{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T11:06:35Z","timestamp":1774263995720,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2012,10,26]],"date-time":"2012-10-26T00:00:00Z","timestamp":1351209600000},"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>We revisit the Frieden\u2013Hawkins\u2019 Fisher order measure with a consideration of temperature effects. To this end, we appeal to the semiclassical approach. The order-measure\u2019s appropriateness is validated in the semiclassical realm with regard to two physical systems. Insight is thereby gained with respect to the relationships amongst semiclassical quantifiers. In particular, it is seen that Wehrl\u2019s entropy is as good a disorder indicator as the Frieden\u2013Hawkins\u2019 one.<\/jats:p>","DOI":"10.3390\/e14112081","type":"journal-article","created":{"date-parts":[[2012,10,26]],"date-time":"2012-10-26T11:07:39Z","timestamp":1351249659000},"page":"2081-2099","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Temperature Effects, Frieden\u2013Hawkins\u2019 Order-Measure, and Wehrl Entropy"],"prefix":"10.3390","volume":"14","author":[{"given":"Flavia","family":"Pennini","sequence":"first","affiliation":[{"name":"La Plata Physics Institute, National University and CCT-CONICET, C. C. 727, La Plata 1900, Argentina"},{"name":"Physics Department, North Catholic University, Av. Angamos 0610, Antofagasta 1240000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angelo","family":"Plastino","sequence":"additional","affiliation":[{"name":"La Plata Physics Institute, National University and CCT-CONICET, C. C. 727, La Plata 1900, Argentina"},{"name":"Physics Department and IFISC-CSIC, University of Balearic Islands, Palma de Mallorca 07122, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gustavo L.","family":"Ferri","sequence":"additional","affiliation":[{"name":"Exact and Natural Sciences Faculty, La Pampa National University, Uruguay 151, Santa Rosa 6300, La Pampa, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Frieden, B.R., and Hawkins, R.J. (2010). Quantifying system order for full and partial coarse graining. Phys. Rev. E, 82.","DOI":"10.1103\/PhysRevE.82.066117"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dimassi, M., and Sjoestrand, J. (1999). Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press.","DOI":"10.1017\/CBO9780511662195"},{"key":"ref_3","unstructured":"Brack, M., and Bhaduri, R.K. (1997). Semiclassical Physics, Addison-Wesley."},{"key":"ref_4","unstructured":"Arnheim, R. (1971). Entropy and Art: An Essay in Disorder and Order, University of California Press."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Darrigol, O. (1992). From c-Numbers to q-Numbers, University of California Press.","DOI":"10.1525\/9780520328280"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rosenkrantz, R.D. (1983). E.T. Jaynes: Papers on Probability, Statistics and Statistical Physics, D. Reidel.","DOI":"10.1007\/978-94-009-6581-2"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"M. Gell-Mann, M., and Tsallis, C. (2004). Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press.","DOI":"10.1093\/oso\/9780195159769.001.0001"},{"key":"ref_8","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, Springer."},{"key":"ref_9","unstructured":"Schr\u00f6dinger, E. (1967). What Is Life? The Physical Aspect of the Living Cell and Mind and Matter, Cambridge University Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2300","DOI":"10.1016\/j.physa.2011.11.061","article-title":"Fisher order measure and Petri\u2019s universe","volume":"391","author":"Frieden","year":"2012","journal-title":"Phys. A"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Frieden, B.R., and Gatenby, R.A. (2011). Order in a multidimensional system. Phys. Rev. E, 84.","DOI":"10.1103\/PhysRevE.84.011128"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.physa.2011.08.005","article-title":"Effect upon universal order of Hubble expansion","volume":"391","author":"Frieden","year":"2011","journal-title":"Phys. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Frieden, B.R. (2004). Science from Fisher Information, Cambridge University Press. [2 ed.].","DOI":"10.1017\/CBO9780511616907"},{"key":"ref_14","first-page":"1","article-title":"Three approaches to the quantitative definition of information","volume":"1","author":"Kolmogorov","year":"1965","journal-title":"Problems Inform. Transmission"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1145\/321526.321530","article-title":"On the simplicity and speed of programs for computing infinite sets of natural numbers","volume":"16","author":"Chaitin","year":"1969","journal-title":"J. ACM"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1103\/PhysRevLett.78.985","article-title":"Bose-Einstein condensation of lithium: Observation of limited condensate number","volume":"78","author":"Bradley","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1126\/science.269.5221.198","article-title":"Observation of Bose-Einstein condensation in a dilute atomic vapor","volume":"269","author":"Anderson","year":"1995","journal-title":"Science,"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3969","DOI":"10.1103\/PhysRevLett.75.3969","article-title":"Bose-Einstein condensation in a gas of sodium atoms","volume":"75","author":"Davis","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the quantum correction for thermodynamic equilibrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02055756","article-title":"Quantenmechanik und gruppentheorie","volume":"46","author":"Weyl","year":"1927","journal-title":"Z. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1111\/j.2517-6161.1949.tb00030.x","article-title":"Stochastic processes and statistical physics","volume":"11","author":"Moyal","year":"1949","journal-title":"J. Roy. Stat. Soc. B"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zachos, C., Fairlie, D., and Curtright, T. (2005). Quantum Mechanics in Phase Space, World Scientific.","DOI":"10.1142\/5287"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0370-1573(95)00007-4","article-title":"Theory and application of the quantum phase-space distribution functions","volume":"259","author":"Lee","year":"1995","journal-title":"Phys. Rep."