{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:45:08Z","timestamp":1740109508687,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T00:00:00Z","timestamp":1594684800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T00:00:00Z","timestamp":1594684800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["636144"],"award-info":[{"award-number":["636144"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2020,12]]},"DOI":"10.1007\/s00332-020-09644-7","type":"journal-article","created":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T19:03:20Z","timestamp":1594753400000},"page":"3057-3087","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Turbulent Energy Spectrum via an Interaction Potential"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8309-1432","authenticated-orcid":false,"given":"Rafail V.","family":"Abramov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,14]]},"reference":[{"issue":"2","key":"9644_CR1","first-page":"162","volume":"2","author":"R Abramov","year":"2019","unstructured":"Abramov, R.: The random gas of hard spheres. Multidiscip. Sci. J. 2(2), 162\u2013205 (2019)","journal-title":"Multidiscip. Sci. J."},{"issue":"8","key":"9644_CR2","first-page":"691","volume":"16","author":"N Bogoliubov","year":"1946","unstructured":"Bogoliubov, N.: Kinetic equations. J. Exp. Theor. Phys. 16(8), 691\u2013702 (1946)","journal-title":"J. Exp. Theor. Phys."},{"key":"9644_CR3","first-page":"275","volume":"66","author":"L Boltzmann","year":"1872","unstructured":"Boltzmann, L.: Weitere Studien \u00fcber das W\u00e4rmegleichgewicht unter Gasmolek\u00fclen. Sitz-Ber Kais Akad Wiss (II) 66, 275\u2013370 (1872)","journal-title":"Sitz-Ber Kais Akad Wiss (II)"},{"key":"9644_CR4","first-page":"10","volume":"188","author":"M Born","year":"1946","unstructured":"Born, M., Green, H.: A general kinetic theory of liquids I: the molecular distribution functions. Proc. R. Soc. A 188, 10\u201318 (1946)","journal-title":"Proc. R. Soc. A"},{"issue":"1","key":"9644_CR5","first-page":"1","volume":"I","author":"J Boussinesq","year":"1877","unstructured":"Boussinesq, J.: Essai sur la th\u00e9orie des eaux courantes. M\u00e9moires pr\u00e9sent\u00e9s par divers savants \u00e0 l\u2019Acad\u00e9mie des Sciences XXII I(1), 1\u2013680 (1877)","journal-title":"M\u00e9moires pr\u00e9sent\u00e9s par divers savants \u00e0 l\u2019Acad\u00e9mie des Sciences XXII"},{"issue":"8","key":"9644_CR6","doi-asserted-by":"publisher","first-page":"085109","DOI":"10.1063\/1.4999102","volume":"29","author":"P Buchhave","year":"2017","unstructured":"Buchhave, P., Velte, C.: Measurement of turbulent spatial structure and kinetic energy spectrum by exact temporal-to-spatial mapping. Phys. Fluids 29(8), 085109 (2017)","journal-title":"Phys. Fluids"},{"key":"9644_CR7","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1098\/rspa.1938.0173","volume":"168","author":"R Buckingham","year":"1938","unstructured":"Buckingham, R.: The classical equation of state of gaseous helium, neon and argon. Proc. R. Soc. Lond. A 168, 264\u2013283 (1938)","journal-title":"Proc. R. Soc. Lond. A"},{"key":"9644_CR8","volume-title":"Theory and Application of the Boltzmann Equation","author":"C Cercignani","year":"1975","unstructured":"Cercignani, C.: Theory and Application of the Boltzmann Equation. Elsevier Science, New York (1975)"},{"key":"9644_CR9","volume-title":"Applied Mathematical Sciences","author":"C Cercignani","year":"1994","unstructured":"Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, vol. 106. Springer, Berlin (1994)"},{"key":"9644_CR10","first-page":"20","volume":"200","author":"S Chandrasekhar","year":"1949","unstructured":"Chandrasekhar, S.: On Heisenberg\u2019s elementary theory of turbulence. Proc. R. Soc. 200, 20\u201333 (1949)","journal-title":"Proc. R. Soc."},{"key":"9644_CR11","series-title":"Cambridge Mathematical Library","volume-title":"The Mathematical Theory of Non-Uniform Gases","author":"S Chapman","year":"1991","unstructured":"Chapman, S., Cowling, T.: The Mathematical Theory of Non-Uniform Gases. Cambridge Mathematical Library, 3rd edn. Cambridge University Press, Cambridge (1991)","edition":"3"},{"issue":"4","key":"9644_CR12","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1063\/1.1699986","volume":"22","author":"S Corrsin","year":"1951","unstructured":"Corrsin, S.: On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22(4), 469\u2013473 (1951)","journal-title":"J. Appl. Phys."