Finite-resolution exhaustive traversal of thermodynamic state spaces has divergent thermodynamic length

Abstract

Mapping an entire multidimensional thermodynamic control region is fundamentally different from driving a system between prescribed endpoints. We formulate such mapping as a finite-resolution coverage problem: a rectifiable protocol must come within thermodynamic distance $\varepsilon$ of every point in a regular $d$-dimensional state-space window. A covering argument shows that the required thermodynamic length grows at least as $\varepsilon^{(1-d)}$. Hilbert- and Peano-type finite traversals attain the same exponent, demonstrating that it is fixed by the codimension of a curve rather than by a particular scanning construction. When the coverage metric is a physical friction tensor, or is uniformly dominated by one, this geometric law implies a slow-driving resource constraint: at fixed duration, the quadratic excess-work cost grows at least as $\varepsilon^{(2(1-d))}$, whereas at fixed excess-work budget the required duration grows with the same power. We derive microscopic friction metrics for a detailed-balance three-state Markov jump process and an overdamped harmonic trap. Raster scans of the trap illustrate how refinement costs can appear either as excess work or as acquisition time, depending on the allocation of time along the protocol. Finite experimental or numerical resolution cuts off the continuum divergence, while singular thermodynamic response can additionally modify the prefactor according to a directional-integrability criterion. Morton/Z-order traversal preserves the universal exponent but increases locality-dependent amplitudes. These results establish finite-resolution state-space coverage as a resource problem distinct from endpoint-to-endpoint thermodynamic control.