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1023\/A:1025439010479","article-title":"Entropy and Wigner distribution functions revisited","volume":"42","author":"Wlodarz","year":"2003","journal-title":"Int. J. Theor. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/0003-4916(90)90277-U","article-title":"Semiclassical expansions for confined n fermion systems","volume":"200","author":"Krivine","year":"1990","journal-title":"Ann. Phys. (NY)"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/S0370-1573(97)00070-7","article-title":"The Weyl representation in classical and quantum mechanics","volume":"295","year":"1998","journal-title":"Phys. Rep."},{"key":"ref_27","first-page":"264","article-title":"Some formal properties of the density matrix","volume":"22","author":"Husimi","year":"1940","journal-title":"Proc. Phys. Math. Soc. Jpn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/0378-4371(84)90130-4","article-title":"Quantum mechanics in the Gaussian wave-packet phase space representation","volume":"127","author":"Mizrahi","year":"1984","journal-title":"Phys. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"972","DOI":"10.3390\/e11040972","article-title":"Fisher information and semiclassical treatments","volume":"11","author":"Pennini","year":"2009","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0375-9601(81)90881-1","article-title":"Some properties of a non-negative quantum-mechanical distribution function","volume":"85","author":"Wigner","year":"1981","journal-title":"Phys. Lett. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/0378-4371(86)90115-9","article-title":"Quantum mechanics in the Gaussian wave-packet phase space representation II: Dynamics","volume":"135","author":"Mizrahi","year":"1986","journal-title":"Phys. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/0378-4371(88)90254-3","article-title":"Quantum mechanics in the gaussian wave-packet phase space representation III: From phase space probability functions to wave-functions","volume":"150","author":"Mizrahi","year":"1988","journal-title":"Phys. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2753","DOI":"10.1103\/PhysRevD.48.2753","article-title":"Information-theoretic measure of uncertainty due to quantum and thermal fluctuations","volume":"48","author":"Anderson","year":"1993","journal-title":"Phys. Rev. D"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1038\/35089017","article-title":"Sub-Planck structure in phase space and its relevance for quantum decoherence","volume":"412","author":"Zurek","year":"2001","journal-title":"Nature"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Karkuszewski, Z.P., Jarzynski, C., and Zurek, W.H. (2002). Quantum chaotic environments, the butterfly effect, and decoherence. Phys. Rev. Lett., 89.","DOI":"10.1103\/PhysRevLett.89.170405"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Wisniacki, D.A. (2003). Short-time decay of the Loschmidt echo. Phys. Rev. E, 67.","DOI":"10.1103\/PhysRevE.67.016205"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1140\/epjd\/e2004-00189-8","article-title":"Smoothed Wigner functions: A tool to resolve semiclassical structures","volume":"32","author":"Rivas","year":"2005","journal-title":"Eur. Phys. J. D"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4665","DOI":"10.1103\/PhysRevE.62.4665","article-title":"Entropy and Wigner functions","volume":"62","author":"Manfredi","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1088\/0951-7715\/15\/3\/309","article-title":"Hyperbolic scar patterns in phase space","volume":"15","author":"Rivas","year":"2002","journal-title":"Nonlinearity"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0034-4877(79)90070-3","article-title":"On the relation between classical and quantum entropy","volume":"16","author":"Wehrl","year":"1979","journal-title":"Rep. Math. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2766","DOI":"10.1103\/PhysRev.131.2766","article-title":"Coherent and incoherent states of the radiation field","volume":"131","author":"Glauber","year":"1963","journal-title":"Phys. Rev."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Klauder, J.R., and Skagerstam, B.S. (1985). Coherent States, World Scientific.","DOI":"10.1142\/0096"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1927","DOI":"10.1016\/j.physleta.2010.02.040","article-title":"Thermal effects in quantum phase-space distributions","volume":"374","author":"Pennini","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2218","DOI":"10.1016\/j.physa.2010.01.043","article-title":"Phase space distributions from variation of information measures","volume":"389","author":"Olivares","year":"2010","journal-title":"Phys. A"},{"key":"ref_45","first-page":"624","article-title":"Statistical, noise-related non-classicality\u2019s indicator","volume":"7","author":"Pennini","year":"2009","journal-title":"Central Eur. J. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"32","DOI":"10.3390\/e11010032","article-title":"Information, Deformed, \u03ba-Wehrl entropies and semiclassical delocalization","volume":"11","author":"Pennini","year":"2009","journal-title":"Entropy"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1140\/epjd\/e2010-00233-2","article-title":"Smoothed Wigner distributions, decoherence, and the temperature dependence of the classical-quantical frontier","volume":"61","author":"Pennini","year":"2011","journal-title":"Eur. Phys. J. D."},{"key":"ref_48","unstructured":"Haus, H. (1984). Waves and Fields Optoelectronics, Prentice Hall."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"7647","DOI":"10.1103\/PhysRevA.44.7647","article-title":"Statistical and phase properties of displaced Kerr states","volume":"44","author":"Knight","year":"1991","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1590\/S0103-97332009000400029","article-title":"Note on semiclassical uncertainty relations","volume":"39","author":"Olivares","year":"2009","journal-title":"Braz. J. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/11\/2081\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:53:08Z","timestamp":1760219588000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/11\/2081"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,26]]},"references-count":50,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2012,11]]}},"alternative-id":["e14112081"],"URL":"https:\/\/doi.org\/10.3390\/e14112081","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,26]]}}}