### Non-equilibrium statistical mechanics of turbulence

David Ruelle

Abstract:
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.