{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T15:44:34Z","timestamp":1783698274495,"version":"3.55.0"},"reference-count":155,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,31]],"date-time":"2025-03-31T00:00:00Z","timestamp":1743379200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princeton University","award":["80NSSC18K0237"],"award-info":[{"award-number":["80NSSC18K0237"]}]},{"name":"Princeton University","award":["80GSFC19C0027"],"award-info":[{"award-number":["80GSFC19C0027"]}]},{"name":"NASA\u2019s Explorer Program","award":["80NSSC18K0237"],"award-info":[{"award-number":["80NSSC18K0237"]}]},{"name":"NASA\u2019s Explorer Program","award":["80GSFC19C0027"],"award-info":[{"award-number":["80GSFC19C0027"]}]},{"name":"NASA\u2019s Solar Terrestrial Probes (STP) Program","award":["80NSSC18K0237"],"award-info":[{"award-number":["80NSSC18K0237"]}]},{"name":"NASA\u2019s Solar Terrestrial Probes (STP) Program","award":["80GSFC19C0027"],"award-info":[{"award-number":["80GSFC19C0027"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Kappa distributions, their statistical framework, and their thermodynamic origin describe systems with correlations among their particle energies, residing in stationary states out of classical thermal equilibrium\/space plasmas, from solar wind to the outer heliosphere, are such systems. We show how correlations from long-range interactions compete with collisions to define the specific shape of particle velocity distributions, using a simple numerical experiment with collisions and a variable amount of correlation among the particles. When the correlations are turned off, collisions drive any initial distribution to evolve toward equilibrium and a Maxwell\u2013Boltzmann (MB) distribution. However, when some correlation is introduced, the distribution evolves toward a different stationary state defined by a kappa distribution with some finite value of the thermodynamic kappa \u03ba (where \u03ba\u2192\u221e corresponds to a MB distribution). Furthermore, the stronger the correlations, the lower the \u03ba value. This simple numerical experiment illuminates the role of correlations in forming stationary state particle distributions, which are described by kappa distributions, as well as the physical interpretation of correlations from long-range interactions and how they are related to the thermodynamic kappa.<\/jats:p>","DOI":"10.3390\/e27040375","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T08:21:41Z","timestamp":1743495701000},"page":"375","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Correlations and Kappa Distributions: Numerical Experiment and Physical Understanding"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6160-1158","authenticated-orcid":false,"given":"David J.","family":"McComas","sequence":"first","affiliation":[{"name":"Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7655-6019","authenticated-orcid":false,"given":"George","family":"Livadiotis","sequence":"additional","affiliation":[{"name":"Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8483-519X","authenticated-orcid":false,"given":"Nicholas V.","family":"Sarlis","sequence":"additional","affiliation":[{"name":"Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA"},{"name":"Physics Department, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Athens, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1080\/14786446008642818","article-title":"Illustrations of the Dynamical Theory of Gases","volume":"19","author":"Maxwell","year":"1860","journal-title":"Philos. Mag."},{"key":"ref_2","first-page":"195","article-title":"\u00dcber die Mechanische Bedeutung des Zweiten Hauptsatzes der W\u00e4rmetheorie","volume":"53","author":"Boltzmann","year":"1866","journal-title":"Wien. Berichte"},{"key":"ref_3","unstructured":"Binsack, J.H. (1966). Plasma Studies with the IMP-2 Satellite. [Ph.D. Thesis, MIT]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/978-94-010-3467-8_23","article-title":"Summary of Experimental Results from M.I.T. Detector on IMP-1","volume":"Volume 10","author":"Carovillano","year":"1968","journal-title":"Physics of the Magnetosphere"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2839","DOI":"10.1029\/JA073i009p02839","article-title":"A survey of low-energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3","volume":"73","author":"Vasyliunas","year":"1968","journal-title":"J. Geophys. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1029\/97GL01760","article-title":"Theory of super-diffusion for the magnetopause","volume":"24","author":"Treumann","year":"1997","journal-title":"Geophys. Res. