Homogeneous nucleation for two-dimensional Kawasaki dynamics
8 affiliations
- 1Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy
- 2Universiteit Leiden = Leiden University (Leiden University | 2300 RA Leiden The Netherlands - Netherlands)
- 3Aix-Marseille Université, CNRS, I2M, 3 Place Victor Hugo, 13003, Marseille, France
- 4I2M - Institut de Mathématiques de Marseille (Centre de Mathématiques et Informatique (CMI) Technopôle Château-Gombert 39, rue Frédéric Joliot Curie 13453 Marseille Cedex 13 - France)
- 5Mathematisch Instituut, Universiteit Leiden, Einsteinweg 55, 2333 CC, Leiden, The Netherlands
- 6Università degli Studi di Firenze, Dipartimento di Matematica e Informatica “Ulisse Dini”, Viale Morgagni 67/a, 50134, Firenze, Italy
- 7Dipartimento di Matematica , Università di Roma Tor Vergata , Via della Ricerca Scientifica , Roma , 00133 , Italy
- 8Dipartimento di Matematica e Fisica, Universit‘a di Roma Tre, Largo S. Leonardo Murialdo 1, 00146, Roma, Italy
Abstract
This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature β > 0 in a large finite box Λ β ⊂ Z 2 whose volume depends on β . Each pair of neighbouring particles has a negative binding energy − U < 0 , while each particle has a positive activation energy Δ > 0. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime Δ ∈ ( U , 2 U ) and in the limit as β → ∞, how and when the system nucleates, i.e., creates a critical droplet somewhere in Λ β that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large , namely, | Λ β | = e Θ β with Δ < Θ < 2 Δ − U . In the present paper we consider the setting where the volume is very large , namely, | Λ β | = e Θ β with Θ < Γ − ( 2 Δ − U ) , where Γ is the energy of the critical droplet in the local model with fixed volume, and we use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. We will see that in very large volumes critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation . This leads to a nucleation time of order e Γ β /|Λ β |. One of the key ingredients in the proof is an estimate showing that no information can travel between the moderately large boxes on relevant time scales. Prior to nucleation, the dynamics inside the moderately large boxes spends most of its time in configurations consisting of a single quasi-square surrounded by free particles, and makes transitions between these configurations at rates that depend on the linear sizes of the quasi-square in a computable manner.
Related
- Large Deviations for the Two-Dimensional Dean-Kawasaki Equation with Coulomb Interactions
- Linear Fluctuation of Interfaces in Glauber–Kawasaki Dynamics
- Dissolution and nucleation of jump discontinuities for one-dimensional periodic peridynamic waves
- The Three-Dimensional Divergence-Curl Problem with Homogeneous Boundary Conditions and Pseudo-Harmonic Fields
- Homogeneous attractive Bose-Einstein condensates with repulsive three-body interactions: the two-dimensional case
BibTeX
@article{baldassarri2026homogeneous,
title = {{Homogeneous nucleation for two-dimensional Kawasaki dynamics}},
author = {Simone Baldassarri and Alexandre Gaudillière and Frank den Hollander and Francesca Romana Nardi and Enzo Olivieri and Elisabetta Scoppola},
journal = {Stochastic Processes and their Applications},
year = {2026},
month = jun,
doi = {10.1016/j.spa.2026.105031},
url = {https://doi.org/10.1016/j.spa.2026.105031},
note = {CC BY},
}
RIS
TY - JOUR
TI - Homogeneous nucleation for two-dimensional Kawasaki dynamics
AU - Simone Baldassarri
AU - Alexandre Gaudillière
AU - Frank den Hollander
AU - Francesca Romana Nardi
AU - Enzo Olivieri
AU - Elisabetta Scoppola
PY - 2026
DA - 2026/06/24
JO - Stochastic Processes and their Applications
DO - 10.1016/j.spa.2026.105031
UR - https://doi.org/10.1016/j.spa.2026.105031
AB - This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature β > 0 in a large finite box Λ β ⊂ Z 2 whose volume depends on β . Each pair of neighbouring particles has a negative binding energy − U < 0 , while each particle has a positive activation energy Δ > 0. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime Δ ∈ ( U , 2 U ) and in the limit as β → ∞, how and when the system nucleates, i.e., creates a critical droplet somewhere in Λ β that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large , namely, | Λ β | = e Θ β with Δ < Θ < 2 Δ − U . In the present paper we consider the setting where the volume is very large , namely, | Λ β | = e Θ β with Θ < Γ − ( 2 Δ − U ) , where Γ is the energy of the critical droplet in the local model with fixed volume, and we use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. We will see that in very large volumes critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation . This leads to a nucleation time of order e Γ β /|Λ β |. One of the key ingredients in the proof is an estimate showing that no information can travel between the moderately large boxes on relevant time scales. Prior to nucleation, the dynamics inside the moderately large boxes spends most of its time in configurations consisting of a single quasi-square surrounded by free particles, and makes transitions between these configurations at rates that depend on the linear sizes of the quasi-square in a computable manner.
SN - 0304-4149
SN - 1879-209X
ER -