The derivative expansion approach to the interaction between close surfaces

TL;DR

This paper presents a comprehensive, formalized approach to the derivative expansion method for calculating interactions between close surfaces, generalizing the proximity force approximation and exploring its broader applications.

Contribution

It introduces a unified, formal framework for the derivative expansion, extending its applicability and connecting it with traditional approximations in various physics fields.

Findings

Derived the formal generalization of the derivative expansion

Applied the approach to multiple physical examples

Discussed potential extensions to nuclear and colloidal physics

Abstract

The derivative expansion approach to the calculation of the interaction between two surfaces, is a generalization of the proximity force approximation, a technique of widespread use in different areas of physics. The derivative expansion has so far been applied to seemingly unrelated problems in different areas; it is our principal aim here to present the approach in its full generality. To that end, we introduce an unified setting, which is independent of any particular application, provide a formal derivation of the derivative expansion in that general setting, and study some its properties. With a view on the possible application of the derivative expansion to other areas, like nuclear and colloidal physics, we also discuss the relation between the derivative expansion and some time-honoured uncontrolled approximations used in those contexts. By putting them under similar terms as the derivative expansion, we believe that the path is open to the calculation of next to leading order corrections also for those contexts. We also review some results obtained within the derivative expansion, by applying it to different concrete examples and highlighting some important points.