Perturbation of an infinite Network of Identical Capacitors

TL;DR

This paper investigates how removing a bond in an infinite square lattice of capacitors affects the capacitance between two sites, using Green's functions and asymptotic analysis, with numerical validation.

Contribution

It introduces a method to relate perturbed lattice capacitance to the perfect lattice using Green's functions and explores asymptotic behavior for large separations.

Findings

Capacitance can be expressed in terms of the lattice Green's function.

Asymptotic behavior of perturbed capacitance is characterized.

Numerical results confirm theoretical predictions.

Abstract

The capacitance between any two arbitrary lattice sites in an infinite square lattice is studied when one bond is removed (i.e. perturbed). A connection is made between the capacitance and the Lattice Green's Function of the perturbed network, where they are expressed in terms of those of the perfect network. The asymptotic behavior of the perturbed capacitance is investigated as the separation between the two sites goes to infinity. Finally, numerical results are obtained along different directions and a comparison is carried out with the perfect capacitances