{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T01:21:37Z","timestamp":1773537697062,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,21]],"date-time":"2017-01-21T00:00:00Z","timestamp":1484956800000},"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>We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching them from the resting to the movement. The results show that in the asymptotic limit of small and long times, the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for intermediate times depends on the rates present in the reaction terms. In this scenario, we show that, in the asymptotic limits, the distributions for this process are given by in terms of power laws which may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a situation characterized by different diffusive regimes, which emerges when the diffusive term is a mixing of linear and nonlinear terms.<\/jats:p>","DOI":"10.3390\/e19010042","type":"journal-article","created":{"date-parts":[[2017,1,23]],"date-time":"2017-01-23T10:40:33Z","timestamp":1485168033000},"page":"42","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Intermittent Motion, Nonlinear Diffusion Equation and Tsallis Formalism"],"prefix":"10.3390","volume":"19","author":[{"given":"Ervin","family":"Lenzi","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR 87030-900, Brazil"},{"name":"National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas F\u00edsicas, Rio de Janeiro, RJ 22290-180, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luciano","family":"Da Silva","sequence":"additional","affiliation":[{"name":"National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas F\u00edsicas, Rio de Janeiro, RJ 22290-180, Brazil"},{"name":"Departamento de F\u00edsica, Universidade Federal do Rio Grande do Norte, Natal, RN 59072-970, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcelo","family":"Lenzi","sequence":"additional","affiliation":[{"name":"Departamento de Engenharia Qu\u00edmica, Universidade Federal do Paran\u00e1, Curitiba, PR 81531-990, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maike","family":"Dos Santos","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR 87030-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haroldo","family":"Ribeiro","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Maring\u00e1, PR 87020-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luiz","family":"Evangelista","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Maring\u00e1, PR 87020-900, Brazil"},{"name":"National Institute of Science and Technology on Complex Fluids, Instituto de F\u00edsica da USP, S\u00e3o Paulo, SP 05508-090, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1039\/C3SM52846D","article-title":"2014 Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity","volume":"10","author":"Cherstvy","year":"2014","journal-title":"Soft Matter"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"046602","DOI":"10.1088\/0034-4885\/76\/4\/046602","article-title":"Anomalous transport in the crowded world of biological cells","volume":"76","author":"Franosch","year":"2013","journal-title":"Rep. Prog. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bressloff, P.C. (2014). Stochastic Processes in Cell Biology, Springer.","DOI":"10.1007\/978-3-319-08488-6"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s11237-014-9351-0","article-title":"Nonexponential Kinetics of Surface Chemical Reactions","volume":"50","author":"Snopok","year":"2014","journal-title":"Theor. Exp. Chem."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1023\/B:THEC.0000041803.99437.8b","article-title":"Macrokinetics of Chemical Processes on Porous Catalysts having regard to Anomalous Diffusion","volume":"40","author":"Strizhak","year":"2004","journal-title":"Theor. Exp. Chem."},{"key":"ref_6","unstructured":"Avnir, D. (1990). The Fractal Approach to Heterogeneous Chemistry, Wiley-Interscience."},{"key":"ref_7","unstructured":"Ben-Avraham, D., and Havlin, S. (2005). Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1103\/RevModPhys.87.483","article-title":"L\u00e9vy walks","volume":"87","author":"Zaburdaev","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Klages, R., Radons, G., and Sokolov, I.M. (2008). Anomalous Transport: Foundations and Applications, Wiley-VCH.","DOI":"10.1002\/9783527622979"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1901","DOI":"10.1029\/WR022i013p01901","article-title":"Particle Transport Through Porous Media","volume":"22","author":"Hunt","year":"1986","journal-title":"Water Resour. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1029\/WR015i005p01137","article-title":"A generalized solution for solute flow in soils with mobile and immobile water","volume":"15","author":"Wierenga","year":"1979","journal-title":"Water Resour. Res."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"McGuffee, S.R., and Elcock, A.H. (2010). Diffusion, Crowding & Protein Stability in a Dynamic Molecular Model of the Bacterial Cytoplasm. PLoS Comput. Biol., 6.","DOI":"10.1371\/journal.pcbi.1000694"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5167","DOI":"10.1002\/2016WR018667","article-title":"Infiltration experiments demonstrate an explicit connection between heterogeneity and anomalous diffusion behavior","volume":"52","author":"Filipovitch","year":"2016","journal-title":"Water Resour. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/s00249-015-1054-5","article-title":"Fractal model of anomalous diffusion","volume":"44","author":"Gmachowski","year":"2015","journal-title":"Eur. Biophys. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"114704","DOI":"10.1063\/1.3637944","article-title":"Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell","volume":"135","author":"Santoro","year":"2011","journal-title":"J. Chem. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"041128","DOI":"10.1103\/PhysRevE.84.041128","article-title":"Anomalous-diffusion approach applied to the electrical response of water","volume":"84","author":"Lenzi","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_17","unstructured":"Robert, A.M. (2011). Mathematics of Complexity and Dynamical Systems, Springer."