Statistical theory of materials with nanoscale phase separation

Materials with nanoscale phase separation are considered. These materials are formed by a mixture of several phases, so that inside one phase there exist nanosize inclusions of other phases, with random shapes and random spatial locations. A general approach is described for treating statistical properties of such materials with nanoscale phase separation. Averaging over the random phase configurations, it is possible to reduce the problem to the set of homogeneous phase replicas, with additional equations defining the geometric weights of different phases in the mixture. The averaging over phase configurations is mathematically realized as the functional integration over the manifold indicator functions. This procedure leads to the definition of an effective renormalized Hamiltonian taking into account the existence of competing phases. Heterophase systems with mesoscopic phase separation can occur for different substances. The approach is illustrated by the model of a high-temperature superconductor with non-superconducting admixture and by the model of a ferroelectric with paraelectric random inclusions.