},{"key":"9644_CR13","first-page":"602","volume":"6","author":"E Fermi","year":"1927","unstructured":"Fermi, E.: A statistical method for determining some properties of the atom. Rend. Accad. Naz. Lincei. 6, 602\u2013607 (1927)","journal-title":"Rend. Accad. Naz. Lincei."},{"key":"9644_CR14","doi-asserted-by":"publisher","DOI":"10.4171\/129","volume-title":"From Newton to Boltzmann: Hard Spheres and Short-Range Potentials","author":"I Gallagher","year":"2014","unstructured":"Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. European Mathematical Society, Z\u00fcrich (2014)"},{"key":"9644_CR15","volume-title":"Introduction to the Theory of Random Processes","author":"I Gikhman","year":"1969","unstructured":"Gikhman, I., Skorokhod, A.: Introduction to the Theory of Random Processes. Courier Dover Publications, Mineola (1969)"},{"key":"9644_CR16","doi-asserted-by":"crossref","unstructured":"Gluskin, E., Miller, S.: On the recovery of the time average of continuous and discrete time functions from their Laplace and $$z$$-transforms (2012). arXiv:1109.3356","DOI":"10.1002\/cta.1877"},{"key":"9644_CR17","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/S1874-5717(06)80006-X","volume-title":"Handbook of Differential Equations: Evolutionary Equations","author":"F Golse","year":"2005","unstructured":"Golse, F.: The Boltzmann equation and its hydrodynamic limits. Handbook of Differential Equations: Evolutionary Equations, vol. 2, pp. 159\u2013301. Elsevier, Amsterdam (2005)"},{"issue":"4","key":"9644_CR18","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1002\/cpa.3160020403","volume":"2","author":"H Grad","year":"1949","unstructured":"Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331\u2013407 (1949)","journal-title":"Commun. Pure Appl. Math."},{"key":"9644_CR19","volume-title":"The Molecular Theory of Gases and Liquids","author":"J Hirschfelder","year":"1964","unstructured":"Hirschfelder, J., Curtiss, C., Bird, R.: The Molecular Theory of Gases and Liquids. Wiley, Hoboken (1964)"},{"key":"9644_CR20","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1063\/1.1724117","volume":"14","author":"J Kirkwood","year":"1946","unstructured":"Kirkwood, J.: The statistical mechanical theory of transport processes I: general theory. J. Chem. Phys. 14, 180\u2013201 (1946)","journal-title":"J. Chem. Phys."},{"key":"9644_CR21","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1143\/JPSJ.10.759","volume":"10","author":"S Kobayashi","year":"1955","unstructured":"Kobayashi, S., Matsukuma, T., Nagai, S., Umeda, K.: Accurate value of the initial slope of the ordinary TF function. J. Phys. Soc. Jpn. 10, 759\u2013762 (1955)","journal-title":"J. Phys. Soc. Jpn."},{"key":"9644_CR22","first-page":"538","volume":"31","author":"A Kolmogorov","year":"1941","unstructured":"Kolmogorov, A.: Decay of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk SSSR 31, 538\u2013541 (1941a)","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"9644_CR23","first-page":"19","volume":"32","author":"A Kolmogorov","year":"1941","unstructured":"Kolmogorov, A.: Energy dissipation in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 19\u201321 (1941b)","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"9644_CR24","first-page":"299","volume":"30","author":"A Kolmogorov","year":"1941","unstructured":"Kolmogorov, A.: Local structure of turbulence in an incompressible fluid at very high Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299\u2013303 (1941c)","journal-title":"Dokl. Akad. Nauk SSSR"},{"issue":"738","key":"9644_CR25","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1098\/rspa.1924.0082","volume":"106","author":"J Lennard-Jones","year":"1924","unstructured":"Lennard-Jones, J.: On the determination of molecular fields.\u2014 II. From the equation of state of a gas. Proc. R. Soc. Lond. A 106(738), 463\u2013477 (1924)","journal-title":"Proc. R. Soc. Lond. A"},{"key":"9644_CR26","doi-asserted-by":"publisher","first-page":"1021","DOI":"10.1007\/BF02179552","volume":"83","author":"C Levermore","year":"1996","unstructured":"Levermore, C.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021\u20131065 (1996)","journal-title":"J. Stat. Phys."