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Self-consistent theory for inhomogeneous systems with mesoscopic fluctuations

We have developed a theory for inhomogeneous systems that allows for incorporation of effects of mesoscopic fluctuations. A hierarchy of equations relating the correlation and direct correlation functions for the local excess $ϕ({\bf r})$ of the volume fraction of particles $ζ$ has been obtained, and an approximation leading to a closed set of equations for the two-point functions has been introduced. We have solved numerically the self-consistent equations for one (1D) and three (3D) dimensional models with short-range attraction and long-rannge repulsion (SALR). Predictions for all the qualitative properties of the 1D model agree with the exact results, but only semi-quantitative agreement is obtained in the simplest version of the theory. The effects of fluctuations in the two considered 3D models are significantly different, despite very similar properties of these models in the mean-field approximation. In both cases we obtain the sequence of large - small - large compressibility for increasing $ζ$. The very small compressibility is accompanied by the oscillatory decay of correlations with the correlation length orders of magnitude larger than the size of particles. Only in one of the two considered models for decreasing temperature the small compressibility becomes very small and the large compressibility becomes very large, and eventually van der Waals loops appear. Further studies are necessary to determine the nature of the strongly inhomogeneous phase present for intermediate volume fractions in 3D.