  In this communication, the closure formulas of von K\'arm\'an--Howarth and
Corrsin equations are obtained through the Liouville theorem and the hypothesis
of homogeneous isotropic incompressible turbulence. Such closures, based on the
concept that, in fully developed turbulence, contiguous fluid particles
trajectories continuously diverge, are of non--diffusive nature, and express a
correlations spatial propagation phenomenon between the several scales which
occurs with a propagation speed depending on length scale and velocity standard
deviation. These closure formulas coincide with those just obtained in previous
works through the finite scale Lyapunov analysis of the fluid act of motion.
Here, unlike the other articles, the present study does not use the Lyapunov
theory, and provides the closures showing first an exact relationship between
the pair spatial correlations calculated with the velocity distribution
function and those obtained using the material separation line distribution
function. As this analysis does not adopt the Lyapunov theory, this does not
need the definition and/or the existence of the Lyapunov exponents.
Accordingly, the present proof of the closures results to be more general and
rigorous than that presented in the other works, corroborating the previous
results. Finally, the conditions of existence of invariants in isotropic
turbulence are studied by means of the proposed closures. In the presence of
such invariants and self--similarity, the sole evolution of velocity and
temperature standard deviations and of the correlation scales is shown to be
adequate to fairly describe the isotropic turbulence.
