Rigorous Entropy - Energy Arguments

One of the most appealing heuristic devices for unders tanding when a phase transition will take place is to consider the balance of energy versus entropy. Probably the simplest example is the well-known (9'27) heuristic a rgument for the absence of phase transitions in one-dimensional systems with short-range forces: Suppose that there were two distinct phases with the same bulk free energy. Then, starting with a single phase, we can insert a droplet of size L of the other phase: the energy cost is finite ( independent of L), since the forces are short range; while the droplet can be inserted in approximately N / L places (where N is the length of the system), leading to an entropy gain on the order of k l n ( N / L ) . Thus, for any temperature T > 0, the free energy change c o n s t k T I n ( N / L ) is negative for sufficiently large N; that is, the putative pure phase is unstable.