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Van der Waals interactions in a dielectric with continuously varying dielectric function.

We formulate and evaluate the van der Waals part of the free energy due to a dielectric profile that varies continuously throughout the space between two interacting bodies. Not considering the work needed to create the inhomogeneous dielectric profile, focusing only on that part of the free energy affected by the inhomogeneity, we compare the ensuing interaction free energy with that of the original Lifshitz formulation with its step function changes at material boundaries and uniform dielectric medium. Rather than the monotonically varying attraction between like bodies given by the original formulation, the inhomogeneous continuous dielectric function leads to attractions as well as repulsions. The Lifshitz result emerges naturally in the limit of separations much larger than the thickness of the interfaces.