Thermodynamic Geometry: Evolution, Correlation and Phase Transition

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.