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(2025) introduced a numerical experiment, where ordinary uncorrelated collisions between collision pairs are followed by other, controlled (correlated) collisions, shedding light on the emergence of kappa distributions through particle correlations in space plasmas. We extend this experiment by introducing correlations indicating that (i) when long-range correlations are interwoven with collision pairs, the resulting thermodynamic kappa are described as that corresponding to an \u2018interatomic\u2019 potential interaction among particles; (ii) searching for a closer description of heliospheric plasmas, we found that pairwise short-range correlations are sufficient to lead to appropriate values of thermodynamic kappa, especially when forming correlated clusters; (iii) multi-particle correlations do not lead to physical stationary states; finally, (iv) an optimal model arises when combining all previous findings. In an excellent match with space plasmas observations, the thermodynamic kappa that describes the stationary state at which the system is stabilized behaves as follows: (a) When correlations are turned off, kappa is turning toward infinity, indicating the state of classical thermal equilibrium (Maxwell-Boltzmann distribution), (b) When collisions are turned off, kappa is turning toward the anti-equilibrium state, the furthest state from the classical thermal equilibrium (\u22125 power-law phase-space distribution), and (c) the finite kappa values are generally determined by the competing factor of collisions and correlations.<\/jats:p>","DOI":"10.3390\/e27060646","type":"journal-article","created":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T05:52:47Z","timestamp":1750139567000},"page":"646","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Exploring Numerical Correlations: Models and Thermodynamic Kappa"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8483-519X","authenticated-orcid":false,"given":"Nicholas V.","family":"Sarlis","sequence":"first","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6160-1158","authenticated-orcid":false,"given":"David J.","family":"McComas","sequence":"additional","affiliation":[{"name":"Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7655-6019","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"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,17]]},"reference":[{"key":"ref_1","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_2","unstructured":"Livadiotis, G. (2017). Kappa Distributions: Theory and Applications in Plasmas, Elsevier."},{"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":"Proceedings of the Physics of the Magnetosphere: Based upon the Proceedings of the Conference Held at Boston College June 19\u201328, 1967"},{"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":"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_7","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_8","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_9","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. Suppl. Ser."},{"key":"ref_10","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_11","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_12","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_13","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_14","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_15","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_16","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_17","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. Suppl. Ser."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"McComas, D.J., Livadiotis, G., and Sarlis, N.V. (2025). Correlations and Kappa Distributions: Numerical Experiment and Physical Understanding. Entropy, 27.","DOI":"10.3390\/e27040375"},{"key":"ref_19","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_20","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_21","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_22","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_23","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_24","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_25","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_26","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_27","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_28","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_29","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_30","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_31","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_32","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s10509-016-2949-z","article-title":"Misestimation of temperature when applying Maxwellian distributions to space plasmas described by kappa distributions","volume":"361","author":"Nicolaou","year":"2016","journal-title":"Astrophys. Space Sci."},{"key":"ref_33","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_34","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_35","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_36","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_37","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_38","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_39","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_40","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_41","doi-asserted-by":"crossref","first-page":"169","DOI":"10.3847\/1538-4357\/adb8d2","article-title":"What Defines Stationarity in Space Plasmas","volume":"982","author":"Livadiotis","year":"2025","journal-title":"Astrophys. J."},{"key":"ref_42","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_43","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_44","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_45","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. Suppl. Ser."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1007\/s10910-009-9524-6","article-title":"Approach on Tsallis statistical interpretation of hydrogen-atom by adopting the generalized radial distribution function","volume":"45","author":"Livadiotis","year":"2009","journal-title":"J. Math. Chem."},{"key":"ref_47","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_48","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_49","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_50","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. Suppl. Ser."},{"key":"ref_51","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_52","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_53","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_54","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. Appl."},{"key":"ref_55","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_56","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_57","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_58","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_59","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_60","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_61","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_62","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_63","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_64","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_65","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_66","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_67","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_68","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_69","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_70","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_71","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_72","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_73","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_74","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_75","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_76","doi-asserted-by":"crossref","first-page":"7343","DOI":"10.1103\/PhysRevE.60.7343","article-title":"Statistical analysis of ionic current fluctuations in membrane channels","volume":"60","author":"Mercik","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"051910","DOI":"10.1103\/PhysRevE.63.051910","article-title":"Stochastic origins of the long-range correlations of ionic current fluctuations in membrane channels","volume":"63","author":"Mercik","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_78","unstructured":"Gradsteyn, I.S., and Ryzhik, I.M. (1980). Table of Integrals, Series and Products, Academic Press."},{"key":"ref_79","unstructured":"Williams, T., and Kelley, C. (2021, December 24). Gnuplot 5.4: An Interactive Plotting Program. Available online: http:\/\/www.gnuplot.info\/."},{"key":"ref_80","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_81","unstructured":"Jackson, J.D. (1975). Classical Electrodynamics, Wiley. [2nd ed.]."},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Malthe-S\u00f8renssen, A. (2024). Percolation Theory Using Python, Springer Nature.","DOI":"10.1007\/978-3-031-59900-2"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of Scaling in Random Networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1038\/nature11486","article-title":"Network science: Luck or reason","volume":"489","year":"2012","journal-title":"Nature"},{"key":"ref_85","unstructured":"Barab\u00e1si, A.L., and P\u00f3sfai, M. (2016). Network Science, Cambridge University Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/646\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:53:21Z","timestamp":1760032401000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/646"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,17]]},"references-count":85,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["e27060646"],"URL":"https:\/\/doi.org\/10.3390\/e27060646","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,17]]}}}