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"211","DOI":"10.5194\/npg-7-211-2000","article-title":"Functional background of the Tsallis entropy: \u201ccoarse-grained\u201d systems and \u201ckappa\u201d distribution functions","volume":"7","author":"Milovanov","year":"2000","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1023\/A:1020990413487","article-title":"A Nonextensive Entropy Approach to Kappa-Distributions","volume":"282","author":"Leubner","year":"2002","journal-title":"Astrophys. Space Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"A11105","DOI":"10.1029\/2009JA014352","article-title":"Beyond kappa distributions: Exploiting Tsallis statistical mechanics in space plasmas","volume":"114","author":"Livadiotis","year":"2009","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4290","DOI":"10.3390\/e16084290","article-title":"\u201cLagrangian Temperature\u201d: Derivation and Physical Meaning for Systems Described by Kappa Distributions","volume":"16","author":"Livadiotis","year":"2014","journal-title":"Entropy"},{"key":"ref_11","unstructured":"Livadiotis, G. (2017). Kappa Distributions: Theory and Applications in Plasmas, Elsevier."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Yoon, P.H. (2019). Classical Kinetic Theory of Weakly Turbulent Nonlinear Plasma Processes, Cambridge University Press.","DOI":"10.1017\/9781316771259"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Tsallis, C. (2023). Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, Springer.","DOI":"10.1007\/978-3-030-79569-6"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s11207-010-9640-2","article-title":"Kappa Distributions: Theory and Applications in Space Plasmas","volume":"267","author":"Pierrard","year":"2010","journal-title":"Sol. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s11214-013-9982-9","article-title":"Understanding Kappa Distributions: A Toolbox for Space Science and Astrophysics","volume":"175","author":"Livadiotis","year":"2013","journal-title":"Space Sci. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1002\/2014JA020825","article-title":"Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas","volume":"120","author":"Livadiotis","year":"2015","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Tsallis, C. (2019). Beyond Boltzmann-Gibbs-Shannon in Physics and Elsewhere. Entropy, 21.","DOI":"10.3390\/e21070696"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Livadiotis, G., and McComas, D.J. (2021). Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect. Entropy, 23.","DOI":"10.3390\/e23121683"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"83","DOI":"10.3847\/1538-4357\/ac99df","article-title":"Physical Correlations Lead to Kappa Distributions","volume":"940","author":"Livadiotis","year":"2022","journal-title":"Astrophys. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1051\/0004-6361:20078826","article-title":"Formation of suprathermal electron distributions in the quiet solar corona","volume":"480","author":"Vocks","year":"2008","journal-title":"Astron. Astrophys."},{"key":"ref_21","unstructured":"Lee, E., Williams, D.R., and Lapenta, G. (2013). Spectroscopic indication of suprathermal ions in the solar corona. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"L31","DOI":"10.1088\/2041-8205\/791\/2\/L31","article-title":"Suprathermal Electrons in the Solar Corona: Can Nonlocal Transport Explain Heliospheric Charge States?","volume":"791","author":"Cranmer","year":"2014","journal-title":"Astrophys. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"A05203","DOI":"10.1029\/2007JA012903","article-title":"Energetic electron distributions fitted with a relativistic kappa-type function at geosynchronous orbit","volume":"113","author":"Xiao","year":"2008","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1088\/0004-637X\/770\/1\/73","article-title":"On the Remote Detection of Suprathermal Ions in the Solar Corona and their Role as Seeds for Solar Energetic Particle Production","volume":"770","author":"Laming","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"33","DOI":"10.3847\/1538-4365\/ab5527","article-title":"Seed Population Preconditioning and Acceleration Observed by the Parker Solar Probe","volume":"246","author":"Schwadron","year":"2020","journal-title":"Astrophys. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"64","DOI":"10.3847\/1538-4357\/ad479d","article-title":"Persistent Behavior in Solar Energetic Particle Time Series","volume":"969","author":"Sarlis","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6","DOI":"10.3847\/1538-4357\/ad5e72","article-title":"Kappa-tail Technique: Modeling and Application to Solar Energetic Particles Observed by Parker Solar Probe","volume":"973","author":"Livadiotis","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"148","DOI":"10.3847\/1538-4357\/ad37f8","article-title":"Observations of the 2022 September 5 Solar Energetic Particle Event at 15 Solar Radii","volume":"966","author":"Cohen","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"76","DOI":"10.3847\/1538-4357\/ad68fd","article-title":"Observations of Kappa Distributions in Solar Energetic Protons and Derived Thermodynamic Properties","volume":"973","author":"Cuesta","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1051\/0004-6361:200810099","article-title":"Generation of highly energetic electrons at reconnection outflow shocks during solar flares","volume":"494","author":"Mann","year":"2009","journal-title":"Astron. Astrophys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.3390\/e15031118","article-title":"Evidence of Large-Scale Quantization in Space Plasmas","volume":"15","author":"Livadiotis","year":"2013","journal-title":"Entropy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1088\/0004-637X\/796\/2\/142","article-title":"The Formation of Kappa-distribution Accelerated Electron Populations in Solar Flares","volume":"796","author":"Bian","year":"2014","journal-title":"Astrophys. