Phase Transitions in Reduced Geometry

Phase transitions in condensed-matter systems are caused by collective, cooperative behavior of a macroscopically large number (N -> rAJ) of atoms or molecules (1, 2). Often, the more ordered phase results from a symmetry breaking of the less ordered phase; the degeneracy between the various ordered states may then be lifted by boundary effects. For example, for an uniaxial magnet undergoing a second order transition from the paramagnetic phase for temperatures T> Te to the ferromagnetic phase in zero external field H at a critical temperature Te, we have a degeneracy between oppositely oriented domains for T 00. For finite N, the transition is rounded and shifted over some region of the independent parameters (T, H, pressure p, etc.). As N -> 00, this region smoothly shrinks to zero.