Kawasaki Dynamics

Abstract

The goal of this chapter is to extend the analysis in Chap. 19 to Kawasaki dynamics. Again, the average time until the appearance of a critical droplet somewhere is inversely proportional to the volume, and is driven by the same quantities as for small volumes. However, in the proof we encounter several difficult issues, all coming from the fact that Kawasaki dynamics is conservative. The first is to understand why the energetic cost to create a critical droplet in a small box with an open boundary, i.e., in a grand-canonical setting, reappears even though we choose our box to have a closed boundary, i.e., we work in a canonical setting. This “mystery” is resolved by the observation that the formation of a critical droplet reduces the entropy of the system: the precise computation of this entropy loss yields the proper scaling via dynamical equivalence of ensembles. The second problem is to control the probability of a particle moving from the gas to the protocritical droplet at the last stage of the nucleation, which plays a key role in understanding how the prefactor in the scaling comes up. This non-locality issue will be dealt with via upper and lower estimates. The latter in fact causes the scaling to be slightly different than for small volumes. Sections 20.1–20.3 develop the key steps of the proof. Sections 20.4–20.5 provide some key ingredients that are needed along the way.

Tout le monde trouve à redire en autrui ce qu’on trouve à redire en lui.  (François de La Rochefoucauld, Réflexions)