Monin-Lundgren hierarchy versus the Hopf equation in the statistical theory of turbulence.

A functional-differential equation equivalent to the infinitely hierarchic system of the coupled equations for many-point velocity distribution functions in turbulence established by Monin and Lundgren is presented, to be compared with the Hopf characteristic functional equation which governs the time evolution of a turbulent velocity field. It turns out that this functional entirely implies the Hopf functional, and vice versa. This functional allows us to view turbulence in a Lagrangian picture simulated by many-particle dynamics.