"Statistical" symmetry with applications to phase transitions

Abstract

Hermann proposed that mesomorphic media should be classified by assigning certain "statistical symmetry groups" to each possible partially ordered array. Two translational groups introduced were called superordinate and subordinate. We find that the average density in such a partially ordered medium has the superordinate symmetry ℒ1, while the pair correlation function has the subordinate symmetry ℒ2. A complete listing is made of all compatible combinations of ℒ1 and ℒ2 in two and three dimensions. This leads to more possible symmetries than Hermann obtained, e.g., also to nonstoichiometric crystals. The order parameter space for the systems is found to be the quotient space ℒ1/ℒ2. In most cases it is identical to the order parameter space of low-dimensional XY spin systems. The Landau free energy is expanded as functional of the two-particle correlation function K; the translation group is found to be ℒ1×ℒ2. A Landau mean-field theory can then be carried out by expanding the system free energy into a series of invariants of the active irreducible representations of K and mapping the free energy onto that for an XY planar spin system. We predict novel critical behavior for transitions between mesomorphic phases and “go nogo” selection rules for continuous transitions. We give the structure factors for X-ray scattering so changes in all such phase transitions are observable. The statistical symmetry groups, which describe point and translational symmetries of the mesophases, are classified. Proposals are made to include quasi-long-range or topological order in the classification scheme.