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"35","DOI":"10.3847\/1538-4357\/836\/1\/35","article-title":"Non-Gaussian Velocity Distributions in Solar Flares from Extreme Ultraviolet Lines: A Possible Diagnostic of Ion Acceleration","volume":"836","author":"Jeffrey","year":"2017","journal-title":"Astrophys. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.asr.2003.02.060","article-title":"Towards a quantitative theory for 2-3 kHz radio emission from beyond the heliopause","volume":"34","author":"Cairns","year":"2004","journal-title":"Adv. Space Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"L34","DOI":"10.1088\/2041-8205\/763\/2\/L34","article-title":"Type III Radio Bursts in Coronal Plasmas with Kappa Particle Distributions","volume":"763","author":"Li","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1002\/2015GL067271","article-title":"Quantitative prediction of type II solar radio emission from the Sun to 1 AU","volume":"43","author":"Schmidt","year":"2016","journal-title":"Geophys. Res. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1088\/0067-0049\/206\/1\/6","article-title":"H to Zn Ionization Equilibrium for the Non-Maxwellian Electron \u03ba-distributions: Updated Calculations","volume":"206","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1088\/0067-0049\/217\/1\/14","article-title":"KAPPA: A Package for Synthesis of Optically Thin Spectra for the Non-Maxwellian \u03ba-distributions Based on the Chianti Database","volume":"217","year":"2015","journal-title":"Astrophys. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"34","DOI":"10.3847\/1538-4357\/ab8010","article-title":"Plasma Diagnostics from Active Region and Quiet-Sun Spectra Observed by Hinode\/EIS: Quantifying the Departures from a Maxwellian Distribution","volume":"893","author":"Mason","year":"2020","journal-title":"Astrophys. J."},{"key":"ref_40","first-page":"725","article-title":"A kinetic model of the solar wind with Kappa distribution functions in the corona","volume":"324","author":"Maksimovic","year":"1997","journal-title":"Astron. Astrophys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"17021","DOI":"10.1029\/1999JA900169","article-title":"Electron velocity distribution functions from the solar wind to the corona","volume":"104","author":"Pierrard","year":"1999","journal-title":"J. Geophys. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1051\/0004-6361:20020866","article-title":"On electron acceleration at CIR related shock waves","volume":"391","author":"Mann","year":"2002","journal-title":"Astron. Astrophys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.12942\/lrsp-2006-1","article-title":"Kinetic Physics of the Solar Corona and Solar Wind","volume":"3","author":"Marsch","year":"2006","journal-title":"Living Rev. Sol. Phys."},{"key":"ref_44","first-page":"A05104","article-title":"Radial evolution of nonthermal electron populations in the low-latitude solar wind: Helios, Cluster, and Ulysses Observations","volume":"114","author":"Maksimovic","year":"2009","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s11214-012-9867-3","article-title":"Langmuir Turbulence and Suprathermal Electrons","volume":"173","author":"Yoon","year":"2012","journal-title":"Space Sci. Rev."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Yoon, P.H., L\u00f3pez, R.A., Salem, C.S., Bonnell, J.W., and Kim, S. (2024). Non-Thermal Solar Wind Electron Velocity Distribution Function. Entropy, 26.","DOI":"10.20944\/preprints202402.1292.v1"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"7074","DOI":"10.1002\/2014JA020353","article-title":"Electron kappa distribution and quasi-thermal noise","volume":"119","author":"Yoon","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"9441","DOI":"10.1002\/2014JA020678","article-title":"Coronal heating and solar wind acceleration for electrons, protons, and minor ions obtained from kinetic models based on kappa distributions","volume":"119","author":"Pierrard","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.physa.2016.07.058","article-title":"Non-extensive statistical analysis of magnetic field during the March 2012 ICME event using a multi-spacecraft approach","volume":"464","author":"Pavlos","year":"2016","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"A08111","DOI":"10.1029\/2007JA012979","article-title":"Measuring suprathermal electron parameters in space plasmas: Implementation of the quasi-thermal noise spectroscopy with kappa distributions using in situ Ulysses\/URAP radio measurements in the solar wind","volume":"113","author":"Zouganelis","year":"2008","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.1002\/2017JA024978","article-title":"Using Kappa Distributions to Identify the