Surface Topography as an Energy State: Hamiltonian Description of Functional Properties in Phase Space and Quantification via Optical Methods

The assessment of tribological processes necessitates comprehensive monitoring and interpretation of surface states, traditionally by topography height values and surface texture parameters. This paper proposes a phase space representation as generic interpretation of surfaces to enhance tribological assessments. By deriving distinctions between two-dimensional and three-dimensional manifolds from phenomenological thermodynamics, we demonstrate the existence of fractal dimensions on physical surfaces. Utilizing the smallest representable wavelength, a coordinate ensemble where each coordinate has a generalized degree of freedom and a differential amount across fractal dimensions is proposed. The metric is interpreted as a time coordinate, allowing the derivative to be seen as velocity. Through geometric dimension analysis, we derive a topographic Hamiltonian via generalized momentum. Each coordinate pair in the ensemble is assigned a location in phase space. Statistical considerations define a symplectic structure and phase space volume. As a use-case, a two-disc experiment demonstrates that topography changes under load can be understood as a self-organization process through thermodynamic reasoning. As a possibility for a direct assessment of the phase space of a surface for monitoring tribological processes, we propose the combination of topographic measuring systems with scattering light measurement. This multi sensor approach offers opportunities for optimizing performance in tribological applications.