Entropy-Time Geodesics as a Universal Framework for Transport and Transition Phenomena

TL;DR

The paper presents a unified geometric framework for irreversible transport by combining a thermodynamic state-space metric, derived from the Hessian of a chosen potential, with a dissipation metric from Onsager reciprocity. Entropy production defines a thermodynamic proper time τ that reparametrizes evolution, and a thermodynamic action yields geodesic transport equations; for an incompressible Newtonian fluid this recovers the Navier–Stokes equations without external constitutive closure. A balance between inertial curvature and dissipation identifies a geometric Kolmogorov length η = (ν^3/ε)^{1/4} as a minimal geometric resolution, and breakdown corresponds to metric degeneracy signaling phase transitions such as cavitation. Beyond fluids, the framework applies to heat conduction, diffusion, and electrochemical processes by selecting appropriate potentials and Onsager matrices, offering a general, geometry-driven perspective on stability, scaling, and breakdown in irreversible systems.

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. 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 and other irreversible processes.

Figures (2)

• Figure 1: Schematic representation of the thermodynamic manifold. Entropy defines a time-like direction with proper time $d\tau=\sqrt{T/C_P}\,dS$, while pressure spans a spatial-like axis. Null curves $ds^2=0$ define thermodynamic causality via $c_{\mathrm{th}}^2=T/(VC_P\kappa_S)$. Contours of constant $\tau$ spread apart (become sparse) near critical points where $C_P\to\infty$, reflecting the divergence of the metric and the physical slowing of relaxation.
• Figure 2: Schematic of geometric breakdown at cavitation. (Left) Liquid Manifold: As the fluid approaches the spinodal limit, the thermodynamic geometry stretches—represented by the sparse spacing of the vertical pressure grid lines—due to diverging adiabatic compressibility ($\kappa_S \to \infty$) and the singularity of the metric component $g_{PP} \to -\infty$. The fluid trajectory (blue curve) is forced to become vertical as the speed of sound vanishes ($c_s \to 0$), creating an infinite barrier in thermodynamic proper time ($d\tau \to \infty$). (Center) The system undergoes a topological transition (gray dashed arrow) across the singular region where the continuum description fails. (Right) Vapor Manifold: The trajectory resumes on a distinct, stable manifold with non-degenerate geometry.