Potential Energy","volume":"123","author":"Livadiotis","year":"2018","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Nicolaou, G., Livadiotis, G., and Wicks, R.T. (2019). On the Calculation of the Effective Polytropic Index in Space Plasmas. Entropy, 21.","DOI":"10.3390\/e21100997"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"A75","DOI":"10.1051\/0004-6361\/202040047","article-title":"Analytic solution for the electrostatic potential of the solar wind","volume":"653","author":"Zouganelis","year":"2021","journal-title":"Astron. Astrophys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"192","DOI":"10.3847\/1538-4357\/ad85d6","article-title":"Radial Distribution of Electron Quasi-thermal Noise in the Inner Heliosphere","volume":"976","author":"Li","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"23107","DOI":"10.1029\/1998JA000015","article-title":"The suprathermal seed population for corotating interaction region ions at 1 AU deduced from composition and spectra of H+, He++, and He+ observed on Wind","volume":"105","author":"Chotoo","year":"2000","journal-title":"J. Geophys. Res."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"3714","DOI":"10.1029\/JA078i019p03714","article-title":"Solar wind interaction with the Earth\u2019s magnetic field: 1. Magnetosheath","volume":"78","author":"Formisano","year":"1973","journal-title":"J. Geophys. Res."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3126","DOI":"10.1002\/jgra.50353","article-title":"Characterizing the dayside magnetosheath using energetic neutral atoms: IBEX and THEMIS observations","volume":"118","author":"Ogasawara","year":"2013","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1002\/2014JA020851","article-title":"Interplanetary magnetic field dependence of the suprathermal energetic neutral atoms originated in subsolar magnetopause","volume":"120","author":"Ogasawara","year":"2015","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3614","DOI":"10.1103\/PhysRevLett.84.3614","article-title":"Generation of Broadband Electrostatic Waves in Earth\u2019s Magnetotail","volume":"84","author":"Grabbe","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/S1364-6826(02)00015-9","article-title":"The ion differential spectra in outer boundary of the ring current: November 17, 1995 case study","volume":"64","author":"Pisarenko","year":"2002","journal-title":"J. Atmos.-Sol.-Terr. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/0019-1035(87)90040-6","article-title":"A comparison of the Mercury and Earth magnetospheres: Electron measurements and substorm time scales","volume":"71","author":"Christon","year":"1987","journal-title":"Icarus"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1029\/2002JA009678","article-title":"Auroral source region: Plasma properties of the high-latitude plasma sheet","volume":"108","author":"Kletzing","year":"2003","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_63","first-page":"1074","article-title":"Modeling the inner plasma sheet protons and magnetic field under enhanced convection","volume":"108","author":"Wang","year":"2003","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1016\/j.pss.2010.06.002","article-title":"The role of suprathermal particle measurements in CrossScale studies of collisionless plasma processes","volume":"59","author":"Hapgood","year":"2011","journal-title":"Planet. Space Sci."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"3475","DOI":"10.1002\/2017GL072715","article-title":"Properties of suprathermal electrons associated with discrete auroral arcs","volume":"44","author":"Ogasawara","year":"2017","journal-title":"Geophys. Res. Lett."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1029\/94GL02997","article-title":"The relationship between kappa and temperature in energetic ion spectra at Jupiter","volume":"22","author":"Collier","year":"1995","journal-title":"Geophys. Res. Lett."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"A09S12","DOI":"10.1029\/2003JA010270","article-title":"Energetic ion characteristics and neutral gas interactions in Jupiter\u2019s magnetosphere","volume":"109","author":"Mauk","year":"2004","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"3463","DOI":"10.1002\/2013JA019665","article-title":"Properties of plasma ions in the distant Jovian magnetosheath using Solar Wind Around Pluto data on New Horizons","volume":"119","author":"Nicolaou","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"8426","DOI":"10.1002\/2014JA020324","article-title":"Using the kappa function to investigate hot plasma in the magnetospheres of the giant planets","volume":"119","author":"Carbary","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"3683","DOI":"10.1002\/2013JA019616","article-title":"Multi-instrument analysis of plasma parameters in Saturn\u2019s equatorial, inner magnetosphere using corrections for corrections for spacecraft potential and penetrating background radiation","volume":"119","author":"Livi","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"8066","DOI":"10.1029\/2018JA025820","article-title":"Energetic Ion Moments and Polytropic Index in Saturn\u2019s Magnetosphere using Cassini\/MIMI Measurements: A Simple Model Based on \u03ba-Distribution Functions","volume":"123","author":"Dialynas","year":"2018","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"15283","DOI":"10.1029\/JA092iA13p15283","article-title":"The hot plasma and radiation environment of the Uranian magnetosphere","volume":"92","author":"Mauk","year":"1987","journal-title":"J. Geophys. Res."