Can one hear the shape of an electrode? II. Theoretical study of the Laplacian transfer

The flux across resistive irregular interfaces driven by a force deriving from a Laplacian potential is computed on a rigorous basis. The theory permits one to relate the size of the active zone A(act). to the derivative of the spectroscopic impedance Z(spect.)(r) with respect to the surface resistivity r through: d(Z(spect.))/dr = A(act)(-1). It is shown that the macroscopic transfer properties through a system of arbitrary shape are determined by the characteristics of a first-passage interface-interface random walk operator. More precisely, it is the distribution of the harmonic measure (or normalized primary current) on the eigenmodes of this linear operator that controls the transfer. In addition, it is also shows that, whatever the dimension, the impedance of a weakly polarizable electrode for any irregular geometry scales under a homothety transformation as Ld-1, L being the size of the-system and d its topological dimension. In this new formalism, the question addressed in the title is transformed in a open mathematical question: "Knowing the distribution of the harmonic measure on the eigenmodes of the self-transport operator, can one retrieve the shape of the interface?"