The generating functional for the probability density functions of Navier-Stokes turbulence

A generating functional for the equal-time spatial probability density functions which represent the ensemble of turbulent incompressible Navier-Stokes fluids is introduced. By formally solving the linear evolution equation satisfied by this functional, the probability densities are represented as functional integrals. It is shown that the generating functional can be regarded as the space characteristic functional of a generalized random field defined on the phase space spanned by the material position and velocity fields of a fluid particle. The interpretation of this random field, which satisfies a dynamical equation similar to Vlasov's, is clarified through the formal analogies between the statistics of molecules and fluid particles at the functional level. A class of statistically realizable and solvable models is also considered within the context of the present formalism.