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.1029\/2001JA900124","article-title":"Latitudinal structure of outer Io plasma torus","volume":"107","author":"Moncuquet","year":"2002","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"2172","DOI":"10.1029\/2002GL015855","article-title":"Saturn: Search for a missing water source","volume":"29","author":"Jurac","year":"2002","journal-title":"Geophys. Res. Lett."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"7407","DOI":"10.1002\/2016JA022972","article-title":"Characterizing cometary electrons with kappa distributions","volume":"121","author":"Broiles","year":"2016","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"A42","DOI":"10.1051\/0004-6361\/201834964","article-title":"Plasma properties of suprathermal electrons near comet 67P\/Churyumov-Gerasimenko with Rosetta","volume":"630","author":"Myllys","year":"2019","journal-title":"Astron. Astrophys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"2020","DOI":"10.1126\/science.1117569","article-title":"Voyager 1 in the Foreshock, Termination Shock, and Heliosheath","volume":"309","author":"Decker","year":"2005","journal-title":"Science"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1086\/588248","article-title":"The Effects of a \u03ba-Distribution in the Heliosheath on the Global Heliosphere and ENA Flux at 1 AU","volume":"682","author":"Heerikhuisen","year":"2008","journal-title":"Astrophys. J."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"1516","DOI":"10.1002\/2014JA020636","article-title":"\u03ba-distributed protons in the solar wind and their charge-exchange coupling to energetic hydrogen","volume":"120","author":"Heerikhuisen","year":"2015","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1088\/0004-637X\/714\/1\/971","article-title":"Exploring Transitions of Space Plasmas Out of Equilibrium","volume":"714","author":"Livadiotis","year":"2010","journal-title":"Astrophys. J."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"035003","DOI":"10.1088\/0031-8949\/82\/03\/035003","article-title":"Measure of the departure of the q-metastable stationary states from equilibrium","volume":"82","author":"Livadiotis","year":"2010","journal-title":"Phys. Scr."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1088\/0004-637X\/741\/2\/88","article-title":"Invariant Kappa Distribution in Space Plasmas Out of Equilibrium","volume":"741","author":"Livadiotis","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1088\/0004-637X\/738\/1\/64","article-title":"The Influence of Pick-up Ions on Space Plasma Distributions","volume":"738","author":"Livadiotis","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1088\/0004-637X\/749\/1\/11","article-title":"Non-equilibrium Thermodynamic Processes: Space Plasmas and the Inner Heliosheath","volume":"749","author":"Livadiotis","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"72","DOI":"10.3847\/1538-4357\/ace1e3","article-title":"Transport Equation of Kappa Distributions in the Heliosphere","volume":"954","author":"Livadiotis","year":"2023","journal-title":"Astrophys. J."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1126\/science.1180906","article-title":"Global Observations of the Interstellar Interaction from the Interstellar Boundary Explorer (IBEX)","volume":"326","author":"McComas","year":"2009","journal-title":"Science"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"1092","DOI":"10.1088\/0004-637X\/708\/2\/1092","article-title":"Microstructure of the Heliospheric Termination Shock: Implications for Energetic Neutral Atom Observations","volume":"708","author":"Zank","year":"2010","journal-title":"Astrophys. J."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/0004-637X\/734\/1\/1","article-title":"First Sky Map of the Inner Heliosheath Temperature Using IBEX Spectra","volume":"734","author":"Livadiotis","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1088\/0004-637X\/751\/1\/64","article-title":"Pick-up Ion Distributions and Their Influence on Energetic Neutral Atom Spectral Curvature","volume":"751","author":"Livadiotis","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1088\/0004-637X\/762\/2\/134","article-title":"Pressure of the Proton Plasma in the Inner Heliosheath","volume":"762","author":"Livadiotis","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"53","DOI":"10.3847\/1538-4365\/ac8b88","article-title":"Thermodynamics of the Inner Heliosheath","volume":"262","author":"Livadiotis","year":"2022","journal-title":"Astrophys. J."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"21","DOI":"10.3847\/1538-4357\/acd1e1","article-title":"Temperature of the Polar Inner Heliosheath: Connection to Solar Activity","volume":"951","author":"Livadiotis","year":"2023","journal-title":"Astrophys. J."