{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:37:24Z","timestamp":1760236644378,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,8]],"date-time":"2021-12-08T00:00:00Z","timestamp":1638921600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Superdiffusive transport of energetic particles in the solar system and in other plasma environments is often inferred; while this can be described in terms of L\u00e9vy walks, a corresponding transport differential equation still calls for investigation. Here, we propose that superdiffusive transport can be described by means of a transport equation for pitch-angle scattering where the time derivative is fractional rather than integer. We show that this simply leads to superdiffusion in the direction parallel to the magnetic field, and we discuss some advantages with respect to approaches based on transport equations with symmetric spatial fractional derivates.<\/jats:p>","DOI":"10.3390\/sym13122368","type":"journal-article","created":{"date-parts":[[2021,12,8]],"date-time":"2021-12-08T23:30:00Z","timestamp":1639006200000},"page":"2368","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Energetic Particle Superdiffusion in Solar System Plasmas: Which Fractional Transport Equation?"],"prefix":"10.3390","volume":"13","author":[{"given":"Gaetano","family":"Zimbardo","sequence":"first","affiliation":[{"name":"Physics Department, University of Calabria, Ponte P. Bucci, Cubo 31C, 87036 Rende, Italy"},{"name":"Istituto Nazionale di Astrofisica (INAF), 84I, 00136 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5554-8765","authenticated-orcid":false,"given":"Francesco","family":"Malara","sequence":"additional","affiliation":[{"name":"Physics Department, University of Calabria, Ponte P. Bucci, Cubo 31C, 87036 Rende, Italy"},{"name":"Istituto Nazionale di Astrofisica (INAF), 84I, 00136 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8399-3268","authenticated-orcid":false,"given":"Silvia","family":"Perri","sequence":"additional","affiliation":[{"name":"Physics Department, University of Calabria, Ponte P. Bucci, Cubo 31C, 87036 Rende, Italy"},{"name":"Istituto Nazionale di Astrofisica (INAF), 84I, 00136 Rome, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","article-title":"TOPICAL REVIEW: 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_3","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/S0370-1573(02)00331-9","article-title":"Chaos, fractional kinetics, and anomalous transport","volume":"371","author":"Zaslavsky","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF00240886","article-title":"Non-relativistic solar electrons","volume":"16","author":"Lin","year":"1974","journal-title":"Space Sci. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"L177","DOI":"10.1086\/525523","article-title":"Evidence of superdiffusive transport of electrons accelerated at interplanetary shocks","volume":"671","author":"Perri","year":"2007","journal-title":"Astrophys. J. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"L34","DOI":"10.1088\/2041-8205\/731\/2\/L34","article-title":"Sign for Super-diffusive Transport of Energetic Ions Associated with a Coronal-mass-ejection-driven Interplanetary Shock","volume":"731","author":"Sugiyama","year":"2011","journal-title":"Astrophys. J. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1088\/0004-637X\/815\/1\/75","article-title":"Short Acceleration Times from Superdiffusive Shock Acceleration in the Heliosphere","volume":"815","author":"Perri","year":"2015","journal-title":"Astrophys. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"A2","DOI":"10.1051\/0004-6361\/201425295","article-title":"Parameter estimation of superdiffusive motion of energetic particles upstream of heliospheric shocks","volume":"578","author":"Perri","year":"2015","journal-title":"Astron. Astrophys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101605","DOI":"10.1016\/j.newast.2021.101605","article-title":"Energetic particle fluxes at heliospheric shocks: Evidences of superdiffusion and comparison between analytical and numerical modeling","volume":"87","author":"Prete","year":"2021","journal-title":"New Astron."},{"key":"ref_11","first-page":"432","article-title":"Anomalous transport of cosmic ray electrons","volume":"327","author":"Ragot","year":"1997","journal-title":"Astron. Astrophys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"A34","DOI":"10.1051\/0004-6361\/201628767","article-title":"Transport of relativistic electrons at shocks in shell-type supernova remnants: Diffusive and superdiffusive regimes","volume":"596","author":"Perri","year":"2016","journal-title":"Astron. Astrophys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"014005","DOI":"10.1088\/1361-6587\/aa8602","article-title":"Superdiffusion of relativistic electrons at supernova remnant shocks","volume":"60","author":"Perri","year":"2018","journal-title":"Plasma Phys. Control Fusion"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"0163","DOI":"10.1038\/s41550-017-0163","article-title":"Superdiffusive shock acceleration at galaxy cluster shocks","volume":"1","author":"Zimbardo","year":"2017","journal-title":"Nat. Astron."