{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:45:27Z","timestamp":1760233527324,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T00:00:00Z","timestamp":1611100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["1898\/17"],"award-info":[{"award-number":["1898\/17"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["305167\/2010-3"],"award-info":[{"award-number":["305167\/2010-3"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","award":["Finance Code 001"],"award-info":[{"award-number":["Finance Code 001"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann\u2013Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker\u2013Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker\u2013Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.<\/jats:p>","DOI":"10.3390\/e23020131","type":"journal-article","created":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T12:16:18Z","timestamp":1611144978000},"page":"131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Non-Normalizable Quasi-Equilibrium Solution of the Fokker\u2013Planck Equation for Nonconfining Fields"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9130-2933","authenticated-orcid":false,"given":"Celia","family":"Anteneodo","sequence":"first","affiliation":[{"name":"Department of Physics, Pontif\u00edcia Universidade Cat\u00f3lica do Rio de Janeiro (PUC-Rio), Rio de Janeiro 22541-900, Brazil"},{"name":"Institute of Science and Technology for Complex Systems, Rio de Janeiro 22290-180, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5860-2205","authenticated-orcid":false,"given":"Lucianno","family":"Defaveri","sequence":"additional","affiliation":[{"name":"Department of Physics, Pontif\u00edcia Universidade Cat\u00f3lica do Rio de Janeiro (PUC-Rio), Rio de Janeiro 22541-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9518-6949","authenticated-orcid":false,"given":"Eli","family":"Barkai","sequence":"additional","affiliation":[{"name":"Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5279-1655","authenticated-orcid":false,"given":"David A.","family":"Kessler","sequence":"additional","affiliation":[{"name":"Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"043088","DOI":"10.1103\/PhysRevResearch.2.043088","article-title":"Regularized Boltzmann-Gibbs statistics for a Brownian particle in a nonconfining field","volume":"2","author":"Defaveri","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1007\/BF01327311","article-title":"Uber die Wahrscheinlichkeit der Quantenzustande","volume":"26","author":"Fermi","year":"1924","journal-title":"Z. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"122169","DOI":"10.1016\/j.physa.2019.122169","article-title":"Resolving the partition function\u2019s paradox of the hydrogen atom","volume":"534","author":"Plastino","year":"2019","journal-title":"Physica A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"200601","DOI":"10.1103\/PhysRevLett.125.200601","article-title":"Freezing Transition in the Barrier Crossing Rate of a Diffusing Particle","volume":"125","author":"Sabhapandit","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1524","DOI":"10.1007\/s10955-011-0363-z","article-title":"Solution of the Fokker\u2013Planck Equation with a Logarithmic Potential","volume":"145","author":"Dechant","year":"2011","journal-title":"J. Stat. Phys."},{"key":"ref_6","unstructured":"van Kampen, N.G. (1981). Stochastic Processes in Physics and Chemistry, North-Holland Personal Library."},{"key":"ref_7","unstructured":"Risken, H. (1989). The Fokker\u2013Planck Equation, Springer."},{"key":"ref_8","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (2007). Numerical Recipes: The Art of Scientific Computing, Third Edition (C++), Cambridge University Press."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Redner, S. (2001). A Guide to First-Passage Processes, Cambridge University Press.","DOI":"10.1017\/CBO9780511606014"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1103\/RevModPhys.62.251","article-title":"Reaction-rate theory: Fifty years after Kramers","volume":"62","author":"Talkner","year":"1990","journal-title":"Rev. Mod. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1515\/zpch-1889-0416","article-title":"\u00dcber die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch S\u00e4uren","volume":"4","author":"Arrhenius","year":"1889","journal-title":"Z. Phys. Chem. (Leipzig)"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"010601","DOI":"10.1103\/PhysRevLett.122.010601","article-title":"From NonNormalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory","volume":"122","author":"Aghion","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109890","DOI":"10.1016\/j.chaos.2020.109890","article-title":"Infinite ergodic theory meets Boltzmann statistics","volume":"138","author":"Aghion","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1103\/PhysRevE.58.107","article-title":"Front propagation: Precursos, cutoffs, and structural stabilitty","volume":"58","author":"Kessler","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2597","DOI":"10.1103\/PhysRevE.56.2597","article-title":"Shift in the velocity of a front due to a cutoff","volume":"56","author":"Brunet","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1063\/1.1750822","article-title":"Molecular distribution","volume":"9","author":"Mayer","year":"1941","journal-title":"J. Chem. Phys."},{"key":"ref_17","unstructured":"(Mathematica, 2016). Mathematica, Version 11."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory, American Mathematical Society.","DOI":"10.1090\/surv\/050"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"032915","DOI":"10.1103\/PhysRevE.87.032915","article-title":"Aging generates regular motions in weakly chaotic systems","volume":"87","author":"Akimoto","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"110601","DOI":"10.1103\/PhysRevLett.112.110601","article-title":"Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem","volume":"112","author":"Rebenshtok","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","first-page":"041055","article-title":"Alkaline-Earth Atoms in Optical Tweezers","volume":"8","author":"Cooper","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"17721","DOI":"10.1038\/srep17721","article-title":"Superdiffusive motion of membrane-targeting C2 domains","volume":"5","author":"Campagnola","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"10628","DOI":"10.1039\/D0SM00712A","article-title":"Characterising the diffusion of biological nanoparticles on fluid and cross-linked membranes","volume":"16","author":"Debets","year":"2020","journal-title":"Soft Matter"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"268001","DOI":"10.1103\/PhysRevLett.119.268001","article-title":"Three-Dimensional Tracking of Interfacial Hopping Diffusion","volume":"119","author":"Wang","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:13:11Z","timestamp":1760159591000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["e23020131"],"URL":"https:\/\/doi.org\/10.3390\/e23020131","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,1,20]]}}}