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"66","DOI":"10.3847\/1538-4357\/ad3e79","article-title":"Thermodynamics of Pickup Ions in the Heliosphere","volume":"968","author":"Livadiotis","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1088\/0004-637X\/784\/2\/89","article-title":"Low Energy Neutral Atoms from the Heliosheath","volume":"784","author":"Fuselier","year":"2014","journal-title":"Astrophys. J."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1146\/annurev-astro-082214-122254","article-title":"Faltering Steps Into the Galaxy: The Boundary Regions of the Heliosphere","volume":"53","author":"Zank","year":"2015","journal-title":"Annu. Rev. Astron. Astrophys."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1088\/0004-637X\/815\/1\/31","article-title":"Using Kappa Functions to Characterize Outer Heliosphere Proton Distributions in the Presence of Charge-exchange","volume":"815","author":"Zirnstein","year":"2015","journal-title":"Astrophys. J."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"10003","DOI":"10.1209\/0295-5075\/113\/10003","article-title":"Curie law for systems described by kappa distributions","volume":"113","author":"Livadiotis","year":"2016","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"254","DOI":"10.3847\/1538-4357\/aafa78","article-title":"Non-equilibrium Distributions of Interstellar Neutrals and the Temperature of the Local Interstellar Medium","volume":"871","author":"Swaczyna","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1088\/0004-637X\/752\/2\/148","article-title":"Resolving the Electron Temperature Discrepancies in H II Regions and Planetary Nebulae: \u03ba-distributed Electrons","volume":"752","author":"Nicholls","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1088\/0067-0049\/207\/2\/21","article-title":"Measuring Nebular Temperatures: The Effect of New Collision Strengths with Equilibrium and \u03ba-distributed Electron Energies","volume":"207","author":"Nicholls","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1088\/0004-637X\/780\/1\/93","article-title":"H I Free-Bound Emission of Planetary Nebulae with Large Abundance Discrepancies: Two-component Models versus \u03ba-distributed Electrons","volume":"780","author":"Zhang","year":"2014","journal-title":"Astrophys. J."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"33","DOI":"10.3847\/1538-4357\/ab9d7e","article-title":"Can the Kappa-distributed Electron Energies Account for the Intensity Ratios of O II Lines in Photoionized Gaseous Nebulae?","volume":"899","author":"Lin","year":"2020","journal-title":"Astrophys. J."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"143","DOI":"10.3847\/1538-4357\/ac8979","article-title":"The Ultraviolet C II Lines as a Diagnostic of \u03ba-distributed Electrons in Planetary Nebulae","volume":"936","author":"Yao","year":"2022","journal-title":"Astrophys. J."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"A10","DOI":"10.1051\/0004-6361\/201732007","article-title":"Photoionization models for extreme Ly\u03b1 \u03bb1216 and Hell \u03bb1640 line ratios in quasar halos, and PopIII vs. AGN diagnostics","volume":"621","author":"Humphrey","year":"2019","journal-title":"Astron. Astrophys."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"1389","DOI":"10.1093\/mnras\/stab1599","article-title":"Electron energy distributions in the extended gas nebulae associated with high-z AGN: Maxwell-Boltzmann versus \u03ba distributions","volume":"506","author":"Morais","year":"2021","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"A2","DOI":"10.1051\/0004-6361\/201936150","article-title":"Modeling non-thermal emission from the jet-launching region of M 87 with adaptive mesh refinement","volume":"632","author":"Davelaar","year":"2019","journal-title":"Astron. Astrophys."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1088\/0004-637X\/712\/2\/901","article-title":"Non-Maxwellian H\u03b1 Profiles in Tycho\u2019s Supernova Remnant","volume":"712","author":"Raymond","year":"2010","journal-title":"Astrophys. J."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3847\/1538-4357\/834\/2\/165","article-title":"Non-extensive Statistics to the Cosmological Lithium Problem","volume":"834","author":"Hou","year":"2017","journal-title":"Astrophys. J."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"49001","DOI":"10.1209\/0295-5075\/ac2e2c","article-title":"Black-body radiation in space plasmas","volume":"135","author":"Livadiotis","year":"2021","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s11214-007-9205-3","article-title":"The Solar Wind Around Pluto (SWAP) Instrument Aboard New Horizons","volume":"140","author":"McComas","year":"2008","journal-title":"Space Sci. Rev."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"45","DOI":"10.3847\/1538-4357\/ad8578","article-title":"Persistent Behavior in Energetic Neutral Atom Time Series from IBEX","volume":"976","author":"Sarlis","year":"2024","journal-title":"Astrophys. J."