Muitifractals in Convection and Aggregation

Abstract

Patterns found in convection and aggregation are often multifractals and can be characterized by a continuum of exponents. This differs from critical phenomena where typically a finite number of relevant exponents is needed. The set of exponents for a multifractal is conveniently presented as an f(α) spectrum, i.e., a spectrum of pointwise dimensions. We shall illustrate the formalism by a Rayleigh-Benard convection experiment developed by Libchaber and co-workers. The convec-tive state exhibits an unstable mode. This mode is coupled to an external oscillation and it is possible to drive the system towards the onset of chaos via quasiperiod-icity, by tuning the ratio between the two frequencies to an irrational number. At the critical point an attractor is extracted from a stroboscopic temperature signal and the corresponding f(α) spectrum is computed. A similar scenario is found in the theory of dynamical systems with two characteristic frequencies. The simplest such systems are discrete mappings expressed in terms of angular variables, circle maps. For circle maps one can tune the ratio of the frequencies to the same irrational value as in the convection experiment. At the critical point where chaos sets in, an attractor is extracted. The associated f(α) spectrum is calculated and a renormalization group analysis establishes it to be universal. This is a theory of no adjustable parameters which can be compared to the experiment. Good agreement is found providing a strong evidence that the convection experiment and circle maps belong to the same universality class.