Polymorphic phase transitions in systems evolving in a two-dimensional discrete space

Polymorphic phase transitions in systems evolving in a two-dimensional disc rete space have been studied. The driving force of the transitions appears to be a difference between two main energetic contributions: one, related t o the thermal activation of the process, and another, being of quantum natu re. The former (high temperature limit) is naturally assigned to the expans ion (melting) part of the transition, while the latter (low temperature lim it) has much in common with the contraction (solidification) part. Between the two main physical states distinguished, there exists a certain state, c orresponding to a discontinuity point (pole) in the morphological phase dia gram, represented by the well-known Bose-Einstein (Planck) formula, in whic h the system blows up. This point is related to an expected situation in wh ich the contour of the object under investigation stands for the Brownian o r purely diffusional path, with the fractal dimension d(w)=2, and the situa tion can be interpreted as some emergence of an intermediate "tetratic" pha se. This, in turn, recalls a certain analogy to the equilibrium (order-diso rder) phase transition of Kosterlitz-Thouless type, characteristic of, e.g. , rough vs rigid interfaces in a two-dimensional space, with some disappear ance of interface correlation length at d(w)=2. Otherwise, the contours of the objects are equivalent to fractional Brownian paths either in superline ar or "turbulent" (d(w)<2; the expansion case), or sublinear, viz., anomalo usly slow (d(w)>2; the contraction case) regimes, respectively. It is hoped that the description offered will serve to reflect properly the main subtl eties of the dynamics of the polymorphic transitions in complex ''soft-matt er'' systems, like formation of lipid mesomorphs or diffusional patterns, w ith nonzero line tension effect. [S1063-651X(99)08706-1].