Critical droplets in Metastable States of Probabilistic Cellular Automata

We consider the problem of metastability in a probabilistic cellular automaton (PCA) with a parallel updating rule which is reversible with respect to a Gibbs measure. The dynamical rules contain two parameters $\beta$ and $h$ which resemble, but are not identical to, the inverse temperature and external magnetic field in a ferromagnetic Ising model; in particular, the phase diagram of the system has two stable phases when $\beta$ is large enough and $h$ is zero, and a unique phase when $h$ is nonzero. When the system evolves, at small positive values of $h$, from an initial state with all spins down, the PCA dynamics give rise to a transition from a metastable to a stable phase when a droplet of the favored $+$ phase inside the metastable $-$ phase reaches a critical size. We give heuristic arguments to estimate the critical size in the limit of zero ``temperature'' ($\beta\to\infty$), as well as estimates of the time required for the formation of such a droplet in a finite system. Monte Carlo simulations give results in good agreement with the theoretical predictions.Comment: 5 LaTeX picture

Dramatic Shape Sensitivity of Directional Emission Patterns from Similarly Deformed Cylindrical Polymer Lasers

Recent experiments on similarly shaped polymer micro-cavity lasers show a dramatic difference in the far-field emission patterns. We show for different deformations of the ellipse, quadrupole and hexadecapole that the large differences in the far-field emission patterns is explained by the differing ray dynamics corresponding to each shape. Analyzing the differences in the appropriate phase space for ray motion, it is shown that the differing geometries of the unstable manifolds of periodic orbits are the decisive factors in determining the far-field pattern. Surprisingly, we find that strongly chaotic ray dynamics is compatible with highly directional emission in the far-field.Comment: 14 pages, 16 figures (eps), RevTeX 4, submitted to JOSA

Metastability in the two-dimensional Ising model with free boundary conditions

We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit $T\to 0$, $h\to 0$ and via Monte Carlo simulations at fixed values of $T$ and $h$ and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.Comment: 24 pages, TeX fil

Metastability and Nucleation for the Blume-Capel Model. Different mechanisms of transition

We study metastability and nucleation for the Blume-Capel model: a ferromagnetic nearest neighbour two-dimensional lattice system with spin variables taking values in -1,0,+1. We consider large but finite volume, small fixed magnetic field h and chemical potential "lambda" in the limit of zero temperature; we analyze the first excursion from the metastable -1 configuration to the stable +1 configuration. We compute the asymptotic behaviour of the transition time and describe the typical tube of trajectories during the transition. We show that, unexpectedly, the mechanism of transition changes abruptly when the line h=2*lambda is crossed.Comment: 96 pages, 44 tex-figures, 7 postscript figure

Warm gas in the rotating disk of the Red Rectangle: accurate models of molecular line emission

We aim to study the excitation conditions of the molecular gas in the rotating disk of the Red Rectangle, the only post-Asymptotic-Giant-Branch object in which the existence of an equatorial rotating disk has been demonstrated. For this purpose, we developed a complex numerical code that accurately treats radiative transfer in 2-D, adapted to the study of molecular lines from rotating disks. We present far-infrared Herschel/HIFI observations of the 12CO and 13CO J=6-5, J=10-9, and J=16-15 transitions in the Red Rectangle. We also present our code in detail and discuss the accuracy of its predictions, from comparison with well-tested codes. Theoretical line profiles are compared with the empirical data to deduce the physical conditions in the disk by means of model fitting. We conclude that our code is very efficient and produces reliable results. The comparison of the theoretical predictions with our observations reveals that the temperature of the Red Rectangle disk is typically ~ 100-150 K, about twice as high as previously deduced from mm-wave observations of lower-J lines. We discuss the relevance of these new temperature estimates for understanding the thermodynamics and dynamics of this prototype object, as well as for interpreting observations of other rarely studied post-AGB disks. Despite our sophisticated treatment of the line formation, our model cannot explain the relatively strong line-wing emission for intermediate-J transitions. We argue that a model including a rotating disk only cannot reproduce these data and suggest that there is an additional extended (probably bipolar) structure expanding at about 7--15 km/s.Comment: 18 pages, 21 figure