  We present a field theory to describe the composition of a surface
spontaneously exchanging matter with its bulk environment. By only assuming
matter conservation in the system, we show with extensive numerical simulations
that, depending on the matter exchange rates, a complex patterned composition
distribution emerges in the surface. For one-dimensional systems we show
analytically and numerically that coarsening is arrested and as a consequence
domains have a characteristic length scale. Our results show that the causes of
heterogeneous lipid composition in cellular membranes may be justified in
simple physical terms.
