Duality in two dimensions

The dielectric breakdown problem can be solved very easily from the solution of the fuse problem in two dimensions using the concept of duality. This concept is largely used in the case of composite materials and in percolation for problems in d = 2 or with cylindrical symmetry (Mendelson 1975, Bowman and Stroud 1989). Here we follow the derivation of Bowman and Stroud. 

One considers a composite material made of a mixture of conductors and insulators. As above, for p Pc, the material is insulating, and conducting for p Pc- We consider first the case with p smaller than Pc- The equations for the induction vector D and the field E are  

We consider now the dual composite in which the conducting parts are replaced by the insulating material and vice versa. We suppose that the conductivity of the parts which were insulating is proportional to the inverse of the dielectric constant of the insulating parts e = 1/a). The material is a conductor and the relevant quantities are the current density i and the field E. The equations for these quantities are 

The last eqn (2.58) is valid on the conducting parts, i.e. the insulating parts of the original problem. 

Since the divergence of i is zero, it may be expressed as the curl of a potential vector V. We choose it such that only the component Vz perpendicular to the plane of the sample is different from zero, Vz = 0(x,y). V satisfies the equation 

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