Phase transitions in one dimension: are they all driven by domain walls?
  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. 
N. Theodorakopoulos