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

[2606.29751] Finite-resolution exhaustive traversal of thermodynamic state spaces has divergent thermodynamic length Skip to main content Search Submit Donate Log in Condensed Matter > Statistical Mechanics arXiv:2606.29751 (cond-mat) [Submitted on 29 Jun 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: Finite-resolution exhaustive traversal of thermodynamic state spaces has divergent thermodynamic length Authors: Satori Tsuzuki View PDF HTML (experimental) 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. Subjects: Statistical Mechanics (cond-mat.stat-mech) ; Mathematical Physics (math-ph) Cite as: arXiv:2606.29751 [cond-mat.stat-mech] (or arXiv:2606.29751v2 [cond-mat.stat-mech] for this version) https://doi.org/10.48550/arXiv.2606.29751 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Satori Tsuzuki Ph.D [ view email ] [v1] Mon, 29 Jun 2026 03:48:59 UTC (209 KB) [v2] Tue, 25 Aug 2026 02:28:38 UTC (209 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.stat-mech < prev | next > new | recent | 2026-06 Change to browse by: cond-mat math math-ph math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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