{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T23:54:32Z","timestamp":1773878072319,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:00:00Z","timestamp":1611187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CHE-1265929"],"award-info":[{"award-number":["CHE-1265929"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CHE- 1738990"],"award-info":[{"award-number":["CHE- 1738990"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1427654"],"award-info":[{"award-number":["PHY-1427654"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000928","name":"Welch Foundation","doi-asserted-by":"publisher","award":["No. C-1570"],"award-info":[{"award-number":["No. C-1570"]}],"id":[{"id":"10.13039\/100000928","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1114, Project A01"],"award-info":[{"award-number":["SFB 1114, Project A01"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006188","name":"Einstein Stiftung Berlin","doi-asserted-by":"publisher","award":["Visiting Professorship C.C."],"award-info":[{"award-number":["Visiting Professorship C.C."]}],"id":[{"id":"10.13039\/501100006188","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007863","name":"Rice University","doi-asserted-by":"publisher","award":["Postdoctoral Fellowship F.N."],"award-info":[{"award-number":["Postdoctoral Fellowship F.N."]}],"id":[{"id":"10.13039\/100007863","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers\u2013Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.<\/jats:p>","DOI":"10.3390\/e23020134","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T11:49:40Z","timestamp":1611229780000},"page":"134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Spectral Properties of Effective Dynamics from Conditional Expectations"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2444-7889","authenticated-orcid":false,"given":"Feliks","family":"N\u00fcske","sequence":"first","affiliation":[{"name":"Center for Theoretical Biological Physics and Department of Chemistry, Rice University, Houston, TX 77005, USA"},{"name":"Institute of Mathematics, Universit\u00e4t Paderborn, 33098 Paderborn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8127-6804","authenticated-orcid":false,"given":"P\u00e9ter","family":"Koltai","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lorenzo","family":"Boninsegna","sequence":"additional","affiliation":[{"name":"Center for Theoretical Biological Physics and Department of Chemistry, Rice University, Houston, TX 77005, USA"},{"name":"Institute for Quantitative and Computational Biosciences and Department of Microbiology, Immunology and Molecular Genetics, University of California Los Angeles, Los Angeles, CA 90095, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cecilia","family":"Clementi","sequence":"additional","affiliation":[{"name":"Center for Theoretical Biological Physics and Department of Chemistry, Rice University, Houston, TX 77005, USA"},{"name":"Department of Physics, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1143\/PTP.33.423","article-title":"Transport, collective motion, and Brownian motion","volume":"33","author":"Mori","year":"1965","journal-title":"Prog. 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