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"046118","DOI":"10.1103\/PhysRevE.63.046118","article-title":"Fractional Fokker-Planck equation, solution, and application","volume":"63","author":"Barkai","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","article-title":"The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics","volume":"37","author":"Metzler","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/j.physa.2014.05.055","article-title":"Reaction on a solid surface supplied by an anomalous mass transfer source","volume":"410","author":"Lenzi","year":"2014","journal-title":"Physica A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"24128","DOI":"10.1039\/C4CP03465A","article-title":"Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking","volume":"16","author":"Metzler","year":"2014","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6634","DOI":"10.1103\/PhysRevE.57.6634","article-title":"Microscopic dynamics of the nonlinear Fokker-Planck equation: A phenomenological model","volume":"57","author":"Borland","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/BF01016796","article-title":"Generalized master equations for continuous-time random walks","volume":"9","author":"Kenkre","year":"1973","journal-title":"J. Stat. Phys."},{"key":"ref_25","first-page":"427383","article-title":"Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion","volume":"2012","author":"Pagnini","year":"2012","journal-title":"Int. J. Stoch. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Klafter, J., and Sokolov, I.M. (2011). First Steps in Random Walks: From Tools to Applications, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199234868.001.0001"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.physa.2006.09.001","article-title":"Fractional nonlinear diffusion equation, solutions and anomalous diffusion","volume":"375","author":"Silva","year":"2007","journal-title":"Physica A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1137\/110834342","article-title":"Continuous dependence estimates for nonlinear fractional convection-diffusion equations","volume":"44","author":"Alibaud","year":"2012","journal-title":"SIAM J. Math. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1088\/0305-4470\/25\/4\/029","article-title":"A nonlinear model of heat conduction","volume":"25","author":"Pascal","year":"1992","journal-title":"J. Phys. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"056303","DOI":"10.1103\/PhysRevE.70.056303","article-title":"Similarity solutions of nonlinear diffusion problems related to mathematical hydraulics and the Fokker-Planck equation","volume":"70","author":"Daly","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/j.physa.2006.10.010","article-title":"A family of evolution equations with nonlinear diffusion, Verhulst growth, and global regulation: Exact time-dependent solutions","volume":"375","author":"Troncoso","year":"2007","journal-title":"Physica A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/0378-4371(95)00211-1","article-title":"Non-Extensive Statistical Mechanics and Generalized Fokker-Planck Equation","volume":"222","author":"Plastino","year":"1995","journal-title":"Physica A"},{"key":"ref_33","first-page":"16","article-title":"Nonlinear diffusion equation, Tsallis formalism and exact solutions","volume":"342","author":"Silva","year":"2004","journal-title":"Physica A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0378-4371(00)00006-6","article-title":"A nonextensive maximum entropy approach to a family of nonlinear reaction-diffusion equations","volume":"280","author":"Plastino","year":"2000","journal-title":"Physica A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1063\/1.2828730","article-title":"On the non-linear Fokker-Planck equation associated with \u03ba-entropy","volume":"965","author":"Wada","year":"2007","journal-title":"AIP Conf. Proc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1140\/epjb\/e2009-00159-6","article-title":"Asymptotic solutions of a nonlinear diffusive equation in the framework of \u03ba-generalized statistical mechanics","volume":"70","author":"Wada","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_37","unstructured":"Frank, T.D. (2005). Nonlinear Fokker-Planck Equations, Springer-Verlag."},{"key":"ref_38","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics, Springer-Verlag."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Murray, J. (1989). Mathematical Biology, Springer.","DOI":"10.1007\/978-3-662-08539-4"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0375-9601(00)00639-3","article-title":"Anomalous diffusion coupled with Verhulst-like growth dynamics: Exact time-dependent solutions","volume":"276","author":"Rigo","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Mendez, V., Campos, D., and Bartumeus, F. (2014). Stochastic Foundations in Movement Ecology, Springer.","DOI":"10.1007\/978-3-642-39010-4"},{"key":"ref_42","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_43","doi-asserted-by":"crossref","unstructured":"Muskat, M. (1937). The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill.","DOI":"10.1063\/1.1710292"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Polubarinova-Kochina, P.Y. (1962). Theory of Ground Water Movement, Princeton University Press.","DOI":"10.1515\/9781400876044"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1017\/S0022112077002328","article-title":"Viscous sheets advancing over dry beds","volume":"81","author":"Buckmaster","year":"1977","journal-title":"J. Fluid Mech."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1051\/jp1:1993117","article-title":"Surface dynamics below the roughening transition","volume":"3","author":"Spohn","year":"1993","journal-title":"J. Phys. France I"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.physa.2016.03.020","article-title":"Fractional diffusion equations coupled by reaction terms","volume":"458","author":"Lenzi","year":"2016","journal-title":"Physica A"},{"key":"ref_48","unstructured":"Press, W.H., Flannery, B.P., and Teukolsky, S.A. (1992). Numerical Recipes in Fortran: The Art of Scientific Computing, Cambridge University Press."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF00276037","article-title":"Spatial Distribution of Dispersing Animals","volume":"9","author":"Shigesada","year":"1980","journal-title":"J. Math. Biology"},{"key":"ref_50","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"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/1\/42\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:26:40Z","timestamp":1760207200000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/1\/42"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,21]]},"references-count":50,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1]]}},"alternative-id":["e19010042"],"URL":"https:\/\/doi.org\/10.3390\/e19010042","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1,21]]}}}