Analysis of Fluctuations Induced by a Scale Hierarchy of Interacting Particles

We investigate the two-point averaging over space of fluctuations arising from a multiscale hierarchy of interacting particles. We assume this will satisfy a condition of homogeneity with respect to scale. We consider the second-order correlation of fluctuations arising from particles of a single scale in the hierarchy; we then form an average over the set of such single-scale correlations. As the hierarchy is refined, a condition of scale continuum is approached. We use the limiting value of this procedure to define a two-point correlation function for the multiscale system as a whole, and identify this with the experimental measurement of correlation in such a multiscale context. In the energy spectrum which emerges in this limit, one term comes to dominate the spectrum for large k; this term has the form ‘k −2 ln k’. In fact, a variety of different shape functions (intended to represent correlation functions) leads to this energy spectrum, which bears a qualitative resemblance to a Kolmogorov power-law. In this sense a degree of universality is exhibited. The ideas are illustrated for two simple one-dimensional test cases before a more general treatment in one and three dimensions is developed.