Abstract
This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics atlow temperature in a large finite box with an open boundary. Each pair of particlesoccupying neighboring sites has a negative binding energy provided their types aredifferent, while each particle has a positive activation energy that depends onits type. There is no binding energy between neighboring particles of the same type.At the boundary of the box particles are created and annihilated in a way thatrepresents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. We identify the region of parameters for which the system is metastable. For thisregion, in the limit as the temperature tends to zero, we show that the firstentrance distribution on the set of critical droplets is uniform, compute theexpected transition time up to a multiplicative factor that tends to one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived under three hypotheses on the energy landscape, which are verified in the second and the third paper for a certain subregion of the metastable region. These hypotheses involve three model-dependent quantities - the energy, the shape and the number of the critical droplets - which are identified in the second and the third paper as well.
Highlights
Introduction and main resultsThe main motivation behind this work is to understand metastability of multi-type particle systems subject to conservative stochastic dynamics
Metastability for Kawasaki dynamics at low temperature with two types of particles Frank den Hollander∗ Francesca Romana Nardi† Alessio Troiani‡. This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary
Theorem 1.7(a) says that C is a gate for the nucleation, i.e., on its way from to the dynamics passes through C
Summary
The main motivation behind this work is to understand metastability of multi-type particle systems subject to conservative stochastic dynamics. The model studied in the present paper falls in the class of variations on Ising spins subject to Glauber dynamics and lattice gas particles subject to Kawasaki dynamics These variations include Blume–Capel, anisotropic interactions, staggered magnetic field, next-nearest-neighbor interactions, and probabilistic cellular automata. Changing the state 0 → 1, 0 → 2, 1 → 0 or 2 → 0 at single sites in ∂−Λ (“creation and annihilation of particles inside ∂−Λ”) This dynamics is ergodic and reversible with respect to the Gibbs measure μβ, i.e., μβ(η)cβ(η, η ) = μβ(η )cβ(η , η) ∀ η, η ∈ X. The dynamics defined by (1.2) and (1.5) models the behavior inside Λ of a lattice gas in Z2, consisting of two types of particles subject to random hopping with hard core repulsion and with binding between different neighboring types. Definitions 1.2–1.3 are canonical in metastability theory and are formalized in Manzo, Nardi, Olivieri and Scoppola [21]
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