  The main objective of this article is to study the order-disorder phase
transition and pattern formation for systems with long-range repulsive
interactions. The main focus is on the Cahn-Hilliard model with a nonlocal term
in the corresponding energy functional, representing the long-range repulsive
interaction. First, we show that as soon as the linear problem loses stability,
the system always undergoes a dynamic transition to one of the three types,
forming different patterns/structures. The types of transition are then
dictated by a nondimensional parameter, measuring the interactions between the
long-range repulsive term and the quadratic and cubic nonlinearities in the
model. The derived explicit form of this parameter offers precise information
for the phase diagrams. Second, we obtain a novel and explicit pattern
selection mechanism associated with the competition between the long-range
repulsive interaction and the short-range attractive interactions. In
particular, the hexagonal pattern is unique to the long-range interaction, and
is associated with a novel two-dimensional reduced transition equations on the
center manifold generated by the unstable modes, consisting of (degenerate)
quadratic terms and non-degenerate cubic terms. Finally, explicit information
on the metastability and basin of attraction of different disordered/ordered
states and patterns are derived as well.
