Fluctuation-induced interactions between rods on a membrane

We consider the interaction between two rods embedded in a fluctuating surface. The modification of fluctuations by the rods leads to an attractive long-range interaction between them. We consider fluctuations governed by either surface tension (films) or bending rigidity (membranes). In both cases the interaction falls off with the separation of the rods as 1R⁴. The orientational part of the interaction is proportional to cos²[θ₁+θ₂] in the former case and to cos²[2θ₁+θ₂] in the latter, where θ₁ and θ₂ are angles between the rods and the line joining them. These interactions are somewhat reminiscent of dipolar forces and will tend to align collections of such rods into chains.