A relativity postulate states the equivalence of rationalized systems of units, constructed as power laws of the scale '. In a scale invariant system, described by a random physical eld , this relativity selects the set of similarity transformations coupling ' and . Acceptable transformations are classied into six possible groups, according to two dimensionless parameters: an exponent C characteristic of the physical system, and describing the small scale / large scale symmetry breaking. Symmetry severely constrains the successive moments of , and hence the shape of its probability distribution. For instance, the Newtonian case C=! 1 corresponds to self-similar statistics, the ultra-relativistic case C= ! 0 to deterministic elds, and the case = 1 to a log-Poisson statistics. These cases are applied to hydrodynamical turbulence in the companion paper.
[1]
Z. She,et al.
Cascade structures and scaling exponents in a dynamical model of turbulence: Measurements and comparison
,
1997
.
[2]
Michael Ghil,et al.
Turbulence and predictability in geophysical fluid dynamics and climate dynamics
,
1985
.
[3]
U. Frisch.
Turbulence: The Legacy of A. N. Kolmogorov
,
1996
.
[4]
Charles M. Grinstead,et al.
Introduction to probability
,
1999,
Statistics for the Behavioural Sciences.
[5]
B. Mandelbrot,et al.
Fractals: Form, Chance and Dimension
,
1978
.
[6]
William Feller,et al.
An Introduction to Probability Theory and Its Applications
,
1967
.
[7]
L. Nottale.
Fractal space-time and microphysics
,
1993
.