{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T04:06:11Z","timestamp":1775793971013,"version":"3.50.1"},"reference-count":114,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,22]],"date-time":"2018-10-22T00:00:00Z","timestamp":1540166400000},"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>The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play\u2014and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components.<\/jats:p>","DOI":"10.3390\/e20100807","type":"journal-article","created":{"date-parts":[[2018,10,23]],"date-time":"2018-10-23T08:43:36Z","timestamp":1540284216000},"page":"807","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["The Role of Data in Model Building and Prediction: A Survey Through Examples"],"prefix":"10.3390","volume":"20","author":[{"given":"Marco","family":"Baldovin","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, \u201cSapienza\u201d Universit\u00e0 di Roma, p.le A. Moro 2, 00185 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8351-248X","authenticated-orcid":false,"given":"Fabio","family":"Cecconi","sequence":"additional","affiliation":[{"name":"Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, 00185 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Massimo","family":"Cencini","sequence":"additional","affiliation":[{"name":"Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, 00185 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7229-9457","authenticated-orcid":false,"given":"Andrea","family":"Puglisi","sequence":"additional","affiliation":[{"name":"CNR-ISC and Dipartimento di Fisica, Sapienza Universit\u00e0 di Roma, p.le A. Moro 2, 00185 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angelo","family":"Vulpiani","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, \u201cSapienza\u201d Universit\u00e0 di Roma, p.le A. Moro 2, 00185 Roma, Italy"},{"name":"Centro Linceo Interdisciplinare \u201cB. Segre\u201d, Accademia dei Lincei, via della Lungara 10, 00165 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1086\/286874","article-title":"The role of models in science","volume":"12","author":"Rosenblueth","year":"1945","journal-title":"Philos. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Morgan, M.S., and Morrison, M. (1999). Models as Mediators: Perspectives on Natural and Social Science, Cambridge University Press.","DOI":"10.1017\/CBO9780511660108"},{"key":"ref_3","unstructured":"R\u00e9nyi, A. (1967). Dialogues on Mathematics, Holden-Day."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic Nonperiodic Flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1038\/s41567-018-0204-y","article-title":"A dynamical systems approach to gross domestic product forecasting","volume":"14","author":"Tacchella","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_6","unstructured":"Volterra, V. (1926). Variazioni e Fluttuazioni del Numero D\u2019individui in Specie Animali Conviventi. Memorie del R. Comitato Talassografico Italiano. Mem. CXXXI, Accademia Nazionale dei Lincei. (In Italian)."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1073\/pnas.6.7.410","article-title":"Analytical Note on Certain Rhythmic Relations in Organic Systems","volume":"6","author":"Lotka","year":"1920","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Holmes, P., Lumley, J.L., Berkooz, G., and Rowley, C.W. (2012). Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press.","DOI":"10.1017\/CBO9780511919701"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF01646553","article-title":"On the nature of turbulence","volume":"20","author":"Ruelle","year":"1971","journal-title":"Commun. Math. Phys."},{"key":"ref_10","unstructured":"Baca\u00ebr, N. (2009). Histoires de Math\u00e9matiques et de Populations, Cassini."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2001). Mathematical Biology: I. An Introduction, Springer. [3rd ed.]. Interdisciplinary Applied Mathematics.","DOI":"10.1007\/b98868"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Cencini, M., Cecconi, F., and Vulpiani, A. (2009). Chaos: From Simple Models to Complex Systems, World Scientific.","DOI":"10.1142\/7351"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"8133","DOI":"10.1038\/ncomms9133","article-title":"Automated adaptive inference of phenomenological dynamical models","volume":"6","author":"Daniels","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"120503","DOI":"10.1103\/PhysRevLett.108.120503","article-title":"Extracting Dynamical Equations from Experimental Data is NP Hard","volume":"108","author":"Cubitt","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.physleta.2017.11.012","article-title":"Reconstruction