{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:33:34Z","timestamp":1760243614058,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2012,5,7]],"date-time":"2012-05-07T00:00:00Z","timestamp":1336348800000},"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>Diffraction methods are used to detect atomic order in solids. While uniquely ergodic systems with pure point diffraction have zero entropy, the relation between diffraction and entropy is not as straightforward in general. In particular, there exist families of homometric systems, which are systems sharing the same diffraction, with varying entropy. We summarise the present state of understanding by several characteristic examples.<\/jats:p>","DOI":"10.3390\/e14050856","type":"journal-article","created":{"date-parts":[[2012,5,7]],"date-time":"2012-05-07T11:16:02Z","timestamp":1336389362000},"page":"856-864","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Comment on the Relation between Diffraction and Entropy"],"prefix":"10.3390","volume":"14","author":[{"given":"Michael","family":"Baake","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, 33501 Bielefeld, Germany"}]},{"given":"Uwe","family":"Grimm","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK"}]}],"member":"1968","published-online":{"date-parts":[[2012,5,7]]},"reference":[{"key":"ref_1","unstructured":"Cowley, J.M. (1995). Diffraction Physics, North-Holland. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.1103\/PhysRevLett.53.1951","article-title":"Metallic phase with long-range orientational order and no translational symmetry","volume":"53","author":"Shechtman","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cornfeld, I.P., Fomin, S.V., and Sinai, Ya.G. (1982). Ergodic Theory, Springer.","DOI":"10.1007\/978-1-4615-6927-5"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s11005-007-0186-7","article-title":"Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies","volume":"82","author":"Baake","year":"2007","journal-title":"Lett. Math. Phys."},{"key":"ref_5","first-page":"441","article-title":"Surprises in diffuse scattering","volume":"215","author":"Baake","year":"2000","journal-title":"Z. Krist."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1524\/zkri.2005.220.12_2005.1027","article-title":"Disorder, structured diffuse scattering and the transmission electron microscope","volume":"220","author":"Withers","year":"2005","journal-title":"Z. Krist."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF02101595","article-title":"On diffraction by aperiodic structures","volume":"169","author":"Hof","year":"1995","journal-title":"Commun. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1088\/0305-4470\/28\/1\/012","article-title":"Diffraction by aperiodic structures at high temperatures","volume":"28","author":"Hof","year":"1995","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_9","first-page":"61","article-title":"Weighted Dirac combs with pure point diffraction","volume":"573","author":"Baake","year":"2004","journal-title":"J. Reine Angew. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1524\/zkri.2011.1389","article-title":"Kinematic diffraction from a mathematical viewpoint","volume":"226","author":"Baake","year":"2011","journal-title":"Z. Krist."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Berg, C., and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups, Springer.","DOI":"10.1007\/978-3-642-66128-0"},{"key":"ref_12","unstructured":"Baake, M., and Grimm, U. Theory of Aperiodic Order: A Mathematical Invitation, Cambridge University Press. in preparation."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"422001","DOI":"10.1088\/1751-8113\/41\/42\/422001","article-title":"The singular continuous diffraction measure of the Thue-Morse chain","volume":"41","author":"Baake","year":"2008","journal-title":"J. Phys. A.: Math. Theor."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"032701","DOI":"10.1063\/1.3688337","article-title":"Spectral and topological properties of a family of generalised Thue-Morse sequences","volume":"53","author":"Baake","year":"2012","journal-title":"J. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9023","DOI":"10.1088\/0305-4470\/31\/45\/003","article-title":"Diffractive point sets with entropy","volume":"31","author":"Baake","year":"1998","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1007\/s00220-009-0942-x","article-title":"Diffraction of stochastic point sets: Explicitly computable examples","volume":"293","author":"Baake","year":"2010","journal-title":"Commun. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"020203(R)","DOI":"10.1103\/PhysRevB.79.020203","article-title":"Kinematic diffraction is insufficient to distinguish order from disorder","volume":"79","author":"Baake","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Queff\u00e9lec, M. (2010). Substitution Dynamical Systems\u2014Spectral Analysis, Springer. [2nd ed.]. LNM 1294.","DOI":"10.1007\/978-3-642-11212-6"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Pytheas Fogg, N. (2002). Substitutions in Dynamics, Arithmetics and Combinatorics, Springer. LNM 1794.","DOI":"10.1007\/b13861"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s10955-011-0163-5","article-title":"Close-packed dimers on the line: Diffraction versus dynamical spectrum","volume":"143","author":"Baake","year":"2011","journal-title":"J. Stat. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1007\/BF01055722","article-title":"How should one define a (weak) crystal?","volume":"66","year":"1992","journal-title":"J. Stat. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Schmidt, K. (1995). Dynamical Systems of Algebraic Origin, Birkh\u00e4user.","DOI":"10.1007\/978-3-0348-0277-2"},{"key":"ref_23","first-page":"A561","article-title":"Un champ markovien peut \u00eatre d\u2019entropie nulle et m\u00e9langeant","volume":"287","author":"Ledrappier","year":"1987","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1007\/s10955-010-9984-x","article-title":"Planar dynamical systems with pure Lebesgue diffraction spectrum","volume":"140","author":"Baake","year":"2010","journal-title":"J. Stat. Phys."},{"key":"ref_25","unstructured":"Meyer, Y. (1972). Algebraic Numbers and Harmonic Analysis, North Holland."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Axel, F., D\u00e9noyer, F., and Gazeau, J.-P. (2000). From Quasicrystals to More Complex Systems, EDP Sciences, Les Ulis, and Springer.","DOI":"10.1007\/978-3-662-04253-3"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/BF02102593","article-title":"Meyer\u2019s concept of quasicrystal and quasiregular sets","volume":"179","author":"Lagarias","year":"1996","journal-title":"Commun. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/s00454-004-1156-9","article-title":"Almost periodic measures and long-range order in Meyer sets","volume":"33","author":"Strungaru","year":"2005","journal-title":"Discr. Comput. Geom."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"483","DOI":"10.4153\/CMB-2002-050-2","article-title":"Diffraction of weighted lattice subsets","volume":"45","author":"Baake","year":"2002","journal-title":"Can. Math. Bulletin"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0012-365X(99)00384-2","article-title":"Diffraction from visible lattice points and k-th power free integers","volume":"221","author":"Baake","year":"2000","journal-title":"Discr. Math."},{"key":"ref_31","unstructured":"Huck, C., and Pleasants, P.A.B. (2011). Entropy and diffraction of the k-free points in n-dimensional lattices. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1017\/S0143385704000318","article-title":"Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra","volume":"24","author":"Baake","year":"2004","journal-title":"Ergod. Theor. Dyn. Syst."},{"key":"ref_33","first-page":"141","article-title":"Diffraction and Palm measure of point processes","volume":"342","year":"2003","journal-title":"C. R. Acad. Sci. Paris"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/5\/856\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:08Z","timestamp":1760219408000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/5\/856"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,7]]},"references-count":33,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2012,5]]}},"alternative-id":["e14050856"],"URL":"https:\/\/doi.org\/10.3390\/e14050856","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2012,5,7]]}}}