The concept of fractal experiments: New possibilities in quantitative description of quasi-reproducible measurements

Abstract

In this paper the authors suggest a new conception of the so-called fractal (self-similar) experiment. Under the fractal experiment (FE) one can imply a cycle of measurements that are subjected by the scaling transformations F(z)→F(zξm) in contrast with conventional scheme F(z)→F(z+mT) (m=0,1,…, M–1), where z defines the controllable (input) variable and can be associated with time, complex frequency, wavelength and etc., T – mean period of time between successive measurements and m defines a number of successive measurements. One can connect a fractal experiment with specific memory effect that arises between successive measurements. The general theory of experiment for quasi-periodic measurements proposed in [1] after some transformations can be applied for the set of the FE, as well. But attentive analysis shown in this paper allows generalizing the previous results for the case when the influence of uncontrollable factors becomes significant. The theory developed for this case allows to consider more real cases when the influence of dynamic (unstable) processes taking place during the cycle of measurements corresponding to some FE is becoming essential. These experiments we define as quasi-reproducible (QR) fractal experiments.The proposed concept opens new possibilities in theory of measurements and numerous applications, especially in different nanotechnologies, when the influence of the scaling factor plays the essential role. This concept allows also to introduce the so-called intermediate model (IM) which can serve as an unified platform for reconciliation of the proposed microscopic theory with reliable experiments “refined” from the influence of the random noise and apparatus function. We forced to consider a modified model experiment in order to demonstrate some common peculiarities that can be appeared in real cases. We know only couple of similar examples of experiments that are close to the proposed concept. Mechanical relaxation and dielectric spectroscopy (based on measurements of the complex susceptibility ε(jω)) represent the branches of physics related to consideration of mechanical and electric relaxation phenomena in different heterogeneous materials. The dielectric spectroscopy can be considered as an instructive example for better understanding of the proposed concept.In cases, when the microscopic model is absent the results of measurements can be expressed in terms of the fitting parameters associated with the generalized Prony spectrum (GPS) belonging to the IM. The authors do hope that this new approach will find an interesting continuation in various applications of different nanotechnologies.