  Two known distinct examples of one-dimensional systems which are known to
exhibit a phase transition are critically examined: (A) a lattice model with
harmonic nearest-neighbor elastic interactions and an on-site Morse potential,
and (B) the ferromagnetic, spin 1/2 Ising model with long-range pair
interactions varying as the inverse square of the distance between pairs. In
both cases it can be shown that the domain wall configurations become
entropically stable at, or very near, the critical temperature. This might
provide a "positive" criterion for the occurrence of a phase transition in
one-dimensional systems.
