DH theory generalized to {\it non}-uniform
densities of various species. The correlation length diverges universally
at low density as (correcting GMSA theory). When
one has as
where the amplitudes compare informatively with experimental data.Comment: 5 pages, REVTeX, 1 ps figure included with epsf. Minor changes,
references added. Accepted for publication in Phys. Rev. Let
Mixing rates of particle systems with energy exchange
A fundamental problem of non-equilibrium statistical mechanics is the
derivation of macroscopic transport equations in the hydrodynamic limit. The
rigorous study of such limits requires detailed information about rates of
convergence to equilibrium for finite sized systems. In this paper we consider
the finite lattice , with an energy \EnergyStateI{i} \in
(0,\infty) associated to each site. The energies evolve according to a Markov
jump process with nearest neighbor interaction such that the total energy is
preserved. We prove that for an entire class of such models the spectral gap of
the generator of the Markov process scales as \Order(N^{-2}). Furthermore, we
provide a complete classification of reversible stationary distributions of
product type. We demonstrate that our results apply to models similar to the
billiard lattice model considered in \cite{10297039,10863485}, and hence
provide a first step in the derivation of a macroscopic heat equation for a
microscopic stochastic evolution of mechanical origin
Three-loop QCD corrections to B_s -> mu^+ mu^-
The decay B_s -> mu^+ mu^- in the Standard Model is generated by the
well-known W-box and Z-penguin diagrams that give rise to an effective
quark-lepton operator Q_A at low energies. We compute QCD corrections of order
alpha_s^2 to its Wilson coefficient C_A. It requires performing three-loop
matching between the full and effective theories. Including the new corrections
makes C_A more stable with respect to the matching scale mu_0 at which the
top-quark mass and alpha_s are renormalized. The corresponding uncertainty in
|C_A|^2 gets reduced from around 1.8% to less than 0.2%. Our results are
directly applicable to all the B_{s(d)} -> l^+ l^- decay modes.Comment: 25 pages, 9 figures; v2: references update
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