A theory of hydrodynamic turbulence based on non-equilibrium statistical mechanics

In earlier papers we have studied the turbulent flow exponents ${\zeta }_p$, where $⟨\mathrm{\mid }\mathrm{\Delta }\mathbf{v}{\mathrm{\mid }}^p⟩\sim {\mathrm{\ell }}^{{\zeta }_p}$ is the contribution to the fluid velocity at small scale $\mathrm{\ell }$. Using ideas of non-equilibrium statistical mechanics we have found $${\zeta }_p=\frac{p}{3}-\frac{1}{\ln \kappa }\ln \mathrm{\Gamma }(\frac{p}{3}+1)$$ where $1\mathrm{/}ln\kappa$ is experimentally $\approx 0.32\pm 0.01$. The purpose of the present note is to propose a somewhat more physical derivation of the formula for ${\zeta }_p$. We also present an estimate $\approx 100$ for the Reynolds number at the onset of turbulence.