Critical droplet for a non-local mean field equation (Q2708438)

The authors study phase changes of a fluid. Specifically, following their 1998 paper in the same journal exploring spectral estimates, they prove the existence of a critical droplet, which marks transition from a metastable state (i.e., a state which is stable under small perturbations) to a stable state. It marks a saddle point in the transition to the stable state. The technique is as follows: Existence is shown of limits forming one-dimensional manifolds which connect the critical droplet to the stable and metastable manifolds. This amounts to the description of tunnelling within the background of Ising systems, and Kac interactions. Their problem now consists of establishing a preferred spin system in the tradition of Leibowitz' and Penrose's treatment of van der Waal equations in the presence of a magnetic field.NEWLINENEWLINENEWLINEThe authors show that in fact the metastable condition may persist for a long time, until a large deviation occurs, whereby the stable phase region expands until it fills all available space. Thus the intuitive argument, that there exists some intermediate phase must be wrong. The transition is instead characterized by formation of a critical droplet which breaks homogeneity of the space producing a so called bump NEWLINE\[NEWLINEm^*_{J(x)}(x)=\text{tanh}(B(J\bigstar m^*_{J(x)}(x)+h).NEWLINE\]NEWLINE Here \(B >1\), \(J\) is a smooth symmetry probability density function, \(\bigstar\) denotes convolution, and \(h\) is energy density due to an external field. (This type of interaction goes back to Penrose's studies in early 1970-s). The authors use Newton's orbit theory and perturbation techniques, as well as much classical and functional analysis. The proof takes more than 30 pages of detailed computations.