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

Abstract

We demonstrate that the thermodynamic friction metric governing dissipation in slowly driven continuous-time Markov chains is equivalent to the commute-time embedding and the resistance distance. This equivalence yields complementary insights: The commute-time embedding demonstrates the intrinsic cost of transporting probability across dynamical bottlenecks, while the resistance distance maps thermodynamic dissipation to Joule heating in an electrical network. We further demonstrate that the linear-response thermodynamic distance is a discrete $L^2$-Wasserstein optimal transport cost evaluated along paths of equilibrium distributions, extending a continuous-state correspondence to discrete networks. This conceptual synthesis of linear-response thermodynamics, random walks on graphs, electrical circuits, and optimal-transport theory connects independently developed geometric frameworks, reduces complex metric calculations to simple circuit algebra, and provides a clear physical picture of dissipation as the energetic cost of routing probability through the state space network.

Figures (2)

• Figure 1: (a) The Markov graph for a linear chain of states and its series circuit representation. The orientation of the edge currents $i_0$ reflect the convention in the main text. (b) Circuit representation of a cyclic Markov graph, with total currents $i(x)$ decomposed into a reference current $i_0(x)$ obtained by a cut along $(n,0)$ and a cycle correction $i_{\text{cyc}}$.
• Figure 2: Three-state cycle with stationary current $j_{\text{ss}}$.