A Thermodynamic-Geometric View of Irreversibility

Fisher-information-based metrics and thermodynamic length have become central tools in modern thermodynamic geometry, where they quantify the sensitivity of equilibrium states to changes in control parameters and bound the minimal dissipation associated with finite-time transformations. In this work we examine the relative diagnostic power of Fisher-information geometry and Gibbs–Shannon entropy in characterizing structural changes along thermodynamic paths. We emphasize that the notion of irreversibility considered here is geometric rather than trajectorylevel entropy production. Specifically, a nonzero thermodynamic length between nearby equilibrium states signals a finite minimal dissipation for any finite-time protocol connecting them. From this perspective, Fisher information provides a local measure of thermodynamic susceptibility that directly determines thermodynamic length and thus the geometric cost of driving a system through parameter space. Using simple equilibrium models—including a two-level system and a driven harmonic oscillator—we show that Gibbs–Shannon entropy, being a state function, may vary smoothly across regions where the Fisher metric and thermodynamic length exhibit sharp structure. These geometric quantities therefore provide a more sensitive diagnostic of parameter regions where finite-time driving would incur enhanced dissipation. The analysis does not challenge the standard role of entropy production in nonequilibrium thermodynamics; rather, it clarifies how equilibrium information geometry anticipates and constrains irreversible behavior under finite-time driving. Our results highlight the complementary roles of entropy and Fisher-information geometry: entropy characterizes state uncertainty, while the Fisher metric and thermodynamic length quantify the geometric susceptibility of equilibrium states and the minimal dissipation associated with thermodynamic transformations.