of a random phase dynamics network from observations","volume":"382","author":"Pikovsky","year":"2018","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"024101","DOI":"10.1103\/PhysRevLett.109.024101","article-title":"Inference of Time-Evolving Coupled Dynamical Systems in the Presence of Noise","volume":"109","author":"Stankovski","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1126\/science.1165893","article-title":"Distilling free-form natural laws from experimental data","volume":"324","author":"Schmidt","year":"2009","journal-title":"Science"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3932","DOI":"10.1073\/pnas.1517384113","article-title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","volume":"113","author":"Brunton","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1175\/1520-0469(1969)26<636:APARBN>2.0.CO;2","article-title":"Atmospheric Predictability as Revealed by Naturally Occurring Analogues","volume":"26","author":"Lorenz","year":"1969","journal-title":"J. Atmos. Sci."},{"key":"ref_20","first-page":"345","article-title":"Three approachs to atmospheric predictability","volume":"50","author":"Lorenz","year":"1969","journal-title":"Bull. Am. Meteorol. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1126\/science.1091277","article-title":"Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication","volume":"304","author":"Jaeger","year":"2004","journal-title":"Science"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"121102","DOI":"10.1063\/1.5010300","article-title":"Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data","volume":"27","author":"Pathak","year":"2017","journal-title":"Chaos"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"024102","DOI":"10.1103\/PhysRevLett.120.024102","article-title":"Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach","volume":"120","author":"Pathak","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"041102","DOI":"10.1063\/1.4979665","article-title":"Reservoir observers: Model-free inference of unmeasured variables in chaotic systems","volume":"27","author":"Lu","year":"2017","journal-title":"Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Takens, F. (1981). Detecting strange attractors in turbulence. Dynamical Systems and Turbulence, Warwick 1980, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/BFb0091924"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/BF01053745","article-title":"Embedology","volume":"65","author":"Sauer","year":"1991","journal-title":"J. Stat. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1126\/science.1227079","article-title":"Detecting Causality in Complex Ecosystems","volume":"338","author":"Sugihara","year":"2012","journal-title":"Science"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1119\/1.4746070","article-title":"Predicting the future from the past: An old problem from a modern perspective","volume":"80","author":"Cecconi","year":"2012","journal-title":"Am. J. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Campbell, L., and Garnett, W. (1882). The Life of James Clerk Maxwell: With a Selection From His Correspondence and Occasional Writings and a Sketch of His Contributions to Science, Macmillan.","DOI":"10.5962\/bhl.title.28848"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kantz, H., and Schreiber, T. (2004). Nonlinear Time Series Analysis, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511755798"},{"key":"ref_31","first-page":"1","article-title":"Sur le probl\u00e8me des trois corps et les \u00e9quations de la dynamique","volume":"13","year":"1890","journal-title":"Acta Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1002","DOI":"10.1090\/S0002-9904-1947-08927-8","article-title":"On the notion of recurrence in discrete stochastic processes","volume":"53","author":"Kac","year":"1947","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_33","unstructured":"Lorenz, E.N. (1995). Predictability: A problem partly solved. ECMWF Semin. Proc. I, 1."