  We study the Weyl formula for the asymptotic number of eigenvalues of the
Laplace-Beltrami operator with Dirichlet boundary condition on a Riemannian
manifold in the context of geometric flows. Assuming the eigenvalues to be the
energies of some associated statistical system, we show that geometric flows
are directly related with the direction of increasing entropy chosen. For a
closed Riemannian manifold we obtain a volume preserving flow of geometry being
equivalent to the increment of Gibbs entropy function derived from the spectrum
of Laplace-Beltrami operator. Resemblance with Arnowitt, Deser, and Misner
(ADM) formalism of gravity is also noted by considering open Riemannian
manifolds, directly equating the geometric flow parameter and the direction of
increasing entropy as time direction.
