Fluid membranes with spatially varying distributions of bending rigidity and spontaneous curvature are considered, which are applicable to inhomogeneous membranes consisting of different components and membranes with inclusions (such as proteins) or adsorbed colloidal particles. Thermally activated shape fluctuations of the membrane induce rather long-ranged interactions between these inhomogeneities, which are free to diffuse laterally and to organize into spatial structures. As a consequence of these interactions, one finds two ordered phases in addition to the disordered phase in which the inhomogeneities are randomly mixed: a segregated phase, where the inclusions tend to aggregate, and a hexagonal phase, where the inclusions maximize their mutual separation. The phase behavior depends crucially on the lateral correlation length within the membrane. Also, the effective bending rigidity of the membrane is calculated for all different phases; in general, the inhomogeneities in the elastic moduli considered here lead to a characteristic softening of the membrane. These results, which are obtained using perturbation theory, are confirmed by Monte Carlo simulations. Experimental applications comprise proteins or adsorbed colloidal particles in vesicular bilayers and lamellar stacks and membranes consisting of lipid mixtures.
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