Topographic temperature: a maximum-entropy state description of running-in surfaces

Abstract Surface topography governs tribological performance, yet conventional parameters describe either amplitude statistics or spectral content in isolation. We introduce a scale-dependent framework that represents surface height and directional gradient as conjugate coordinates of a structural phase space. The elastic reference energy derived from Persson’s contact mechanics theory defines a metric that couples surface geometry to the elastic half-space response. A maximum-entropy formulation yields a canonical state density. In the Gaussian limit this formulation recovers Persson’s spectral description exactly, showing that the power spectral density is a complete contact mechanical descriptor only under Gaussian statistics. The associated Lagrange multiplier defines a topographic temperature in the sense of Grmela’s multiscale thermodynamics and embeds the areal subsystem within a scale-dependent boundary potential. The framework is validated experimentally using ground and honed AISI 52 100 steel discs before and after running in. The ground surface contracts toward lower entropy and elastic energy, whereas the honed surface expands into previously unoccupied states. These opposite trajectories become transparent only in the coupled height-gradient representation and highlight the role of the principal directional gradient for scale-aware surface metrology.