Parametric ergodicity to measure roughness from a single sample

We introduce a concrete random model system to study the concept of parametric ergodicity. It consists of a continuum mechanical cavity with an embedded random mass distribution, constrained by a parametrized boundary condition. The interest is twofold. On one hand, there is the practical interest of obtaining ensemble averages of physical quantities from a small number of experimentally available samples, in many cases only one. This is typically the case in studies on conductance fluctuations through disorder mesosocopic systems. On the other hand we want to develop more insight into the meaning of parametric ergodicity. For this, we focus on the statistical distribution of resonant frequency generated by the ensemble of random samples, and how to produce the same distribution from a single sample subject to changing a boundary condition — the external parameter. The paper shows how the changing of the boundary condition is equivalent to scanning the ensemble of equivalent samples