{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:59:48Z","timestamp":1760151588409,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T00:00:00Z","timestamp":1648598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Possible implications and consequences of using SL(2R) as invariance groups in the description at any scale resolution of the dynamics of any complex system are analyzed. From this perspective and based on Jaynes\u2019 remark (any circumstance left unspecified in the description of any complex system dynamics has the concrete expression in the existence of an invariance group), in the present paper one specifies such unspecified circumstances that result directly from the consideration of the canonical formalism induced by the SL(2R) as invariance group. It follows that both the Hamiltonian function and the Guassian distribution acquire the status of invariant group functions, the parameters that define the Hamiltonian acquire statistical significances based on a principle of maximizing informational energy, the class of statistical hypotheses specific to Gaussians of the same average acts as transitivity manifolds of the group (transitivity manifolds which can be correlated with the multifractal-non-multifractal scale transitions), joint invariant functions induced through SL(2R) groups isomorphism (the SL(2R) variables group, and the SL(2R) parameters group, etc.). For an ensemble of oscillators of the same frequency, the unspecified circumstances return to the ignorance of the amplitude and phase of each of the oscillators, which forces the recourse to a statistical ensemble traversed by the transformations of the Barbilian-type group. Finally, the model is validated based on numerical simulations and experimental results that refer to transient phenomena in ablation plasmas. The novelty of our model resides in the fact that fractalization through stochasticization is imposed through group invariance, situation in which the group\u2019s transitivity manifolds can be correlated with the scale resolution.<\/jats:p>","DOI":"10.3390\/e24040484","type":"journal-article","created":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T21:22:14Z","timestamp":1648675334000},"page":"484","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Implications and Consequences of SL(2R) as Invariance Group in the Description of Complex Systems Dynamics from a Multifractal Perspective of Motion"],"prefix":"10.3390","volume":"24","author":[{"given":"Lucian","family":"Dobreci","sequence":"first","affiliation":[{"name":"Department of Physical and Occupational Therapy, \u201cVasile Alecsandri\u201d University of Bacau, 600115 Bacau, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Oana","family":"Rusu","sequence":"additional","affiliation":[{"name":"Faculty of Material Science and Engineering, \u201cGheorghe Asachi\u201d Technical University, 700050 Iasi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Decebal","family":"Vasincu","sequence":"additional","affiliation":[{"name":"Faculty of Dental Medicine, \u201cGrigore T. Popa\u201d University of Medicine and Pharmacy of Ia\u0219i, 700050 Iasi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9885-1425","authenticated-orcid":false,"given":"Mihaela","family":"Jarc\u0103u","sequence":"additional","affiliation":[{"name":"Faculty of Food Engineering, Stefan cel Mare University of Suceava, 720229 Suceava, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cristina Marcela","family":"Rusu","sequence":"additional","affiliation":[{"name":"Physics Department, \u201cGheorghe Asachi\u201d Technical University, 700050 Iasi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2248-9954","authenticated-orcid":false,"given":"Silviu","family":"Gurlui","sequence":"additional","affiliation":[{"name":"Physics Faculty, Alexandru Ioan Cuza University of Ia\u0219i, 700050 Iasi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vlad","family":"Ghizdovat","sequence":"additional","affiliation":[{"name":"Department of Biophysics and Medical Physics, \u201cGrigore T. Popa\u201d University of Medicine and Pharmacy of Iasi, 700115 Iasi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alina","family":"Gavrilut","sequence":"additional","affiliation":[{"name":"Mathematics Faculty, Alexandru Ioan Cuza University of Ia\u0219i, 700050 Iasi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maricel","family":"Agop","sequence":"additional","affiliation":[{"name":"Physics Department, \u201cGheorghe Asachi\u201d Technical University, 700050 Iasi, Romania"},{"name":"Romanian Scientists Academy, 54 Splaiul Independentei, 050094 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,30]]},"reference":[{"key":"ref_1","unstructured":"Hsiang, W.