Possible statistics of scale invariant systems

Journal de Physique II

Volumn 6, Issue 5, 1996, Pages 797-816

  Dubrulle, B ^a,b   Graner, F ^c  

^a UNIVERSITÉ PARIS DIDEROT   (France)

^b OBSERVATOIRE MIDI PYRÉNÉES   (France)

^c UNIV GRENOBLE ALPES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EQUIVALENCE CLASSES; HYDRODYNAMICS; INVARIANCE; MATHEMATICAL TRANSFORMATIONS; PROBABILITY; RANDOM PROCESSES; RELATIVITY; TURBULENCE;

HYDRODYNAMICAL TURBULENCE; POISSON STATISTICS; SCALE INVARIANT SYSTEMS; SYMMETRY BREAKING; TRANSFORMATIONS COUPLING;

FLUID DYNAMICS;

EID: 0030145469     PISSN: 11554312     EISSN: None     Source Type: Journal    
DOI: 10.1051/jp2:1996211     Document Type: Article

References (23)

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21. Proof: if there were two parameters α, β (or more), for any arbitrary log-coordinates (X,T) and (X′,T′) it would be generically possible to find a solution α, β of equation (A.1), ie an interval could have arbitrary coordinates; on the other hand, without any parameter there would also be no interesting physics. More generally, in a p-dimensional space, there can be p - 1 free parameters coupling the p independent basis coordinates. For less than p - 1, the independence of the coordinates is not fully exploited, and if there were p free parameters or more it would mean that the coordinates do not form a complete basis. For a more detailed discussion see [8].
22. Proof: the group structure is then associated to the only one-dimensional Lie group, which is the same as for the set R of real numbers; see e.g. Chevalley C., "Theory of Lie Groups" (Princeton University Press, 1946). This is also easy to check in a pedestrian way for any V, starting from V′ = 0 by successive infinitesimal increments.
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