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"043105","DOI":"10.1063\/1.3496397","article-title":"Extensive chaos in the Lorenz-96 model","volume":"20","author":"Karimi","year":"2010","journal-title":"Chaos"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1103\/PhysRevLett.50.346","article-title":"Characterization of Strange Attractors","volume":"50","author":"Grassberger","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","first-page":"241","article-title":"The Claude Bernard Lecture, 1989. Deterministic Chaos: The Science and the Fiction","volume":"427","author":"Ruelle","year":"1990","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0370-1573(01)00025-4","article-title":"Predictability: A way to characterize complexity","volume":"356","author":"Boffetta","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1016\/S0375-9601(98)00350-8","article-title":"Analysing local observations of weakly coupled maps","volume":"244","author":"Olbrich","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_39","unstructured":"Bohr, T., Jensen, M.H., Paladin, G., and Vulpiani, A. (2005). Dynamical Systems Approach to Turbulence, Cambridge University Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1126\/science.181.4096.223","article-title":"Principles that govern the folding of protein chains","volume":"181","author":"Anfinsen","year":"1973","journal-title":"Science"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Finkelstein, A.V., and Ptitsyn, O. (2016). Protein Physics: A Course of Lectures, Academic Press. [2nd ed.].","DOI":"10.1016\/B978-0-12-809676-5.00021-1"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1146\/annurev.physchem.48.1.545","article-title":"Theory of protein folding: The energy landscape perspective","volume":"48","author":"Onuchic","year":"1997","journal-title":"Annu. Rev. Phys. Chem."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1017\/S0033583502003761","article-title":"Understanding protein folding with energy landscape theory. Part I: Basic concepts","volume":"35","author":"Plotkin","year":"2002","journal-title":"Q. Rev. Biophys."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Schlick, T. (2010). Molecular Modeling and Simulation: An Interdisciplinary Guide, Springer. [2nd ed.]. Interdisciplinary Applied Mathematics.","DOI":"10.1007\/978-1-4419-6351-2"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Gremlich, H.U., and Yan, B. (2000). Infrared and Raman Spectroscopy of Biological Materials, CRC Press. [1st ed.].","DOI":"10.1201\/b16936"},{"key":"ref_46","unstructured":"Chandler, D. (1987). Introduction to Modern Statistical Mechanics, OUP USA."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3599","DOI":"10.1103\/PhysRevA.28.3599","article-title":"Pair potentials in disordered many-particle systems: A study for liquid gallium","volume":"28","author":"Schommers","year":"1983","journal-title":"Phys. Rev. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/j.sbi.2005.02.005","article-title":"Coarse-grained models for proteins","volume":"15","author":"Tozzini","year":"2005","journal-title":"Curr. Opin. Struct. Biol."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.sbi.2007.10.005","article-title":"Coarse-grained models of protein folding: toy models or predictive tools?","volume":"18","author":"Clementi","year":"2008","journal-title":"Curr. Opin. Struct. Biol."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"217","DOI":"10.2174\/157489310792006729","article-title":"Coarse Grained Modeling and Approaches to Protein Folding","volume":"5","author":"Guardiani","year":"2010","journal-title":"Curr. Bioinform."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"7898","DOI":"10.1021\/acs.chemrev.6b00163","article-title":"Coarse-Grained Protein Models and Their Applications","volume":"116","author":"Kmiecik","year":"2016","journal-title":"Chem. Rev."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/00018732.2017.1341604","article-title":"Inverse statistical problems: from the inverse Ising problem to data science","volume":"66","author":"Nguyen","year":"2017","journal-title":"Adv. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1021\/ma60054a013","article-title":"Medium- and Long-Range Interaction Parameters between Amino Acids for Predicting Three-Dimensional Structures of Proteins","volume":"9","author":"Tanaka","year":"1976","journal-title":"Macromolecules"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1111\/j.1399-3011.1975.tb02465.x","article-title":"Studies on protein folding, unfolding and fluctuations by computer simulation. I. The effect of specific amino acid sequence represented by specific inter-unit interactions","volume":"7","author":"Taketomi","year":"1975","journal-title":"Int. J. Pept. Protein Res."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1006\/jmbi.2000.3693","article-title":"Topological and energetic factors: what determines the structural details of the transition state ensemble and \u201cen-route\u201d intermediates for protein folding? An investigation for small globular proteins","volume":"298","author":"Clementi","year":"2000","journal-title":"J. Mol. Biol."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1529\/biophysj.105.069138","article-title":"Testing Simplified Proteins Models of the hPin1 WW Domain","volume":"91","author":"Cecconi","year":"2006","journal-title":"Biophys. J."