Entropy-time geodesics as a universal framework for transport and transition phenomena

Abstract We develop a geometric framework for irreversible transport phenomena in which macroscopic evolution equations arise from the combined structure of a thermodynamic state metric and an Onsager-based dissipation metric. The construction begins by defining a pseudo-Riemannian manifold from the Hessian of an appropriate thermodynamic potential. When the enthalpy is used and written in variables ( S , P ), the resulting metric possesses a Lorentzian-type signature: entropy acts as a time-like coordinate, while pressure forms a spatial-like coordinate associated with mechanical response. Local irreversible dynamics are incorporated through the inverse Onsager matrix, which defines a positive-definite dissipation metric on the space of fluxes and gradients. A thermodynamic action integrating these two geometric layers yields geodesic evolution equations. For a Newtonian fluid with constant viscosity, the resulting Euler–Lagrange equations reproduce the incompressible Navier–Stokes equations without requiring an externally imposed constitutive closure. Within this framework, turbulence scaling emerges from competition between inertial curvature and dissipation metric stiffness. The Kolmogorov length scale appears as a minimum geometric resolution length where these contributions balance, providing a geometric interpretation of energy cascade termination and dissipation onset. Finite-time singularities in the classical PDE formulation correspond to curvature divergences in the transport geometry; however, the thermodynamic proper time diverges in such limits, suggesting that blow-up is dynamically suppressed in single-phase continua. Breakdown occurs only when the thermodynamic metric itself becomes degenerate, such as at cavitation or critical points where compressibility or heat capacity diverge, corresponding to a change in manifold topology. Although derived explicitly for fluid flow, the framework is general: by choosing different thermodynamic potentials and Onsager matrices, the same geometric formulation applies to heat conduction, diffusion, electrochemical transport, and other irreversible processes. The results provide a unified thermodynamic geometric basis for transport equations, clarify the conditions under which continuum models remain valid, and offer a new interpretation of turbulence scaling and phase-transition-induced breakdown.