{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T19:35:18Z","timestamp":1770492918861,"version":"3.49.0"},"reference-count":112,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T00:00:00Z","timestamp":1692835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Messina","award":["56199"],"award-info":[{"award-number":["56199"]}]},{"name":"University of Messina","award":["56210"],"award-info":[{"award-number":["56210"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we ask ourselves how non-local effects affect the description of thermodynamic systems with internal variables. Usually, one assumes that the internal variables are local, but that their evolution equations are non-local, i.e., for instance, that their evolution equations contain non-local differential terms (gradients, Laplacians) or integral terms with memory kernels. In contrast to this typical situation, which has led to substantial progress in several fields, we ask ourselves whether in some cases it would be convenient to start from non-local internal variables with non-local evolution equations. We examine this point by considering three main lengths: the observation scale R defining the elementary volumes used in the description of the system, the mean free path l of the microscopic elements of the fluid (particles, phonons, photons, and molecules), and the overall characteristic size L of the global system. We illustrate these ideas by considering three-dimensional rigid heat conductors within the regime of phonon hydrodynamics in the presence of thermal vortices. In particular, we obtain a generalization of the Guyer\u2013Krumhansl equation, which may be of interest for heat transport in nanosystems or in systems with small-scale inhomogeneities.<\/jats:p>","DOI":"10.3390\/e25091259","type":"journal-article","created":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T10:06:48Z","timestamp":1692871608000},"page":"1259","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Non-Local Vectorial Internal Variables and Generalized Guyer-Krumhansl Evolution Equations for the Heat Flux"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-6175","authenticated-orcid":false,"given":"Liliana","family":"Restuccia","sequence":"first","affiliation":[{"name":"Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d\u2019Alcontres, Salita Sperone 31, 98166 Messina, Italy"}]},{"given":"David","family":"Jou","sequence":"additional","affiliation":[{"name":"Grup de Fis\u00edca Estad\u00edstica, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Spain"},{"name":"Institut d\u2019Estudis Catalans, Carme 47, 08001 Barcelona, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1103\/PhysRev.148.766","article-title":"Solution of the linearized Boltzmann equation","volume":"148","author":"Guyer","year":"1966","journal-title":"Phys. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1103\/PhysRev.148.778","article-title":"Thermal Conductivity, Second Sound, and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals","volume":"148","author":"Guyer","year":"1966","journal-title":"Phys. Rev."},{"key":"ref_3","unstructured":"Maxwell, J.C. (1872). 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