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4922","DOI":"10.1093\/mnras\/sty1438","article-title":"Understanding the radio spectral indices of galaxy cluster relics by superdiffusive shock acceleration","volume":"478","author":"Zimbardo","year":"2018","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1086\/307452","article-title":"The transport of cosmic rays across a turbulent magnetic field","volume":"520","author":"Giacalone","year":"1999","journal-title":"Astrophys. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"L127","DOI":"10.1088\/2041-8205\/720\/2\/L127","article-title":"A unified particle diffusion theory for cross-field scattering: Subdiffusion, recovery of diffusion, and diffusion in three-dimensional turbulence","volume":"720","author":"Shalchi","year":"2010","journal-title":"Astrophys. J. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"13325","DOI":"10.1029\/93JA00812","article-title":"Electron-whistler interaction at the Earth\u2019s bow shock: 1. Whistler instability","volume":"98","author":"Veltri","year":"1993","journal-title":"J. Geophys. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"13335","DOI":"10.1029\/93JA01144","article-title":"Electron-whistler interaction at the Earth\u2019s bow shock: 2. Electron pitch angle diffusion","volume":"98","author":"Veltri","year":"1993","journal-title":"J. Geophys. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1103\/PhysRevLett.58.1100","article-title":"Levy dynamics of enhanced diffusion\u2014Application to turbulence","volume":"58","author":"Shlesinger","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3081","DOI":"10.1103\/PhysRevA.35.3081","article-title":"Stochastic pathway to anomalous diffusion","volume":"35","author":"Klafter","year":"1987","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1088\/0004-637X\/750\/2\/87","article-title":"Superdiffusive shock acceleration","volume":"750","author":"Perri","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1088\/0004-637X\/778\/1\/35","article-title":"From L\u00e9vy walks to superdiffusive shock acceleration","volume":"778","author":"Zimbardo","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"012020","DOI":"10.1088\/1742-6596\/642\/1\/012020","article-title":"Evidence for superdiffusive shock acceleration at interplanetary shock waves","volume":"642","author":"Perri","year":"2015","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3854","DOI":"10.1063\/1.1767097","article-title":"Fractional diffusion in plasma turbulence","volume":"11","author":"Carreras","year":"2004","journal-title":"Phys. Plasmas"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"045101","DOI":"10.1103\/PhysRevLett.119.045101","article-title":"Fractional Transport in Strongly Turbulent Plasmas","volume":"119","author":"Isliker","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"35","DOI":"10.3847\/1538-4357\/aa8ee8","article-title":"Particle Acceleration and Fractional Transport in Turbulent Reconnection","volume":"849","author":"Isliker","year":"2017","journal-title":"Astrophys. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"84","DOI":"10.3847\/1538-4357\/abf3c6","article-title":"A Focused Transport-based Kinetic Fractional Diffusion-advection Equation for Energetic Particle Trapping and Reconnection-related Acceleration by Small-scale Magnetic Flux Ropes in the Solar Wind","volume":"913","author":"Zank","year":"2021","journal-title":"Astrophys. J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s11214-013-9966-9","article-title":"Nonclassical transport and particle-field coupling: From laboratory plasmas to the solar wind","volume":"178","author":"Perrone","year":"2013","journal-title":"Space Sci. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"065003","DOI":"10.1103\/PhysRevLett.94.065003","article-title":"Nondiffusive Transport in Plasma Turbulence: A Fractional Diffusion Approach","volume":"94","author":"Carreras","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"105","DOI":"10.3847\/1538-4357\/abb951","article-title":"Non-Markovian Pitch-angle Scattering as the Origin of Particle Superdiffusion Parallel to the Magnetic Field","volume":"903","author":"Zimbardo","year":"2020","journal-title":"Astrophys. J."},{"key":"ref_32","first-page":"238459","article-title":"A Review of Definitions for Fractional Derivatives and Integrals","volume":"2014","author":"Machado","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_33","first-page":"269","article-title":"Time-fractional derivatives in relaxation processes: A tutorial survey","volume":"10","author":"Mainardi","year":"2007","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1103\/PhysRevE.47.851","article-title":"Scale-invariant motion in intermittent chaotic systems","volume":"47","author":"Zumofen","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"010101","DOI":"10.1103\/PhysRevE.67.010101","article-title":"Towards deterministic equations for L\u00e9vy walks: The fractional material derivative","volume":"67","author":"Sokolov","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Schlickeiser, R. (2002). Cosmic Ray Astrophysics, Springer.","DOI":"10.1007\/978-3-662-04814-6"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Gurnett, D.A., and Bhattacharjee, A. (2005). Introduction to Plasma Physics, Cambridge University Press.","DOI":"10.1017\/CBO9780511809125"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1051\/0004-6361:20053664","article-title":"Analytical investigation of the two-dimensional cosmic ray Fokker-Planck equation","volume":"448","author":"Shalchi","year":"2006","journal-title":"Astron. Astrophys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s11214-012-9932-y","article-title":"Shock Acceleration of Ions in the Heliosphere","volume":"173","author":"Lee","year":"2012","journal-title":"Space Sci. Rev."