{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T02:57:34Z","timestamp":1777431454891,"version":"3.51.4"},"reference-count":39,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T00:00:00Z","timestamp":1688083200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11735005"],"award-info":[{"award-number":["11735005"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.<\/jats:p>","DOI":"10.3390\/e25071012","type":"journal-article","created":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T16:06:37Z","timestamp":1688141197000},"page":"1012","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Ergodic Measure and Potential Control of Anomalous Diffusion"],"prefix":"10.3390","volume":"25","author":[{"given":"Bao","family":"Wen","sequence":"first","affiliation":[{"name":"Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Public Policy and Management, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4657-0960","authenticated-orcid":false,"given":"Ming-Gen","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Physics, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Physics, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing-Dong","family":"Bao","sequence":"additional","affiliation":[{"name":"Department of Physics, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,30]]},"reference":[{"key":"ref_1","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. 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