Critical region for droplet formation in the two-dimensional Ising model

Limiting Shapes of Ising Droplets, Ising Fingers, and Ising Solitons

We examine the evolution of an Ising ferromagnet endowed with zero-temperature single spin-flip dynamics. A large droplet of one phase in the sea of the opposite phase eventually disappears. An interesting behavior occurs in the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size. In this regime the shape of the droplet is essentially deterministic (fluctuations are negligible in comparison with characteristic size). In two dimensions the shape is also universal, that is, independent on the initial shape.…

探索 Ising 模型中的亚稳态：临界液滴、能量势垒与退出时间

Exploring Metastability in Ising models: critical droplets, energy barriers and exit time

本文综述 Ising 模型亚稳态转变，通过临界构型与能量势垒刻画退出时间，并推广到多种晶格、相关模型及动力学。

The scaling limit for zero temperature planar Ising droplets: with and without magnetic fields

We consider the continuous time, zero-temperature heat-bath dynamics for the nearest-neighbor Ising model on $Z^2$ with positive magnetic field. For a system of size $L\in N$, we start with initial condition $σ$ such that $σ_x=-1$ if $x\in[-L,L]^2$ and $σ_x=+1$ and investigate the scaling limit of the set of $-$ spins when both time and space are rescaled by $L$. We compare the obtained result and its proof with the case of zero-magnetic fields, for which a scaling result was proved in arXiv:1112.3160. In that case, the time-scaling is diffusive and the scaling limit is given by anisotropic mo…

The heat equation shrinks Ising droplets to points

Let D be a bounded, smooth enough domain of R^2. For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on (Z/L)^2 (the square lattice with lattice spacing 1/L) with initial condition such that σ_x=-1 if x\in D and σ_x=+ otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L^2 and L tends to infinity, the boundary of the droplet of "-" spins follows a deterministic anisotropic curve-shortening flow, such that the normal velocity is given by the local curvature times an ex…

A proof of the Gibbs-Thomson formula in the droplet formation regime

We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs-Thomson formula for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general--albeit heuristic--reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.

On the formation/dissolution of equilibrium droplets

We consider liquid-vapor systems in finite volume $V\subset\R^d$ at parameter values corresponding to phase coexistence and study droplet formation due to a fixed excess $δN$ of particles above the ambient gas density. We identify a dimensionless parameter $Δ\sim(δN)^{(d+1)/d}/V$ and a \textrm{universal} value $\Deltac=\Deltac(d)$, and show that a droplet of the dense phase occurs whenever $Δ>\Deltac$, while, for $Δ<\Deltac$, the excess is entirely absorbed into the gaseous background. When the droplet first forms, it comprises a non-trivial, \textrm{universal} fraction of excess particles. Si…