Nucleation and growth in two dimensions

We consider a dynamical process on a graph $G$, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph $\mathbb{Z}^2$, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.Comment: 35 pages, Section 6 update

Nucleation scaling in jigsaw percolation

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD(log N) is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.Comment: 39 pages, 3 figures. Moved main results to the introduction and improved exposition of section

Statistical Physics of Fracture Surfaces Morphology

Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, succeeding to reproduce the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up with the proposition of new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.Comment: A review paper submitted to J. Stat. Phy

Investigation of pulsed laser induced dewetting in nanoscopic metal films

Hydrodynamic pattern formation (PF) and dewetting resulting from pulsed laser induced melting of nanoscopic metal films have been used to create spatially ordered metal nanoparticle arrays with monomodal size distribution on SiO_{\text{2}}/Si substrates. PF was investigated for film thickness h\leq7 nm < laser absorption depth \sim11 nm and different sets of laser parameters, including energy density E and the irradiation time, as measured by the number of pulses n. PF was only observed to occur for E\geq E_{m}, where E_{m} denotes the h-dependent threshold energy required to melt the film. Even at such small length scales, theoretical predictions for E_{m} obtained from a continuum-level lumped parameter heat transfer model for the film temperature, coupled with the 1-D transient heat equation for the substrate phase, were consistent with experimental observations provided that the thickness dependence of the reflectivity of the metal-substrate bilayer was incorporated into the analysis. The spacing between the nanoparticles and the particle diameter were found to increase as h^{2} and h^{5/3} respectively, which is consistent with the predictions of the thin film hydrodynamic (TFH) dewetting theory. These results suggest that fast thermal processing can lead to novel pattern formation, including quenching of a wide range of length scales and morphologies.Comment: 36 pages, 11 figures, 1 tabl

Dynamical obstruction in a constrained system and its realization in lattices of superconducting devices

Hard constraints imposed in statistical mechanics models can lead to interesting thermodynamical behaviors, but may at the same time raise obstructions in the thoroughfare to thermal equilibration. Here we study a variant of Baxter's 3-color model in which local interactions and defects are included, and discuss its connection to triangular arrays of Josephson junctions of superconductors and \textit{kagom\'e} networks of superconducting wires. The model is equivalent to an Ising model in a hexagonal lattice with the constraint that the magnetization of each hexagon is $\pm 6$ or 0. For ferromagnetic interactions, we find that the system is critical for a range of temperatures (critical line) that terminates when it undergoes an exotic first order phase transition with a jump from a zero magnetization state into the fully magnetized state at finite temperature. Dynamically, however, we find that the system becomes frozen into domains. The domain walls are made of perfectly straight segments, and domain growth appears frozen within the time scales studied with Monte Carlo simulations. This dynamical obstruction has its origin in the topology of the allowed reconfigurations in phase space, which consist of updates of closed loops of spins. As a consequence of the dynamical obstruction, there exists a dynamical temperature, lower than the (avoided) static critical temperature, at which the system is seen to jump from a ``supercooled liquid'' to a ``polycrystalline'' phase. In contrast, for antiferromagnetic interactions, we argue that the system orders for infinitesimal coupling because of the constraint, and we observe no interesting dynamical effects

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let \b denote the inverse temperature and let \L_\b \subset \Z^2 be a square box with periodic boundary conditions such that \lim_{\b\to\infty}|\L_\b|=\infty. We run the dynamics on \L_\b starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large \b, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in \L_\b. Using potential-theoretic methods, we compute the \emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as \b\to\infty. It turns out that this time grows as Ke^{\Gamma\b}/|\L_\b| for Glauber dynamics and K\b e^{\Gamma\b}/|\L_\b| for Kawasaki dynamics, where $\Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |\L_\b|). The fact that the average nucleation time is inversely proportional to |\L_\b| is referred to as \emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition \L_\b.Comment: 45 pages, 11 figure