Thermodynamics Across Scales | Zenodo Skip to main Communities My dashboard Log in Sign up Published April 24, 2026 | Version v1 Preprint Open Thermodynamics Across Scales Authors/Creators Kreder III, Karl J (Contact person) Description On exact-normalized CH^2, the inherited one-point contrast profile is E(z) = -(3/2) log(1 - ||z||^2) = 3 log cosh(r(z)/2), and the canonical family with respect to the exact-normalized realized measure is P_beta(z) proportional to exp(-beta E(z)). Relative to flat volume this gives the exact density p_beta(z) proportional to (1 - ||z||^2)^((3 beta / 2) - 4). This note studies the scale structure already present in that exact law, without invoking coarse graining. We show that there are two distinct exact thresholds. The first is the saturation value beta_sat = 2, which is the tail normalizability threshold already obtained in the thermodynamic note. The second is the shell-core crossover value beta_G = 8/3, with T_G = 3/8, at which the canonical density becomes exactly uniform relative to flat volume. For beta > beta_G, the distinguished point becomes a true quadratic well and the exact dynamics admit a local Ornstein-Uhlenbeck approximation with drift rate 3 beta - 8. Thus the exact law remains exponential in the contrast variable E, while a local Gaussian regime emerges intrinsically in fluctuation coordinates. This provides a concrete example of scale separation by dominant balance within one exact law rather than by coarse graining. Files Thermodynamics_Across_Scales-1.pdf Files (266.9 kB) Name Size Download all Thermodynamics_Across_Scales-1.pdf md5:cc755b649579e656e55ae7f3a1af200e 266.9 kB Preview Download 30 Views 26 Downloads Show more details All versions This version Views Total views 30 30 Downloads Total downloads 26 26 Data volume Total data volume 7.5 MB 7.5 MB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Details DOI DOI Badge DOI 10.5281/zenodo.19716256 Markdown [](https://doi.org/10.5281/zenodo.19716256) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19716256.svg :target: https://doi.org/10.5281/zenodo.19716256 HTML <a href="https://doi.org/10.5281/zenodo.19716256"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19716256.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19716256.svg Target URL https://doi.org/10.5281/zenodo.19716256 Resource type Preprint Publisher Zenodo Rights License Creative Commons Attribution 4.0 International The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited. Read more Citation Export Technical metadata Created April 24, 2026 Modified April 24, 2026 Jump up About About Policies Infrastructure Principles Projects Roadmap Contact Blog Blog Support Help FAQ Developers REST API OAI-PMH Contribute GitHub Donate Funded by Powered by CERN Data Centre & InvenioRDM Status Privacy policy Cookie policy Terms of Use This site uses cookies. Find out more on how we use cookies Accept all cookies Accept only essential cookies