{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T22:16:56Z","timestamp":1780525016137,"version":"3.54.1"},"reference-count":39,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T00:00:00Z","timestamp":1701820800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The complexity measure for the distribution in space-time of a finite-velocity diffusion process is calculated. Numerical results are presented for the calculation of Fisher\u2019s information, Shannon\u2019s entropy, and the Cram\u00e9r\u2013Rao inequality, all of which are associated with a positively normalized solution to the telegrapher\u2019s equation. In the framework of hyperbolic diffusion, the non-local Fisher\u2019s information with the x-parameter is related to the local Fisher\u2019s information with the t-parameter. A perturbation theory is presented to calculate Shannon\u2019s entropy of the telegrapher\u2019s equation at long times, as well as a toy model to describe the system as an attenuated wave in the ballistic regime (short times).<\/jats:p>","DOI":"10.3390\/e25121627","type":"journal-article","created":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T03:48:41Z","timestamp":1701834521000},"page":"1627","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Fisher and Shannon Functionals for Hyperbolic Diffusion"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2135-3126","authenticated-orcid":false,"given":"Manuel O.","family":"C\u00e1ceres","sequence":"first","affiliation":[{"name":"Comision Nacional de Energia Atomica, Centro Atomico Bariloche and Instituto Balseiro, Universidad Nacional de Cuyo, Av. E. Bustillo 9500, Bariloche CP 8400, Argentina"},{"name":"CONICET, Centro Atomico Bariloche, Av. E. Bustillo 9500, Bariloche CP 8400, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4551-6161","authenticated-orcid":false,"given":"Marco","family":"Nizama","sequence":"additional","affiliation":[{"name":"Departamento de Fisica, Facultad de Ingenieria and CONICET, Universidad Nacional del Comahue, Neuquen CP 8300, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Flavia","family":"Pennini","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Facultad de Ingenier\u00eda, Universidad Nacional de Mar del Plata (UNMDP), CONICET, Mar del Plata CP 7600, Argentina"},{"name":"Departamento de F\u00edsica, Universidad Cat\u00f3lica del Norte, Av. Angamos 0610, Antofagasta 1270709, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,6]]},"reference":[{"key":"ref_1","unstructured":"Pearson, J.M. (1966). A Theory of Waves, Allyn and Bacon, Inc.. Chapter 1.7."},{"key":"ref_2","unstructured":"Landau, L.D., Bell, J.S., and Kearsley, M.J. (2013). Electrodynamics of Continuous Media, Elsevier Science. Chapter 71."},{"key":"ref_3","unstructured":"Heaviside, O. (1970). Electrical Papers of Oliver Heaviside, Chelsea."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4334","DOI":"10.1063\/1.530856","article-title":"The behavior of hyperbolic heat equations\u2019 solutions near their parabolic limits","volume":"35","author":"Nagy","year":"1994","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7277","DOI":"10.1088\/0305-4470\/30\/21\/006","article-title":"The generalized Cattaneo equation for the description of anomalous transport processes","volume":"30","author":"Compte","year":"1997","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"022128","DOI":"10.1103\/PhysRevE.102.022128","article-title":"Generalized Cattaneo (telegrapher\u2019s) equations in modeling anomalous diffusion phenomena","volume":"102","author":"Horzela","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_7","unstructured":"Bear, M.F., Connors, B.W., and Paradiso, M.A. (2016). Neuroscience: Exploring the Brain, Wolters Kluwer. [4th ed.]."},{"key":"ref_8","unstructured":"Martinsen, \u00f8., and Jensen, \u00f8. (2009). An Anthology of Developments in Clinical Engineering and Bioimpedance: Festschrift for Sverre Grimnes, Unipub Forlag."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Graaff, R., and Hoenders, B.J. (2008, January 16\u201319). Telegrapher\u2019s Equation for Light Transport in Tissue with Substantial Absorption. Proceedings of the Biomedical Optics 2008, St. Petersburg, FL, USA.","DOI":"10.1364\/BIOMED.2008.BSuE52"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Shlepnev, Y. (2012, January 21\u201324). Coupled 2D telegrapher\u2019s equations for PDN analysis. Proceedings of the 2012 IEEE 21st Conference on Electrical Performance of Electronic Packaging and Systems, Tempe, AZ, USA.","DOI":"10.1109\/EPEPS.2012.6457870"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"054107","DOI":"10.1103\/PhysRevE.107.054107","article-title":"Penetration of waves in global stochastic conducting media","volume":"107","author":"Nizama","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"17","DOI":"10.13182\/NSE09-77","article-title":"Asymptotic telegrapher\u2019s equation (P1) approximation for the transport equation","volume":"166","author":"Heizler","year":"2010","journal-title":"Nucl. Sci. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1016\/0378-4371(87)90043-4","article-title":"Non-Markovian diffusion-like equation for transport processes with anisotropic scattering","volume":"142","author":"Wio","year":"1987","journal-title":"Physica A"},{"key":"ref_14","first-page":"044131","article-title":"Stochastic telegrapher\u2019s approach for solving the random Boltzmann-Lorentz gas","volume":"2022 105","author":"Nizama","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.enganabound.2019.11.010","article-title":"Solving the telegraph equation in 2-D and 3-D using generalized finite difference method (GFDM)","volume":"112","author":"Gavete","year":"2020","journal-title":"Eng. Anal. Boundary Elem. V."