POSSIBLE STATISTICS OF SCALE-INVARIANT SYSTEMS

A relativity postulate states the equivalence of rationalized systems of units, constructed as power laws of the scale l. In a scale invaria nt system, described by a random physical field phi, this relativity s elects the set of similarity transformations coupling l and phi. Accep table transformations are classified into six possible groups, accordi ng to two dimensionless parameters: an exponent C characteristic of th e physical system, and Lambda describing the small scale / large scale symmetry breaking. Symmetry severely constrains the successive moment s of phi, and hence the shape of its probability distribution. For ins tance, the Newtonian case C/Lambda --> infinity corresponds to self-si milar statistics, the ultra-relativistic case C/Lambda --> 0 to determ inistic fields, and the case Lambda = 1 to a log-Poisson statistics. T hese cases are applied to hydrodynamical turbulence in the companion p aper.