English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we study metastability in large volumes at low temperatures. Weconsider both Ising spins subject to Glauber spin-flip dynamics and lattice gasparticles subject to Kawasaki hopping dynamics. Let $\b$ denote the inversetemperature and let $\L_\b \subset \Z^2$ be a square box with periodic boundaryconditions 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. Forlarge $\b$, and for interaction parameters that correspond to the metastableregime, we investigate how the transition from the metastable state (with onlysmall droplets) to the stable state (with one or more large droplets) takesplace under the dynamics. This transition is triggered by the appearance of asingle \emph{critical droplet} somewhere in $\L_\b$. Using potential-theoreticmethods, we compute the \emph{average nucleation time} (= the first time acritical droplet appears and starts growing) up to a multiplicative factor thattends to one as $\b\to\infty$. It turns out that this time grows as$Ke^{\Gamma\b}/|\L_\b|$ for Glauber dynamics and $K\b e^{\Gamma\b}/|\L_\b|$ forKawasaki dynamics, where $\Gamma$ is the local canonical, respectively,grand-canonical energy to create a critical droplet and $K$ is a constantreflecting the geometry of the critical droplet, provided these times tend toinfinity (which puts a growth restriction on $|\L_\b|$). The fact that theaverage nucleation time is inversely proportional to $|\L_\b|$ is referred toas \emph{homogeneous nucleation}, because it says that the critical droplet forthe transition appears essentially independently in small boxes that partition$\L_\b$.

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