Dynamical ensembles equivalence in fluid mechanics

An attempt to put together various theoretical, mathematical, or experimental results recently developed in apparently unrelated subjects, Namely Ruelle's (1980) approach to turbulence, the body of Nose-Hoover type of molecular dynamics experiments, (Evans and Morriss, 1990; Evans et al., 1993; Bonetto et al., 1996), mathematical results on Lyapunov exponents (the pairing rule, Dressier (1988) and Dettman and Morriss (1995)) and experimental results on them (Sarman et al., 1992; Bohr et al., 1996), theoretical as well as mathematical results on fluctuations, Frisch and Parisi (1984) (multifractality), and Gallavoti and Cohen (1995) (chaotic hypothesis). The key idea that we try to clarify is that of ''dynamical ensembles'', as a generalization of the classical ''equilibrium ensembles'', arguing that they should be identified with the SRB distributions and that they share several properties with the classical ensembles. Most of the results invoked here did not deal directly with the Navier-Stokes equations and yet they seem to have a lot to do with them (as we shall argue): here the discussion will focus on the Navier-Stokes and dissipative Euler equations with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis. The ideas developed, to a great extent, from the efforts put in interpreting the experimental results in (Bonetto et al., 1996).