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1006\/jmbi.1998.1645","article-title":"Contact order, transition state placement and the refolding rates of single domain proteins","volume":"277","author":"Plaxco","year":"1998","journal-title":"J. Mol. Biol."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1038\/14890","article-title":"Mutational analysis of acylphosphatase suggests the importance of topology and contact order in protein folding","volume":"6","author":"Chiti","year":"1999","journal-title":"Nat. Struct. Mol. Biol."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/j.jmb.2008.07.007","article-title":"Subdomain competition, cooperativity, and topological frustration in the folding of CheY","volume":"382","author":"Hills","year":"2008","journal-title":"J. Mol. Biol."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1529\/biophysj.107.106625","article-title":"Stability and kinetic properties of C5-domain from myosin binding protein C and its mutants","volume":"94","author":"Guardiani","year":"2008","journal-title":"Biophys. J."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2054","DOI":"10.1529\/biophysj.106.081257","article-title":"The Mechanical Unfolding of Ubiquitin through All-Atom Monte Carlo Simulation with a G\u014d-Type Potential","volume":"92","author":"Kleiner","year":"2007","journal-title":"Biophys. J."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1529\/biophysj.106.087684","article-title":"Refolding upon force quench and pathways of mechanical and thermal unfolding of ubiquitin","volume":"92","author":"Li","year":"2007","journal-title":"Biophys. J."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"3032","DOI":"10.1016\/S0006-3495(02)75308-3","article-title":"Validity of G\u014d models: Comparison with a solvent-shielded empirical energy decomposition","volume":"83","author":"Paci","year":"2002","journal-title":"Biophys. J."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1750","DOI":"10.1110\/ps.03580104","article-title":"The effects of nonnative interactions on protein folding rates: Theory and simulation","volume":"13","author":"Clementi","year":"2004","journal-title":"Protein Sci."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2351","DOI":"10.1110\/ps.0205402","article-title":"The origins of asymmetry in the folding transition states of protein L and protein G","volume":"11","author":"Karanicolas","year":"2002","journal-title":"Protein Sci."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.jmb.2003.09.069","article-title":"Folding of Cu, Zn superoxide dismutase and familial amyotrophic lateral sclerosis","volume":"334","author":"Khare","year":"2003","journal-title":"J. Mol. Biol."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Oppenheim, I., Castiglione, P., Falcioni, M., Lesne, A., and Vulpiani, A. (2008). Chaos and Coarse Graining in Statistical Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511535291"},{"key":"ref_68","first-page":"367","article-title":"Heterogeneous multiscale methods: A review","volume":"2","author":"E","year":"2007","journal-title":"Commun. Comput. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"R55","DOI":"10.1088\/0951-7715\/17\/6\/R01","article-title":"Extracting macroscopic dynamics: Model problems and algorithms","volume":"17","author":"Givon","year":"2004","journal-title":"Nonlinearity"},{"key":"ref_70","unstructured":"Kampen, N.G.V. (2007). Stochastic Processes in Physics and Chemistry, North Holland. [3rd ed.]."},{"key":"ref_71","unstructured":"Gardiner, C. (2009). Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer. [4th ed.]."},{"key":"ref_72","first-page":"530","article-title":"Sur la th\u00e9orie du mouvement brownien","volume":"146","author":"Langevin","year":"1908","journal-title":"C. R. Acad. Sci."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1119\/1.18725","article-title":"Paul Langevin\u2019s 1908 paper \u201con the Theory of Brownian Motion\u201d [\u201cSur la th\u00e9orie du mouvement brownien,\u201d C. R. Acad. Sci. (Paris) 146, 530\u2013533 (1908)]","volume":"65","author":"Lemons","year":"1997","journal-title":"Am. J. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"126001","DOI":"10.1088\/0034-4885\/75\/12\/126001","article-title":"Stochastic thermodynamics, fluctuation theorems and molecular machines","volume":"75","author":"Seifert","year":"2012","journal-title":"Rep. Prog. Phys."