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1007\/s11214-018-0550-1","article-title":"Interstellar Mapping and Acceleration Probe (IMAP): A New NASA Mission","volume":"214","author":"McComas","year":"2018","journal-title":"Space Sci. Rev."},{"key":"ref_113","doi-asserted-by":"crossref","unstructured":"Livadiotis, G., and McComas, D.J. (2023). Entropy defect in thermodynamics. Sci. Rep., 13.","DOI":"10.1038\/s41598-023-36080-w"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"2062","DOI":"10.3390\/e17042062","article-title":"Kappa and q Indices: Dependence on the Degrees of Freedom","volume":"17","author":"Livadiotis","year":"2015","journal-title":"Entropy"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"16","DOI":"10.3847\/1538-4365\/abd4ed","article-title":"Anisotropic Kappa Distributions. I. Formulation Based on Particle Correlations","volume":"253","author":"Livadiotis","year":"2021","journal-title":"Astrophys. J."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"50001","DOI":"10.1209\/0295-5075\/122\/50001","article-title":"Thermodynamic origin of kappa distributions","volume":"122","author":"Livadiotis","year":"2018","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"21001","DOI":"10.1209\/0295-5075\/ad0764","article-title":"Entropy defect: Algebra and thermodynamics","volume":"144","author":"Livadiotis","year":"2023","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"123101","DOI":"10.1088\/1742-5468\/aa967f","article-title":"Uniqueness and characterization theorems for generalized entropies","volume":"12","author":"Enciso","year":"2017","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_120","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_121","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/S0378-4371(99)00064-3","article-title":"Correlation induced by Tsallis\u2019 nonextensivity","volume":"269","author":"Abe","year":"1999","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.physa.2007.02.033","article-title":"Quasi-additivity of Tsallis entropies and correlated subsystems","volume":"379","author":"Asgarani","year":"2007","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"10","DOI":"10.3847\/1538-4357\/ab05b7","article-title":"On the Origin of Polytropic Behavior in Space and Astrophysical Plasmas","volume":"874","author":"Livadiotis","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.physa.2016.09.046","article-title":"On the average uncertainty for systems with nonlinear coupling","volume":"468","author":"Nelson","year":"2017","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"88","DOI":"10.3847\/1538-4357\/acf45d","article-title":"Connection between Polytropic Index and Heating","volume":"956","author":"Livadiotis","year":"2023","journal-title":"Astrophys. J."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"105605","DOI":"10.1088\/1402-4896\/acf795","article-title":"Extensive entropy: The case of zero entropy defect","volume":"98","author":"Livadiotis","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"41003","DOI":"10.1209\/0295-5075\/ad4415","article-title":"Universality of kappa distributions","volume":"146","author":"Livadiotis","year":"2024","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_128","doi-asserted-by":"crossref","unstructured":"Livadiotis, G., and McComas, D.J. (2024). The theory of thermodynamic relativity. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-72779-0"},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"061105","DOI":"10.1103\/PhysRevE.63.061105","article-title":"General pseudoadditivity of composable entropy prescribed by the existence of equilibrium","volume":"63","author":"Abe","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"014127","DOI":"10.1103\/PhysRevE.109.014127","article-title":"Superstatistics from a dynamical perspective: Entropy and relaxation","volume":"109","author":"Ourabah","year":"2024","journal-title":"Phys. Rev. E"},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.physa.2004.03.082","article-title":"A possible deformed algebra and calculus inspired in nonextensive thermostatistics","volume":"340","author":"Borges","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"34","DOI":"10.3847\/1538-4357\/ab7010","article-title":"Evolution of Entropy and Mediation of the Solar Wind by Turbulence","volume":"891","author":"Adhikari","year":"2020","journal-title":"Astrophys. J."},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1002\/andp.19113411505","article-title":"Die Anwendung der kinetischen Theorie der Gase auf chemische Probleme","volume":"341","author":"Sackur","year":"1911","journal-title":"Ann. Der Phys."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1002\/andp.19123430708","article-title":"Die chemische Konstante der Gase und das elementare Wirkungsquantum","volume":"343","author":"Tetrode","year":"1912","journal-title":"Ann. Der Phys."