  Under the fluctuation of the electric charge and atomic mass, this paper
considers the theory of the thin film depletion layer formation of an ensemble
of finitely excited, non-empty $d/f$-orbital heavy materials, from the
thermodynamic geometric perspective. At each state of the local adiabatic
evolutions, we examine the nature of the thermodynamic parameters,
\textit{viz.}, electric charge and mass, changing at each respective
embeddings. The definition of the intrinsic Riemannian geometry and
differential topology offers the properties of (i) local heat capacities, (ii)
global stability criterion and (iv) global correlation length. Under the
Gaussian fluctuations, such an intrinsic geometric consideration is anticipated
to be useful in the statistical coating of the thin film layer of a desired
quality-fine high cost material on a low cost durable coatant. From the
perspective of the daily-life applications, the thermodynamic geometry is thus
intrinsically self-consistent with the theory of the local and global economic
optimizations. Following the above procedure, the quality of the thin layer
depletion could self-consistently be examined to produce an economic, quality
products at a desired economic value.
