{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:41:27Z","timestamp":1760150487467,"version":"build-2065373602"},"reference-count":61,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T00:00:00Z","timestamp":1700697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CNPq","award":["301715\/2022-0","304634\/2020-4","465259\/2014-6","FAPESP\u20132014\/50983-3"],"award-info":[{"award-number":["301715\/2022-0","304634\/2020-4","465259\/2014-6","FAPESP\u20132014\/50983-3"]}]},{"name":"the National Council for Scientific and Technological Development","award":["301715\/2022-0","304634\/2020-4","465259\/2014-6","FAPESP\u20132014\/50983-3"],"award-info":[{"award-number":["301715\/2022-0","304634\/2020-4","465259\/2014-6","FAPESP\u20132014\/50983-3"]}]},{"name":"the National Institute of Science and Technology Complex Fluids (INCT-FCx)","award":["301715\/2022-0","304634\/2020-4","465259\/2014-6","FAPESP\u20132014\/50983-3"],"award-info":[{"award-number":["301715\/2022-0","304634\/2020-4","465259\/2014-6","FAPESP\u20132014\/50983-3"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker\u2013Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.<\/jats:p>","DOI":"10.3390\/e25121578","type":"journal-article","created":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T06:06:23Z","timestamp":1700719583000},"page":"1578","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Entropy Production in a Fractal System with Diffusive Dynamics"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4930-5262","authenticated-orcid":false,"given":"Rafael S.","family":"Zola","sequence":"first","affiliation":[{"name":"Departmento de F\u00edsica, Universidade Tecnol\u00f3gica Federal do Paran\u00e1\u2014Campus de Apucarana, Apucarana 86812-460, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3853-1790","authenticated-orcid":false,"given":"Ervin K.","family":"Lenzi","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6875-1965","authenticated-orcid":false,"given":"Luciano R.","family":"da Silva","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Rio Grande do Norte, Natal 59078-900, RN, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0103-9017","authenticated-orcid":false,"given":"Marcelo K.","family":"Lenzi","sequence":"additional","affiliation":[{"name":"Departamento de Engenharia Qu\u00edmica, Universidade Federal do Paran\u00e1, Curitiba 81531-980, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,23]]},"reference":[{"key":"ref_1","unstructured":"Boltzmann, L. (2003). The Kinetic Theory of Gases, World Scientific."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gibbs, J.W. (2010). Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics, Cambridge Library Collection-Mathematics, Cambridge University Press.","DOI":"10.1017\/CBO9780511686948"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/S0378-4371(00)00007-8","article-title":"Statistical mechanics based on Renyi entropy","volume":"280","author":"Lenzi","year":"2000","journal-title":"Physica A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1103\/PhysRevLett.83.1711","article-title":"Implications of form invariance to the structure of nonextensive entropies","volume":"83","author":"Rajagopal","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"052112","DOI":"10.1103\/PhysRevE.95.052112","article-title":"Composition law of \u03ba-entropy for statistically independent systems","volume":"95","author":"Kaniadakis","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1140\/epja\/i2009-10799-0","article-title":"Nonadditive entropy: The concept and its use","volume":"40","author":"Tsallis","year":"2009","journal-title":"Eur. Phys. J. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1140\/epjb\/e2006-00268-8","article-title":"Relaxation of a test particle in systems with long-range interactions: Diffusion coefficient and dynamical friction","volume":"52","author":"Chavanis","year":"2006","journal-title":"Eur. Phys. J. B-Condens. Matter Complex Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1103\/PhysRevE.62.2213","article-title":"Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions","volume":"62","author":"Bologna","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"214501","DOI":"10.1103\/PhysRevLett.109.214501","article-title":"Superfast nonlinear diffusion: Capillary transport in particulate porous media","volume":"109","author":"Lukyanov","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","unstructured":"Aronson, D.G. (2006). Nonlinear Diffusion Problems: Lectures Given at the 2nd 1985 Session of the Centro Internazionale Matermatico Estivo (CIME) Held at Montecatini Terme, Italy 10\u201318 June 1985, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"R2197","DOI":"10.1103\/PhysRevE.54.R2197","article-title":"Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis","volume":"54","author":"Tsallis","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1103\/PhysRevE.61.1417","article-title":"Anomalous diffusion with absorption: Exact time-dependent solutions","volume":"61","author":"Drazer","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_13","unstructured":"Frank, T.D. (2005). Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer Science & Business Media."