},{"key":"ref_135","doi-asserted-by":"crossref","unstructured":"Baierlein, R. (1999). Thermal Physics, Cambridge University Press.","DOI":"10.1017\/CBO9780511840227"},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/S0378-4371(96)00440-2","article-title":"Generalization of the mean-field Ising model within Tsallis thermostatistics","volume":"238","author":"Demirhan","year":"1997","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"1386","DOI":"10.1088\/0004-637X\/713\/2\/1386","article-title":"Superposition of Stochastic Processes and the Resulting Particle Distributions","volume":"713","author":"Schwadron","year":"2010","journal-title":"Astrophys. J."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"6390","DOI":"10.1073\/pnas.1103539108","article-title":"Generalized entropies and the transformation group of superstatistics","volume":"108","author":"Hanel","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1111\/nrm.12084","article-title":"Kappa Function as a Unifying Framework for Discrete Population Modeling","volume":"29","author":"Livadiotis","year":"2016","journal-title":"Nat. Resour. Model."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"3","DOI":"10.3847\/1538-4357\/ab487a","article-title":"Rankine-Hugoniot Shock Conditions for Space and Astrophysical Plasmas Described by Kappa Distributions","volume":"886","author":"Livadiotis","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"30005","DOI":"10.1209\/0295-5075\/130\/30005","article-title":"Physical meaning of temperature in superstatistics","volume":"130","author":"Gravanis","year":"2020","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1140\/epjb\/s10051-021-00066-2","article-title":"Bivariate superstatistics: An application to statistical plasma physics","volume":"94","year":"2021","journal-title":"Eur. Phys. J. B"},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"A06108","DOI":"10.1029\/2005JA011524","article-title":"Particle acceleration at perpendicular shock waves: Model and observations","volume":"111","author":"Zank","year":"2006","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"8733","DOI":"10.1002\/2014JA020426","article-title":"The case for a common spectrum of particles accelerated in the heliosphere: Observations and theory","volume":"119","author":"Fisk","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"7025","DOI":"10.1002\/2014JA020095","article-title":"Simulations of plasma obeying Coulomb\u2019s law and the formation of suprathermal ion tails in the solar wind","volume":"119","author":"Randol","year":"2014","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"e2022JA030673","DOI":"10.1029\/2022JA030673","article-title":"RCM Modeling of Bubble Injections Into the Inner Magnetosphere: Spectral Properties of Plasma Sheet Particles","volume":"128","author":"Sadeghzadeh","year":"2023","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"e2023JA031983","DOI":"10.1029\/2023JA031983","article-title":"Loading Loss-Cone Distributions in Particle Simulations","volume":"128","author":"Zenitani","year":"2023","journal-title":"J. Geophys. Res. (Space Phys.)"},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"20380","DOI":"10.1073\/pnas.1320578110","article-title":"A maximum entropy framework for nonexponential distributions","volume":"110","author":"Peterson","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1006\/icar.1995.1121","article-title":"Temperature inversion in the Io plasma torus","volume":"116","author":"Moncuquet","year":"1995","journal-title":"Icarus"},{"key":"ref_151","doi-asserted-by":"crossref","first-page":"52","DOI":"10.3847\/1538-4357\/ab31ad","article-title":"Long-term Correlations of Polytropic Indices with Kappa Distributions in Solar Wind Plasma near 1 au","volume":"884","author":"Nicolaou","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"26","DOI":"10.3847\/1538-4357\/abaaae","article-title":"Polytropic Behavior of Solar Wind Protons Observed by Parker Solar Probe","volume":"901","author":"Nicolaou","year":"2020","journal-title":"Astrophys. J."},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"L10","DOI":"10.1051\/0004-6361\/202450217","article-title":"Proton polytropic behavior of periodic density structures in the solar wind","volume":"686","author":"Katsavrias","year":"2024","journal-title":"Astron. Astrophys."},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"142","DOI":"10.3847\/1538-4357\/aaa713","article-title":"Generation of Kappa Distributions in Solar Wind at 1 au","volume":"853","author":"Livadiotis","year":"2018","journal-title":"Astrophys. J."},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"052218","DOI":"10.1103\/PhysRevE.103.052218","article-title":"Self-consistent model of the plasma staircase and nonlinear Schr\u00f6dinger equation with subquadratic power nonlinearity","volume":"103","author":"Milovanov","year":"2021","journal-title":"Phys. Rev. E"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/375\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:07:14Z","timestamp":1760029634000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/375"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,31]]},"references-count":155,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040375"],"URL":"https:\/\/doi.org\/10.3390\/e27040375","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,31]]}}}