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s11214-011-9819-3","article-title":"Cosmic-Ray Modulation Equations","volume":"176","author":"Moraal","year":"2013","journal-title":"Space Sci. Rev."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"13","DOI":"10.3847\/1538-4357\/aaffd2","article-title":"Pitch-angle Diffusion and Bohm-type Approximations in Diffusive Shock Acceleration","volume":"873","author":"Riordan","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3860","DOI":"10.1093\/mnras\/stz3259","article-title":"Superdiffusive stochastic Fermi acceleration in space and energy","volume":"491","author":"Sioulas","year":"2020","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"L111","DOI":"10.1086\/593145","article-title":"Particle Acceleration in a Model of a Turbulent Reconnecting Plasma: A Fractional Diffusion Approach","volume":"687","author":"Bian","year":"2008","journal-title":"Astrophys. J. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"40003","DOI":"10.1209\/0295-5075\/78\/40003","article-title":"Position and velocity space diffusion of test particles in stochastic electromagnetic fields","volume":"78","author":"Perri","year":"2007","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"L04103","DOI":"10.1029\/2008GL036619","article-title":"Stochastic and direct acceleration mechanisms in the Earth\u2019s magnetotail","volume":"36","author":"Perri","year":"2009","journal-title":"Geophys. Res. Lett."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"900","DOI":"10.1086\/154239","article-title":"A nonlinear theory of cosmic-ray pitch-angle diffusion in homogeneous magnetostatic turbulence","volume":"204","author":"Goldstein","year":"1976","journal-title":"Astrophys. J."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1532","DOI":"10.1016\/j.asr.2017.06.035","article-title":"Solutions of the cosmic ray velocity diffusion equation","volume":"60","author":"Lasuik","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1088\/0004-637X\/754\/1\/8","article-title":"Magnetic variances and pitch-angle scattering times upstream of interplanetary shocks","volume":"754","author":"Perri","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"3395","DOI":"10.1093\/mnras\/stw877","article-title":"Energetic particle transport in the presence of magnetic turbulence: Influence of spectral extension and intermittency","volume":"459","author":"Pucci","year":"2016","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s11207-019-1421-y","article-title":"On the Power-Law Distribution of Pitch-Angle Scattering Times in Solar Wind Turbulence","volume":"294","author":"Perri","year":"2019","journal-title":"Sol. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"025208","DOI":"10.1103\/PhysRevE.104.025208","article-title":"Charged-particle chaotic dynamics in rotational discontinuities","volume":"104","author":"Malara","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Perri, S., Prete, G., Malara, F., Pucci, F., and Zimbardo, G. (2021). The Influence of Magnetic Turbulence on the Energetic Particle Transport Upstream of Shock Waves. Atmosphere, 12.","DOI":"10.3390\/atmos12040508"},{"key":"ref_53","first-page":"2267","article-title":"On some properties of the Mittag-Leffler function E\u03b1(t), completely monotone for t > 0 with 0 < \u03b1 < 1","volume":"19","author":"Mainardi","year":"2014","journal-title":"Discret. Contin. Dyn. Syst.-Ser. B (DCDS-B)"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1051\/0004-6361\/200912755","article-title":"Analytical description of nonlinear cosmic ray scattering: Isotropic and quasilinear regimes of pitch-angle diffusion","volume":"507","author":"Shalchi","year":"2009","journal-title":"Astron. Astrophys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"046402","DOI":"10.1103\/PhysRevE.83.046402","article-title":"Applicability of the Taylor-Green-Kubo formula in particle diffusion theory","volume":"83","author":"Shalchi","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"021103","DOI":"10.1103\/PhysRevE.81.021103","article-title":"Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries","volume":"81","author":"Jeon","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"4972","DOI":"10.1093\/mnras\/staa3131","article-title":"Power-law spectra from stochastic acceleration","volume":"499","author":"Lemoine","year":"2020","journal-title":"Mon. Not. R. Astron. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2368\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:43:45Z","timestamp":1760168625000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2368"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,8]]},"references-count":57,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122368"],"URL":"https:\/\/doi.org\/10.3390\/sym13122368","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,8]]}}}