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Author: Bovier, Anton × Subject: capacity ×

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    Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bovier, Anton; Hollander, Frank den; Spitoni, Cristian
    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 $lim_btoinfty L_b =infty$. 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 emphcritical droplet somewhere in $L_b$. Using potential-theoretic methods, we compute the emphaverage nucleation time (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $btoinfty$. It turns out that this time grows as $Ke^Gammab/ L_b $ for Glauber dynamics and $Kb e^Gammab/ 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 emphhomogeneous nucleation, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $L_b$.
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    Metastability : a potential theoretic approach
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2005) Bovier, Anton
    Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems. In this talk, I describe recent attempts towards a model-independent approach to metastability in the context of reversible Markov processes. I will present an outline of a general theory, based on careful use of potential theoretic ideas and indicate a number of concrete examples where this theory was used very successfully. I will also indicate some challenges for future work.
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    Sharp asymptotics for metastability in the random field Curie-Weiss model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bianchi, Alessandra; Bovier, Anton; Ioffe, Dmitry
    In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.