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Livi, R., and Politi, P. (2017). Nonequilibrium Statistical Physics: A Modern Perspective, Cambridge University Press.","DOI":"10.1017\/9781107278974"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Zwanzig, R. (2001). Nonequilibrium Statistical Mechanics, OUP USA.","DOI":"10.1093\/oso\/9780195140187.001.0001"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1002\/andp.19063261405","article-title":"Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen","volume":"326","year":"1906","journal-title":"Ann. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"P12001","DOI":"10.1088\/1742-5468\/2007\/12\/P12001","article-title":"Transport properties of chaotic and non-chaotic many particle systems","volume":"2007","author":"Cecconi","year":"2007","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1139\/p61-056","article-title":"A power series expansion of the master equation","volume":"39","year":"1961","journal-title":"Can. J. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"34001","DOI":"10.1209\/0295-5075\/92\/34001","article-title":"Irreversible dynamics of a massive intruder in dense granular fluids","volume":"92","author":"Sarracino","year":"2010","journal-title":"EPL"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/BF02046762","article-title":"A mechanical model of Brownian motion","volume":"78","author":"Goldstein","year":"1981","journal-title":"Commun. Math. Phys."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1063\/1.1703664","article-title":"Statistical Dynamics of Simple Cubic Lattices. Model for the Study of Brownian Motion","volume":"1","author":"Rubin","year":"1960","journal-title":"J. Math. Phys."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0031-8914(60)90022-7","article-title":"Motion of a heavy particle in a one dimensional chain","volume":"26","author":"Turner","year":"1960","journal-title":"Physica"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF01008729","article-title":"Nonlinear generalized Langevin equations","volume":"9","author":"Zwanzig","year":"1973","journal-title":"J. Stat. Phys."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"064004","DOI":"10.1088\/1361-648X\/29\/6\/064004","article-title":"The modified Langevin description for probes in a nonlinear medium","volume":"29","author":"Maes","year":"2017","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"250601","DOI":"10.1103\/PhysRevLett.114.250601","article-title":"How Statistical Forces Depend on the Thermodynamics and Kinetics of Driven Media","volume":"114","author":"Basu","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"12002","DOI":"10.1088\/1742-5468\/2015\/12\/P12002","article-title":"A consistent description of fluctuations requires negative temperatures","volume":"2015","author":"Cerino","year":"2015","journal-title":"J. Stat. Mech."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.09.001","article-title":"Temperature in and out of equilibrium: A review of concepts, tools and attempts","volume":"709\u2013710","author":"Puglisi","year":"2017","journal-title":"Phys. Rep."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"113202","DOI":"10.1088\/1742-5468\/aa933e","article-title":"About thermometers and temperature","volume":"2017","author":"Baldovin","year":"2017","journal-title":"J. Stat. Mech."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"043207","DOI":"10.1088\/1742-5468\/aab687","article-title":"Langevin equation in systems with also negative temperatures","volume":"2018","author":"Baldovin","year":"2018","journal-title":"J. Stat. Mech."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1126\/science.1227831","article-title":"Negative absolute temperature for motional degrees of freedom","volume":"339","author":"Braun","year":"2013","journal-title":"Science"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.physrep.2011.05.003","article-title":"Approaching complexity by stochastic methods: From biological systems to turbulence","volume":"506","author":"Friedrich","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.physleta.2005.07.077","article-title":"An iterative procedure for the estimation of drift and diffusion coefficients of Langevin processes [rapid communication]","volume":"346","author":"Kleinhans","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"254501","DOI":"10.1103\/PhysRevLett.87.254501","article-title":"Indispensable Finite Time Corrections for Fokker-Planck Equations from Time Series Data","volume":"87","author":"Ragwitz","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_95","unstructured":"Brin, M., Hasselblatt, B., and Pesin, Y. (2004). Some recent advances in averaging. Modern Dynamical Systems and Applications, Cambridge University Press."