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1007\/s10955-019-02395-0","article-title":"Random spherical hyperbolic diffusion","volume":"177","author":"Broadbridge","year":"2019","journal-title":"J. Stat. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Broadbridge, P., Kolesnik, A.D., Leonenko, N., Olenko, A., and Omari, A.D. (2020). Spherically restricted random hyperbolic diffusion. Entropy, 22.","DOI":"10.3390\/e22020217"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1007\/s10955-020-02553-9","article-title":"Finite-velocity diffusion in random media","volume":"179","year":"2020","journal-title":"J. Stat. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"20005","DOI":"10.1209\/0295-5075\/124\/20005","article-title":"Finite-velocity diffusion on a comb","volume":"124","author":"Sandev","year":"2018","journal-title":"Europhys. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1120","DOI":"10.1073\/pnas.0307052101","article-title":"Diffusion at finite speed and random walks","volume":"101","author":"Keller","year":"2004","journal-title":"Proc. Nac. Acd. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s10955-021-02699-0","article-title":"Comments on wave-like propagation with binary disorder","volume":"182","year":"2021","journal-title":"J. Stat. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"045218","DOI":"10.1063\/5.0049572","article-title":"Surface gravity waves on randomly irregular floor and the telegrapher\u2019s equation","volume":"11","year":"2021","journal-title":"AIP Adv."},{"key":"ref_23","first-page":"1","article-title":"On thermal waves\u2019 velocity: Some open questions in thermal waves\u2019 physics","volume":"62","year":"2016","journal-title":"Rev. Mex. Fis. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"L83","DOI":"10.1088\/0143-0807\/34\/5\/L83","article-title":"On thermal waves","volume":"34","author":"Marin","year":"2013","journal-title":"Eur. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_26","unstructured":"Fisher, R.A. (1959). Statistical Methods and Scientific Inference, Oliver and Boyd. [2nd ed.]. Chapter IV."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2428","DOI":"10.1016\/j.physleta.2007.12.008","article-title":"The Fisher-Shannon information plane for atoms","volume":"372","author":"Sen","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1140\/epjd\/e2009-00251-1","article-title":"Configuration complexities of hydrogenic atoms","volume":"55","author":"Dehesa","year":"2009","journal-title":"Eur. Phys. J. D"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"C\u00e1ceres, M.O. (2017). Non-Equilibrium Statistical Physics with Application to Disordered Systems, Springer.","DOI":"10.1007\/978-3-319-51553-3"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cram\u00e9r, H. (1946). Mathematical Methods of Statistics, Princeton University Press. Chapter 32.","DOI":"10.1515\/9781400883868"},{"key":"ref_31","unstructured":"Rao, C.R., and Das Gupta, S. (1994). Selected Papers of C. R. Rao, Wiley."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"053037","DOI":"10.1088\/1367-2630\/acd321","article-title":"Fisher information of correlated stochastic processes","volume":"25","author":"Radaelli","year":"2023","journal-title":"New J. Phys."},{"key":"ref_33","unstructured":"Yoshida, N. (2004). Stochastic Processes and Applications to Mathematical Finance, Proceedings of the March 3rd-7th 2004 Symposium at Ritsumeikan University BKC, Biwako Kusatsu Campus, Ritsumeikan University."},{"key":"ref_34","unstructured":"Morse, P.M., and Feshbach, H. (1953). Methods of Theoretical Physics, McGraw-Hill Book Company, Inc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1216\/RMJ-1974-4-3-497","article-title":"A Stochastic model related to the telegraphers equation","volume":"4","author":"Kac","year":"1974","journal-title":"Rocky Mt. J. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1088\/0143-0807\/17\/4\/008","article-title":"Finite-velocity diffusion","volume":"17","author":"Masoliver","year":"1996","journal-title":"Eur. J. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"129","DOI":"10.2528\/PIER99071601","article-title":"Wave packets and group velocity in absorbing media: Solutions of the telegrapher\u2019s equation","volume":"27","author":"Sonnenschein","year":"2000","journal-title":"Prog. Electromagn. Res. PIER"},{"key":"ref_38","unstructured":"Spanier, J., and Oldham, K. (1987). An Atlas of Functions, Springer."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1007\/BF01210677","article-title":"Sur une propri\u00e9t\u00e9 de la loi de Gau\u03b2","volume":"44","author":"Marcinkiewicz","year":"1939","journal-title":"Math. Z."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1627\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:33:56Z","timestamp":1760132036000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1627"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,6]]},"references-count":39,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["e25121627"],"URL":"https:\/\/doi.org\/10.3390\/e25121627","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,6]]}}}