},{"issue":"9","key":"9644_CR27","doi-asserted-by":"publisher","first-page":"950","DOI":"10.1175\/1520-0469(1985)042<0950:ACOAWS>2.0.CO;2","volume":"42","author":"G Nastrom","year":"1985","unstructured":"Nastrom, G., Gage, K.: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42(9), 950\u2013960 (1985)","journal-title":"J. Atmos. Sci."},{"key":"9644_CR28","first-page":"453","volume":"5","author":"A Obukhov","year":"1941","unstructured":"Obukhov, A.: On the distribution of energy in the spectrum of a turbulent flow. Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz. 5, 453\u2013466 (1941)","journal-title":"Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz."},{"key":"9644_CR29","first-page":"58","volume":"13","author":"A Obukhov","year":"1949","unstructured":"Obukhov, A.: Structure of the temperature field in turbulent flow. Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz. 13, 58\u201369 (1949)","journal-title":"Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz."},{"issue":"8","key":"9644_CR30","doi-asserted-by":"publisher","first-page":"3011","DOI":"10.1029\/JZ067i008p03011","volume":"67","author":"A Obukhov","year":"1962","unstructured":"Obukhov, A.: Some specific features of atmospheric turbulence. J. Geophys. Res. 67(8), 3011\u20133014 (1962)","journal-title":"J. Geophys. Res."},{"key":"9644_CR31","series-title":"Universitext","volume-title":"Stochastic Differential Equations: An Introduction with Applications","author":"B \u00d8ksendal","year":"2010","unstructured":"\u00d8ksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Universitext, 6th edn. Springer, Berlin (2010)","edition":"6"},{"issue":"224\u2013226","key":"9644_CR32","first-page":"84","volume":"35","author":"O Reynolds","year":"1883","unstructured":"Reynolds, O.: An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Proc. R. Soc. Lond. 35(224\u2013226), 84\u201399 (1883)","journal-title":"Proc. R. Soc. Lond."},{"key":"9644_CR33","first-page":"123","volume":"186","author":"O Reynolds","year":"1895","unstructured":"Reynolds, O.: On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos. Trans. R. Soc. A 186, 123\u2013164 (1895)","journal-title":"Philos. Trans. R. Soc. A"},{"key":"9644_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61544-3","volume-title":"The Fokker\u2013Planck Equation","author":"H Risken","year":"1989","unstructured":"Risken, H.: The Fokker\u2013Planck Equation, 2nd edn. Springer, New York (1989)","edition":"2"},{"key":"9644_CR35","first-page":"1","volume-title":"Handbook of Geophysics and the Space Environment","author":"R Sagalyn","year":"1985","unstructured":"Sagalyn, R., Burke, H., Fitzgerald, D.: Atmospheric electricity. In: Jursa, A. (ed.) Handbook of Geophysics and the Space Environment, pp. 1\u201337. National Technical Information Service, Springfield (1985)"},{"key":"9644_CR36","first-page":"788","volume":"15","author":"A Sommerfeld","year":"1932","unstructured":"Sommerfeld, A.: Asymptotic integration of the Thomas\u2013Fermi differential equation. Rend. Accad. Naz. Lincei. 15, 788\u2013792 (1932)","journal-title":"Rend. Accad. Naz. Lincei."},{"issue":"5","key":"9644_CR37","doi-asserted-by":"publisher","first-page":"542","DOI":"10.1017\/S0305004100011683","volume":"23","author":"L Thomas","year":"1927","unstructured":"Thomas, L.: The calculation of atomic fields. Proc. Camb. Philos. Soc. 23(5), 542\u2013548 (1927)","journal-title":"Proc. Camb. Philos. Soc."},{"key":"9644_CR38","volume-title":"The Stopping and Range of Ions in Solids, Stopping and Ranges of Ions in Matter","author":"J Ziegler","year":"1985","unstructured":"Ziegler, J., Biersack, J., Littmark, U.: The Stopping and Range of Ions in Solids, Stopping and Ranges of Ions in Matter, vol. 1. Pergamon, Oxford (1985)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09644-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-020-09644-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09644-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T23:32:45Z","timestamp":1626219165000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-020-09644-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,14]]},"references-count":38,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["9644"],"URL":"https:\/\/doi.org\/10.1007\/s00332-020-09644-7","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2020,7,14]]},"assertion":[{"value":"6 February 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 July 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with Ethical Standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}