Random multifractals: negative dimensions and the resulting limitations of the thermodynamic formalism

The story described in this paper has started with the ‘death or survival’ criterion, which the author published in 1972-1974 and had obtained in 1968 while investigating Kolmogorov’s hypothesis that the turbulent dissipation ϵ(dx) in a box is log-normally distributed. Using this criterion, the present paper discusses the concrete significance of negative fractal dimensions. They arise in those random multifractal measures, for which the Cramèr function f(α) (the ‘spectrum of singularities’) satisfies f(α) < 0 for certain values of α. It is shown that in that case the strict ‘thermodynamical formalism’ solely involves the form of f(α) in the range where f(α) > 0, and concerns three aspects of such measures: (a) the fine-grained multifractal properties, which are non-random and the same for (almost) all realizations; (b) the properties obtained by using the ‘partition function’ formalism ; and (c) the ‘typical’ coarse-grained multifractal properties. However, the f(α)s in the range where f(α) > 0 say nothing about the variability of coarse-grained properties between samples. A description of these fluctuations, hence a fuller multifractal description of the measure, is shown to be provided by the values of f(α) in the range where f(α) < 0. We prefer to reserve the term ‘thermodynamic’ for the fine-grained and partition-functional properties, and to say that the coarse-grained properties go beyond the thermodynamics, i.e. are not macroscopic but ‘mesoscopic'.