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Harmonic field distribution on self-affine surfaces

The aim of this study is to analyse the statistical properties of harmonic fields V in the vicinity of a self-affine Gaussian equipotential boundary. It is shown that the statistical distribution of g = log (|∇V|), in the limit of a vanishing amplitude, is a normal law. As the amplitude increases the distribution develops an exponential tail, hence the field gradient displays a power law distribution. The exponent of the power law varies continuously with the lower scale cut-off ℓ of the self-affine regime, and the roughness amplitude A as τ ∝ A−2ℓ2(1 − ζ), where ζ is the roughness exponent. The latter form is revealed from a second-order perturbation expansion on the roughness amplitude, and directly through numerical simulations in two dimensions using a conformal mapping technique.