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1903","DOI":"10.1016\/j.physleta.2018.05.009","article-title":"A nonlinear Fokker\u2013Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics","volume":"382","author":"Marin","year":"2018","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Liang, Y., Chen, W., and Cai, W. (2019). Hausdorff Calculus: Applications to Fractal Systems, Fractional Calculus in Applied Sciences and Engineering, De Gruyter.","DOI":"10.1515\/9783110608526"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1111\/ffe.12110","article-title":"Bridge fatigue life prediction using Mittag\u2013Leffler distribution","volume":"37","author":"Liang","year":"2014","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.sigpro.2012.07.035","article-title":"A survey on computing L\u00e9vy stable distributions and a new MATLAB toolbox","volume":"93","author":"Liang","year":"2013","journal-title":"Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.cam.2014.10.016","article-title":"On conformable fractional calculus","volume":"279","author":"Abdeljawad","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/j.chaos.2005.08.199","article-title":"Time\u2013space fabric underlying anomalous diffusion","volume":"28","author":"Chen","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"024125","DOI":"10.1103\/PhysRevE.108.024125","article-title":"Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations","volume":"108","author":"Wei","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear Models of Dissipation whose Q is almost Frequency Independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.chaos.2013.03.013","article-title":"Fokker\u2013Planck equation on fractal curves","volume":"52","author":"Satin","year":"2013","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.1515\/fca-2019-0079","article-title":"A time-space Hausdorff derivative model for anomalous transport in porous media","volume":"22","author":"Liang","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"122943","DOI":"10.1016\/j.molliq.2023.122943","article-title":"Probing modulated liquid crystal media with dielectric spectroscopy","volume":"390","author":"Rosseto","year":"2023","journal-title":"J. Mol. Liq."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"037801","DOI":"10.1103\/PhysRevLett.111.037801","article-title":"Fractional Hoppinglike Motion in Columnar Mesophases of Semiflexible Rodlike Particles","volume":"111","author":"Naderi","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2960","DOI":"10.1529\/biophysj.104.051078","article-title":"Anomalous Diffusion of Proteins Due to Molecular Crowding","volume":"89","author":"Banks","year":"2005","journal-title":"Biophys. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"083510","DOI":"10.1063\/1.4892155","article-title":"Anisotropic fractal media by vector calculus in non-integer dimensional space","volume":"55","author":"Tarasov","year":"2014","journal-title":"J. Math. Phys."},{"key":"ref_28","unstructured":"V\u00e1zquez, J.L. (2007). The Porous Medium Equation: Mathematical Theory, Oxford University Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"053301","DOI":"10.1063\/1.5001189","article-title":"Nonlinear Fokker-Planck equations in super-diffusive and sub-diffusive regimes","volume":"60","author":"Casas","year":"2019","journal-title":"J. Math. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Plastino, A.R., Wedemann, R.S., and Tsallis, C. (2021). Nonlinear fokker-planck equation for an overdamped system with drag depending on direction. Symmetry, 13.","DOI":"10.3390\/sym13091621"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"137752","DOI":"10.1016\/j.physletb.2023.137752","article-title":"From the Boltzmann equation with non-local correlations to a standard non-linear Fokker-Planck equation","volume":"839","author":"Deppman","year":"2023","journal-title":"Phys. Lett. B"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Evangelista, L.R., and Lenzi, E.K. (2023). Nonlinear Fokker\u2013Planck Equations, H-Theorem and Generalized Entropy of a Composed System. Entropy, 25.","DOI":"10.3390\/e25091357"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"041123","DOI":"10.1103\/PhysRevE.76.041123","article-title":"Consequences of the H theorem from nonlinear Fokker-Planck equations","volume":"76","author":"Nobre","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1294","DOI":"10.1016\/j.cjph.2017.07.003","article-title":"Nonlinear Fokker\u2013Planck equations, H\u2013theorem, and entropies","volume":"55","author":"Lenzi","year":"2017","journal-title":"Chin. J. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"052109","DOI":"10.1103\/PhysRevE.96.052109","article-title":"Entropic nonadditivity, H theorem, and nonlinear Klein-Kramers equations","volume":"96","author":"Lenzi","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1140\/epjb\/e2008-00142-9","article-title":"Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations","volume":"62","author":"Chavanis","year":"2008","journal-title":"Eur. Phys. J. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1140\/epjb\/e2007-00217-1","article-title":"A general nonlinear Fokker-Planck equation and its associated entropy","volume":"58","author":"Curado","year":"2007","journal-title":"Eur. Phys. J. B"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/S0378-4371(01)00121-2","article-title":"Thermodynamics\u2019 zeroth law in a nonextensive scenario","volume":"295","author":"Pennini","year":"2001","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1016\/j.physa.2006.04.001","article-title":"Temperature of nonextensive systems: Tsallis entropy as Clausius entropy","volume":"368","author":"Abe","year":"2006","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"061147","DOI":"10.1103\/PhysRevE.83.061147","article-title":"Zeroth law compatibility of nonadditive thermodynamics","volume":"83","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"061136","DOI":"10.1103\/PhysRevE.86.061136","article-title":"Entropy production and nonlinear Fokker-Planck equations","volume":"86","author":"Casas","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"11002","DOI":"10.1209\/0295-5075\/ac7c30","article-title":"H-theorems for systems of coupled nonlinear Fokker-Planck equations","volume":"139","author":"Plastino","year":"2022","journal-title":"Europhys. Lett."},{"key":"ref_43","unstructured":"Penrose, O. (2005). Foundations of Statistical Mechanics: A Deductive Treatment, Courier Corporation."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0003-4916(57)90002-7","article-title":"Irreversible Gibbsian ensembles","volume":"1","author":"Lebowitz","year":"1957","journal-title":"Ann. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjc\/s10052-013-2487-6","article-title":"Black hole thermodynamical entropy","volume":"73","author":"Tsallis","year":"2013","journal-title":"Eur. Phys. J. C"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1080\/00107514.2014.900977","article-title":"An introduction to nonadditive entropies and a thermostatistical approach to inanimate and living matter","volume":"55","author":"Tsallis","year":"2014","journal-title":"Contemp. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"022120","DOI":"10.1103\/PhysRevE.97.022120","article-title":"From the nonlinear fokker-planck equation to the vlasov description and back: Confined interacting particles with drag","volume":"97","author":"Plastino","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"031104","DOI":"10.1103\/PhysRevE.67.031104","article-title":"Crossover in diffusion equation: Anomalous and normal behaviors","volume":"67","author":"Lenzi","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"011147","DOI":"10.1103\/PhysRevE.85.011147","article-title":"Different diffusive regimes, generalized Langevin and diffusion equations","volume":"85","author":"Tateishi","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1140\/epjb\/e2005-00368-y","article-title":"On different q-systems in nonextensive thermostatistics","volume":"48","author":"Li","year":"2005","journal-title":"Eur. Phys. J. B-Condens. Matter Complex Syst."},{"key":"ref_51","unstructured":"Wang, Q.A., Nivanen, L., and M\u00e9haut\u00e9, A.L. (2006). A composition of different q nonextensive systems with the normalized expectation based on escort probability. arXiv."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1016\/j.chaos.2004.09.064","article-title":"Applying incomplete statistics to nonextensive systems with different q indices","volume":"24","author":"Nivanen","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"030101","DOI":"10.1103\/PhysRevE.63.030101","article-title":"Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution","volume":"63","author":"Malacarne","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2398","DOI":"10.1103\/PhysRevE.60.2398","article-title":"Nonextensive foundation of L\u00e9vy distributions","volume":"60","author":"Prato","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3589","DOI":"10.1103\/PhysRevLett.75.3589","article-title":"Statistical-mechanical foundation of the ubiquity of L\u00e9vy distributions in nature","volume":"75","author":"Tsallis","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1103\/PhysRevLett.54.455","article-title":"Analytical solutions for diffusion on fractal objects","volume":"54","author":"Procaccia","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3073","DOI":"10.1103\/PhysRevA.32.3073","article-title":"Diffusion on fractals","volume":"32","author":"Procaccia","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2150071","DOI":"10.1142\/S0218348X21500717","article-title":"Diffusion on fractal objects modeling and its physics-informed neural network solution","volume":"29","author":"Zhao","year":"2021","journal-title":"Fractals"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0378-4371(94)00052-2","article-title":"On nonlinear diffusion in fractal structures","volume":"208","author":"Pascal","year":"1994","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_60","unstructured":"Crank, J. (1979). The Mathematics of Diffusion, Oxford University Press."},{"key":"ref_61","unstructured":"Burden, R.L., and Faires, J.D. (2011). Numerical Analysis, PWS Publishing Co.. [9th ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1578\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:28:11Z","timestamp":1760131691000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1578"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,23]]},"references-count":61,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["e25121578"],"URL":"https:\/\/doi.org\/10.3390\/e25121578","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,11,23]]}}}