Nucleation theory, the escaping processes, and nonlinear stability

We study the solutions describing the critical “germs” in nucleation theory, escaping processes, and fractures. We present systems with exact solutions in $D=1,2,3\mathrm{}$ dimensions. We show that when there exist connections between the particles in more than one dimension, the stability is much more increased. In systems where the potential well is a degenerate point, the critical germ solution has a power-law behavior. For $D=3\mathrm{}$ there can exist a continuum of stationary states where all the points of the order parameter take values that are out of the stability zone (i.e., the potential well) leading to effective marginal stability. We discuss the relevance of these results for different physical systems and its connections with recently intensively studied phenomena like sand-pile dynamics, self-organized criticality, noise-induced synchronization in extended systems, and quantum tunneling in the framework of field theory.