{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T16:22:27Z","timestamp":1770481347541,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2008,6,17]],"date-time":"2008-06-17T00:00:00Z","timestamp":1213660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Ecosystems and quantum gases share a number of superficial similarities including enormous numbers of interacting elements and the fundamental role of energy in such interactions. A theory for the synthesis of data and prediction of new phenomena is well established in quantum statistical mechanics. The premise of this paper is that the reason a comparable unifying theory has not emerged in ecology is that a proper role for entropy has yet to be assigned. To this end, a phase space entropy model of ecosystems is developed. Specification of an ecosystem phase space cell size based on microbial mass, length, and time scales gives an ecosystem uncertainty parameter only about three orders of magnitude larger than Planck\u2019s constant. Ecosystem equilibria is specified by conservation of biomass and total metabolic energy, along with the principle of maximum entropy at equilibria. Both Bose - Einstein and Fermi - Dirac equilibrium conditions arise in ecosystems applications. The paper concludes with a discussion of some broader aspects of an ecosystem phase space.<\/jats:p>","DOI":"10.3390\/entropy-e10020058","type":"journal-article","created":{"date-parts":[[2008,6,17]],"date-time":"2008-06-17T07:55:42Z","timestamp":1213689342000},"page":"58-70","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Quantum and Ecosystem Entropies"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2103-2146","authenticated-orcid":false,"suffix":"Jr.","given":"A. 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Macroecology, University of Chicago Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1890\/03-9000","article-title":"Toward a metabolic theory of ecology","volume":"85","author":"Brown","year":"2004","journal-title":"Ecology"},{"key":"ref_6","unstructured":"Brown, J.H., West, G.B., and Enquist, B.J. (2005). Scaling in Biology."},{"key":"ref_7","unstructured":"Cyr, H. (2005). Scaling in Biology."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1006\/jtbi.2000.2238","article-title":"Re-examination of the \u201c3\/4-la\u201d of metabolism","volume":"209","author":"Dodds","year":"2001","journal-title":"J. Theor. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Doetsch, R.N., and Cook, T.M. (1973). Introduction to bacteria and their ecobiology, University Park Press.","DOI":"10.1007\/978-94-015-1135-3"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/0022-5193(83)90427-7","article-title":"Biological application of the statistical concepts used in the Second Law","volume":"105","author":"Elsasser","year":"1983","journal-title":"J. Theor. Biol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1038\/nature01671","article-title":"Scaling metabalism from organisms to ecosystems","volume":"423","author":"Enquist","year":"2003","journal-title":"Nature"},{"key":"ref_12","unstructured":"Enquist, B.J., West, G.B., and Brown, J.H. (2000). Scaling in Biology."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ginzburg, L., and Golyvan, M. (2004). Ecological orbits, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780195168167.001.0001"},{"key":"ref_14","unstructured":"Ginzburg, L. (1998). Material Effects as Adaptations, Oxford University Press."},{"key":"ref_15","first-page":"9","article-title":"Can we scale the Planck scale?","volume":"6","author":"Gross","year":"1989","journal-title":"Phys. Today"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.jtbi.2003.12.007","article-title":"Testing the allometric scaling laws","volume":"228","author":"Kaitaniemi","year":"2004","journal-title":"J. Theor. Biol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF02477883","article-title":"A statistical mechanics of interacting biological species","volume":"19","author":"Kerner","year":"1957","journal-title":"Bull. Math. Biophys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/BF02476361","article-title":"Further considerations on the statistical mechanics of biological associations","volume":"21","author":"Kerner","year":"1959","journal-title":"Bull. Math. Biophys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1016\/j.ijengsci.2003.09.005","article-title":"Intrinsic photon entropy? The dark side of light","volume":"42","author":"Kirwan","year":"2004","journal-title":"Int. J. Eng. Sci."},{"key":"ref_20","first-page":"1","article-title":"A phytoplankton community in a temperate reservoir in New South Wales, Australia: relationships between similarity and diversity indices and measures of hydrological disturbance","volume":"56","author":"Kobayashi","year":"2005","journal-title":"Mar. Fish. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0375-9601(80)90072-9","article-title":"Entropies need not be concave","volume":"78","author":"Landsberg","year":"1980","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/BF01017372","article-title":"Is equilibrium always an entropy maximum?","volume":"35","author":"Landsberg","year":"1984a","journal-title":"J. Stat. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/0375-9601(84)90934-4","article-title":"Can entropy and \u201corder\u201d increase together?","volume":"102","author":"Landsberg","year":"1984b","journal-title":"Phys. Lett. A"},{"key":"ref_24","unstructured":"Lenski, R.E., and Mongold, J.R. (2000). Scaling in Biology."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1126\/science.1114488","article-title":"The illusion of invariant quantities in life histories","volume":"309","author":"Nee","year":"2005","journal-title":"Science"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0304-3800(99)00185-4","article-title":"From theoretical ecology to statistical physics and back: self-similar landscape metrics as a synthesis of ecological diversity and geometrical complexity","volume":"125","author":"Ricotta","year":"2000","journal-title":"Ecol. Modell."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1006\/jtbi.1998.0799","article-title":"Darwinian daisyworld","volume":"195","author":"Robertson","year":"1998","journal-title":"J. Theor. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Schmidt-Nielsen, K. (1984). Scaling:  why is animal size so important?, Cambridge University Press.","DOI":"10.1017\/CBO9781139167826"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1575","DOI":"10.1242\/jeb.01589","article-title":"The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization","volume":"208","author":"West","year":"2005","journal-title":"J. Exp. Biol."},{"key":"ref_30","unstructured":"West, G.B., Brown, J.H., and Enquist, B.J. (2000). Scaling in Biology."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1126\/science.276.5309.122","article-title":"A general model for the origin of allometric scaling laws in biology","volume":"276","author":"West","year":"1997","journal-title":"Science"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1073\/pnas.012579799","article-title":"Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals","volume":"99","author":"West","year":"2002","journal-title":"Proc. Nat. Acad. Sci. (USA)"},{"key":"ref_33","first-page":"12","article-title":"Scaling Mount Planck: A view from the bottom","volume":"6","author":"Wilczek","year":"2001","journal-title":"Phys. Today"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/2\/58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:20:50Z","timestamp":1760221250000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/2\/58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,6,17]]},"references-count":33,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2008,6]]}},"alternative-id":["e10020058"],"URL":"https:\/\/doi.org\/10.3390\/entropy-e10020058","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,6,17]]}}}