Thermodynamic geometry of friction on graphs: Resistance, commute times, and optimal transport

Abstract: We demonstrate that for slowly driven reversible Markov chains, the thermodynamic friction metric governing dissipation is equivalent to two independently developed graph-theoretic geometries: commute-time geometry and resistance distance. This equivalence yields complementary physical insights: the commute-time metric provides the local Euclidean description of the thermodynamic manifold, while the resistance distance maps dissipation to power loss in an electrical network. We further show that this metric arises from a discrete $L^2$-Wasserstein transport cost evaluated along paths of equilibrium distributions, extending a correspondence previously shown for continuous-state processes. These results unify linear-response, electrical-resistance, random-walk, and optimal-transport frameworks, revealing linear-response dissipation as the energetic cost of transporting probability through the intrinsic geometry of the state-space network.