},{"key":"ref_96","doi-asserted-by":"crossref","unstructured":"Thiel, M., Kurths, J., Romano, M.C., K\u00e1rolyi, G., and Moura, A. (2010). Langevin Equation for Slow Degrees of Freedom of Hamiltonian Systems. Nonlinear Dynamics and Chaos: Advances and Perspectives, Springer. Understanding Complex Systems.","DOI":"10.1007\/978-3-642-04629-2"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"226101","DOI":"10.1103\/PhysRevLett.91.226101","article-title":"Stochastic Analysis and Regeneration of Rough Surfaces","volume":"91","author":"Jafari","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1017\/S0022112001003597","article-title":"Experimental indications for Markov properties of small-scale turbulence","volume":"433","author":"Renner","year":"2001","journal-title":"J. Fluid Mech."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"041107","DOI":"10.1103\/PhysRevE.64.041107","article-title":"Modeling of nonlinear L\u00e9vy processes by data analysis","volume":"64","author":"Siegert","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1143\/PTPS.23.177","article-title":"Continuum Approximation for the Motion of a Heavy Particle in One- and Three-Dimensional Lattices","volume":"23","author":"Takeno","year":"1962","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"1973","DOI":"10.1016\/0031-8914(64)90017-5","article-title":"On the statistical mechanical theory of brownian motion","volume":"30","author":"Mazur","year":"1964","journal-title":"Physica"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1063\/1.1704304","article-title":"Statistical Mechanics of Assemblies of Coupled Oscillators","volume":"6","author":"Ford","year":"1965","journal-title":"J. Math. Phys."},{"key":"ref_103","doi-asserted-by":"crossref","unstructured":"Ma, S. (1985). Statistical Mechanics, World Scientific.","DOI":"10.1142\/0073"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"198001","DOI":"10.1103\/PhysRevLett.114.198001","article-title":"Cages and Anomalous Diffusion in Vibrated Dense Granular Media","volume":"114","author":"Scalliet","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"064511","DOI":"10.1063\/1.4928456","article-title":"An itinerant oscillator model with cage inertia for mesorheological granular experiments","volume":"143","author":"Lasanta","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"1505","DOI":"10.1103\/PhysRev.91.1505","article-title":"Fluctuations and Irreversible Processes","volume":"91","author":"Onsager","year":"1953","journal-title":"Phys. Rev."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s004070000032","article-title":"History and Epistemology of Models: Meteorology (1946\u20131963) as a Case Study","volume":"55","author":"Dalmedico","year":"2001","journal-title":"Arch. Hist. Exact Sci."},{"key":"ref_108","doi-asserted-by":"crossref","unstructured":"Hosni, H., and Vulpiani, A. (2017). Forecasting in Light of Big Data. Philos. Technol., 1\u201313.","DOI":"10.1007\/s13347-017-0265-3"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1126\/science.222.4627.971","article-title":"Studying scientific discovery by computer simulation","volume":"222","author":"Bradshaw","year":"1983","journal-title":"Science"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1090\/S0273-0979-1986-15461-3","article-title":"Computers and the Nature of Man: A Historian\u2019s Perspective on Controversies about Artificial Intelligence","volume":"15","author":"Grabiner","year":"1986","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_111","unstructured":"Hansen, J.P., and McDonald, I.R. (1990). Theory of Simple Liquids, Elsevier."},{"key":"ref_112","unstructured":"Hey, T., Tansley, S., and Tolle, K. (2009). The Fourth Paradigm: Data-Intensive Scientific Discovery, Microsoft Research."},{"key":"ref_113","unstructured":"Anderson, C. (2018, October 20). The End of Theory: The Data Deluge Makes the Scientific Method Obsolete | WIRED. Available online: https:\/\/www.wired.com\/2008\/06\/pb-theory\/."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s11071-005-2823-y","article-title":"Low-dimensional modelling of turbulence using the proper orthogonal decomposition: A tutorial","volume":"41","author":"Smith","year":"2005","journal-title":"Nonlinear Dyn."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/10\/807\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:25:30Z","timestamp":1760196330000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/10\/807"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,22]]},"references-count":114,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["e20100807"],"URL":"https:\/\/doi.org\/10.3390\/e20100807","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,22]]}}}