  Over the moduli space of rank $n$ semi-stable lattices is a universal family
of tori. Along the fibers, there are natural differential operators and
differential equations, particularly, the heat equations and the Fokker-Planck
equations in statistical mechanics. In this paper, we explain why, by taking
averages over the moduli spaces, all these are connected with the zeros of rank
$n$ non-abelian zeta functions of the field of rationals, which are known lie
on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$,
our current work recovers that of Armitage, which from the beginning motivates
ours. However, we reverse the order of the results and the hypothesis in their
works, i.e. we construct averaged versions of Fokker-Planck equations using the
above structure of non-abelian zeta zeros. This then leads to an infinite
dimensional Hilbert vector bundle with smooth sections parametrized by
non-abelian zeta zeros. We conjectures that the above structure of
Fokker-Planck equation comes naturally from an \lq essential projectively flat
connection' of the infinite dimensional Hilbert bundle and the above family of
smooth sections are its \lq essential pro-flat sections'.
