Non-equilibrium statistical mechanics of turbulence
David Ruelle

The macroscopic study of hydrodynamic turbulence is equivalent, 
at an abstract level, to the microscopic study of a heat flow for a 
suitable mechanical system.  Turbulent fluctuations (intermittency) 
then correspond to thermal fluctuations, and this allows to estimate 
the exponents tau_p and zeta_p associated with moments of dissipation 
fluctuations and velocity fluctuations.  This approach, initiated in 
an earlier note, is pursued here more carefully.  In particular we 
derive probability distributions at finite Reynolds number for the 
dissipation and velocity fluctuations, and the latter permit an 
interpretation of numerical experiments.  Specifically, if p(z)dz 
is the probability distribution of the radial velocity gradient we can 
explain why, when the Reynolds number increases, log p(z) passes from 
a concave to a linear then to a convex profile for large z as observed.  
We show that the central limit theorem applies to the dissipation 
and velocity distribution functions, so that a logical relation with 
the lognormal theory of Kolmogorov and Obukhov is established.  We find 
however that the lognormal behavior of the distribution functions fails
 at large value of the argument, so that a lognormal theory cannot 
correctly predict the exponents tau_p and zeta_p.