Logarithmic fractals and hierarchical deposition of debris

We study geometrical objects on the borderline between standard Euclidean forms and fractals. The length (or area) increases with an additive rather than multiplicative constant, upon reducing the ruler length by a fixed rescaling factor. This leads to a logarithmic law instead of the usual power law for fractals. The fractal dimension DF equals the topological dimension DT, and a fractal amplitude AF is proposed for characterizing the objects. We introduce a model for the random deposition of debris consisting of a hierarchy of fragments with a hyperbolic size distribution (similar to meteors in space) that fall onto a D-dimensional surface (D=1 or 2). The deposition takes place in air or another viscous medium so that the fragments hit the surface in order of size, the large ones first. Employing both numerical simulation and analytic solution we verify that the rough landscape after impact is a logarithmic fractal for both D=1 and 2, and determine the amplitude AF as a function of the probabilities P for piling up hills, and Q for digging holes, with P+Q⩽1.