-Y. (2017). Lectures on Lie Groups, World Scientific."},{"key":"ref_2","unstructured":"Varopoulos, N.T. (2018). Geometric and Potential Theoretic Results on Lie Groups, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mazilu, N., Agop, M., and Merches, I. (2020). The Mathematical Principles of Scale Relativity Physics: The Concept of Interpretation, CRC Press Taylor & Francis Group.","DOI":"10.1201\/9780429329050"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mazilu, N., Agop, M., and Merches, I. (2021). Scale Transitions as Foundations of Physics, World Scientific.","DOI":"10.1142\/12151"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jaynes, E.T. (2003). Probability Theory: The Logic of Science, Cambridge University Press.","DOI":"10.1017\/CBO9780511790423"},{"key":"ref_6","unstructured":"Mazilu, N., and Agop, M. (2012). Skyrmions: A Great Finishing Touch to Classical Newtonian Philosophy, Nova."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lawrence, A. (2019). Probability in Physics: An Introductory Guide, Springer.","DOI":"10.1007\/978-3-030-04544-9"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Saviuc, A., Girtu, M., Topliceanu, L., Petrescu, T.C., and Agop, M. (2021). \u201cHolographic Implementations\u201d in the Complex Fluid Dynamics through a Fractal Paradigm. Mathematics, 9.","DOI":"10.3390\/math9182273"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gavrilut, G., Topliceanu, L., Girtu, M., Rotundu, A.M., Irimiciuc, S.A., and Agop, M. (2021). Assessment of Complex System Dynamics via Harmonic Mapping in a Multifractal Paradigm. Mathematics, 9.","DOI":"10.3390\/math9243298"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Agop, M., and Merches, I. (2019). Operational Procedures describing Physical Systems, CRC Press Taylor & Francis Group.","DOI":"10.1201\/9780429399589"},{"key":"ref_11","unstructured":"Okubo, S., and Das, A. (2014). Lie Groups and Lie Algebras for Physicists, World Scientific."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hall, B.C. (2015). Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Springer.","DOI":"10.1007\/978-3-319-13467-3"},{"key":"ref_13","unstructured":"Udriste, C.N., and Calin, O.V. (2014). Geometric Modeling in Probability and Statistics, World Scientific."},{"key":"ref_14","unstructured":"Mathai, A.M. (1999). An Introduction to Geometrical Probability: Distributional Aspects with Applications, CRC Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Popa, M.N., and Pricop, V.V. (2021). Lie Algebra of Operators of Centro-Affine Group Representation in the Coefficient Space of Polynomial Differential Systems, Chapman and Hall, CRC.","DOI":"10.1201\/9781003193074-2"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Merches, I., and Agop, M. (2016). Differentiability and Fractality in Dynamics of Physical Systems, World Scientific.","DOI":"10.1142\/9606"},{"key":"ref_17","unstructured":"Stoka, M. (1967). Integral Geometry, Romanian Academy Publishing House. (In Romanian)."},{"key":"ref_18","unstructured":"Agop, M., and Paun, V.P. (2017). On the new perspectives of fractal theory. Fundaments and Applications, Romanian Academy Publishing House."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"105774","DOI":"10.1016\/j.sab.2020.105774","article-title":"Multiple Structure Formation and Molecule Dynamics in Transient Plasmas Generated by Laser Ablation of Graphite","volume":"165","author":"Irimiciuc","year":"2020","journal-title":"Spectrochim. Acta Part B At. Spectrosc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2000136","DOI":"10.1002\/ppap.202000136","article-title":"Investigation of Laser-Produced Plasma Multistructuring by Floating Probe Measurements and Optical Emission Spectroscopy","volume":"17","author":"Irimiciuc","year":"2020","journal-title":"Plasma Processes Polym."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"083301","DOI":"10.1063\/1.4977010","article-title":"A Compact Non-Differential Approach for Modeling Laser Ablation Plasma Dynamics","volume":"121","author":"Irimiciuc","year":"2017","journal-title":"J. Appl. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113511","DOI":"10.1063\/1.4936106","article-title":"On the Interaction between Two Fireballs in Low-Temperature Plasma","volume":"22","author":"Dimitriu","year":"2015","journal-title":"Phys. Plasmas"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"105904","DOI":"10.1016\/j.sab.2020.105904","article-title":"Space-and Time-Resolved Optical Investigations on Ns-Laser Produced Plasmas on Various Geological Samples","volume":"170","author":"Irimiciuc","year":"2020","journal-title":"Spectrochim. Acta Part B At. Spectrosc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3342","DOI":"10.1016\/j.physleta.2009.07.044","article-title":"Plume Splitting in Pico-Second Laser? Material Interaction under the Influence of Shock Wave","volume":"373","author":"Gacek","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1103\/PhysRevB.58.1533","article-title":"Dynamics of Plume Propagation and Splitting during Pulsed-Laser Ablation of Si in He and Ar","volume":"58","author":"Wood","year":"1998","journal-title":"Phys. Rev. B\u2014Condens. Matter Mater. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1571","DOI":"10.1103\/PhysRevLett.79.1571","article-title":"Dynamics of Plume Propagation and Splitting during Pulsed-Laser Ablation","volume":"79","author":"Wood","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"113512","DOI":"10.1063\/1.4835255","article-title":"Understanding Plume Splitting of Laser Ablated Plasma: A View from Ion Distribution Dynamics","volume":"20","author":"Wu","year":"2013","journal-title":"Phys. Plasmas"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Heil, C., and Walnut, D.F. (2009). Fundamental Papers in Wavelet Theory, Princeton University Press.","DOI":"10.1515\/9781400827268"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2773","DOI":"10.1007\/s10773-015-2910-x","article-title":"The Classical Theory of Light Colors: A Paradigm for Description of Particle Interactions","volume":"55","author":"Mazilu","year":"2016","journal-title":"Int. J. Theor. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1650153","DOI":"10.1142\/S0217984916501530","article-title":"From Kepler problem to skyrmions","volume":"30","author":"Mazilu","year":"2016","journal-title":"Modern Phys. Lett. B"},{"key":"ref_31","first-page":"139","article-title":"Role of surface gauging in extended particle interactions: The case for spin","volume":"131","author":"Mazilu","year":"2016","journal-title":"European"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Baddour, N. (2019). Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules. Mathematics, 7.","DOI":"10.20944\/preprints201907.0151.v1"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Arfaoui, S., Ben Mabrouk, A., and Cattani, C. (2021). Wavelet Analysis. Basic Concepts and Applications, CRC Press.","DOI":"10.1201\/9781003096924"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gray, R.M. (2011). Entropy and Information Theory, Springer.","DOI":"10.1007\/978-1-4419-7970-4"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Nottale, L. (1993). Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity, World Scientific.","DOI":"10.1142\/1579"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Nottale, L. (2011). Scale Relativity and Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics, Imperial College Press.","DOI":"10.1142\/9781848166516"},{"key":"ref_37","unstructured":"Mandelbrot, B.B. (1982). The Fractal Geometry of Nature, W.H. Freeman and Co."},{"key":"ref_38","unstructured":"Einstein, A., Lawson, R.W., and Raine, D.J. (2014). Relativity: The Special and the General Theory, Routledge."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3174","DOI":"10.1103\/PhysRevB.4.3174","article-title":"Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture","volume":"4","author":"Wilson","year":"1971","journal-title":"Phys. Rev. B"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/484\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:46:39Z","timestamp":1760136399000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/484"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,30]]},"references-count":39,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["e24040484"],"URL":"https:\/\/doi.org\/10.3390\/e24040484","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,3,30]]}}}