  The present work analyzes the distribution function of the finite scale local
Lyapunov exponent of a pair fluid particles trajectories in fully developed
incompressible homogeneous isotropic turbulence. According to the hypothesis of
fully developed chaos, this PDF is reasonably estimated by maximizing the
entropy associated to such distribution, resulting to be an uniform
distribution function in a proper interval of variation of the local Lyapunov
exponents. From this PDF, we determine the relationship between the average and
maximum Lyapunov exponents and the longitudinal velocity correlation function.
This link, which leads to the closure of von K\`arm\`an--Howarth and Corrsin
equations, agrees with the relation obtained in the previous work, supporting
the proposed PDF calculation, at least for the purposes of the energy cascade
effect estimation. Furthermore, through the property that the Lyapunov vectors
tend to align to the direction of the maximum growth rate of trajectories
distance, we obtain the link between maximum and average Lyapunov exponents in
line with the previous result.
