{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:51:27Z","timestamp":1760151087214,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T00:00:00Z","timestamp":1643414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology","doi-asserted-by":"publisher","award":["109-2112-M-009-015-MY3"],"award-info":[{"award-number":["109-2112-M-009-015-MY3"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The resonant modes generated from the modern Chladni experiment are systematically confirmed to intimately correspond to the maximum entropy states obtained from the inhomogeneous Helmholtz equation for the square and equilateral triangle plates. To investigate the origin of maximum entropy states, the inhomogeneous Helmholtz equation is modified to consider the point interaction coming from the driving oscillator. The coupling strength associated with the point interaction is characterized by a dimensionless factor \u03b1. The \u03b4 potential of the point interaction is numerically modelled by a truncated basis with an upper index N. The asymptotic behavior for the upper index N is thoroughly explored to verify that the coupling strength of \u03b1 = 1.0 can make the theoretical resonant modes agree excellently with the maximum entropy states as N\u2192\u221e. It is further authenticated that nearly the same resonant modes can be obtained by using a larger coupling strength \u03b1 when a smaller upper index N is exploited in the calculation.<\/jats:p>","DOI":"10.3390\/e24020215","type":"journal-article","created":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T23:02:06Z","timestamp":1643497326000},"page":"215","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Exploring the Origin of Maximum Entropy States Relevant to Resonant Modes in Modern Chladni Plates"],"prefix":"10.3390","volume":"24","author":[{"given":"Yu-Hsin","family":"Shu","sequence":"first","affiliation":[{"name":"Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan"}]},{"given":"Yu-Chen","family":"Tseng","sequence":"additional","affiliation":[{"name":"Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan"}]},{"given":"Yu-Hsiang","family":"Lai","sequence":"additional","affiliation":[{"name":"Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan"}]},{"given":"Yan-Ting","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan"}]},{"given":"Kai-Feng","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1435-5366","authenticated-orcid":false,"given":"Yung-Fu","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,29]]},"reference":[{"key":"ref_1","unstructured":"Chladni, E.F.F. (1787). Entdeckungen \u00fcber die Theorie des Klanges, Breitkopf und H\u00e4rtel."},{"key":"ref_2","unstructured":"Chladni, E.F.F. (1802). Die Akustik, Breitkopf und H\u00e4rtel."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"045005","DOI":"10.1103\/RevModPhys.89.045005","article-title":"Nodal portraits of quantum billiards: Domains, lines, and statistics","volume":"89","author":"Jain","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"026102","DOI":"10.1103\/PhysRevLett.98.026102","article-title":"Chladni figures revisited based on nanomechanics","volume":"98","author":"Dorrestijn","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"076102","DOI":"10.1103\/PhysRevLett.106.076102","article-title":"Nanoscale self-organization using standing surface acoustic waves","volume":"106","author":"Taillan","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1103\/RevModPhys.83.647","article-title":"Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics","volume":"83","author":"Friend","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6675","DOI":"10.1038\/srep06675","article-title":"Musical interfaces: Visualization and reconstruction of music with a microfluidic two-phase flow","volume":"4","author":"Mak","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"045803","DOI":"10.1088\/0143-0807\/37\/4\/045803","article-title":"Patterns beyond Faraday waves: Observation of parametric crossover from Faraday instabilities to the formation of vortex lattices in open dual fluid strata","volume":"37","author":"Ohlin","year":"2016","journal-title":"Eur. J. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"23929","DOI":"10.1038\/srep23929","article-title":"Cymatics for the cloaking of flexural vibrations in a structured plate","volume":"6","author":"Misseroni","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1038\/s41467-018-03154-7","article-title":"Rotating robots move collectively and self-organize","volume":"9","author":"Scholz","year":"2018","journal-title":"Nat. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"198003","DOI":"10.1103\/PhysRevLett.118.198003","article-title":"Velocity distribution of a homogeneously driven two-dimensional granular gas","volume":"118","author":"Scholz","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8686","DOI":"10.1038\/ncomms9686","article-title":"Two-dimensional single-cell patterning with once cell per well driven by surface acoustic waves","volume":"6","author":"Collins","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"11105","DOI":"10.1073\/pnas.1209288109","article-title":"On-chip manipulation of single microparticles, cells, and organisms using surface acoustic waves","volume":"109","author":"Ding","year":"2012","journal-title":"Proc. Natl Acad. Sci. USA"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"12764","DOI":"10.1038\/ncomms12764","article-title":"Controlling the motion of multiple objects on a Chladni plate","volume":"7","author":"Zhou","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"053501","DOI":"10.1063\/1.3298741","article-title":"Collection of suspended particles in a drop using low frequency vibration","volume":"96","author":"Whitehill","year":"2010","journal-title":"Appl. Phys. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2113","DOI":"10.1121\/1.4916704","article-title":"Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers","volume":"137","author":"Tuan","year":"2015","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"062906","DOI":"10.1103\/PhysRevE.92.062906","article-title":"Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by using RLC networks","volume":"92","author":"Tuan","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"64004","DOI":"10.1209\/0295-5075\/111\/64004","article-title":"Resolving the formation of modern Chladni figures","volume":"111","author":"Tuan","year":"2015","journal-title":"Europhys. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"10844","DOI":"10.1038\/s41598-018-29244-6","article-title":"Point-driven modern Chladni figures with symmetry breaking","volume":"8","author":"Tuan","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"022911","DOI":"10.1103\/PhysRevE.89.022911","article-title":"Exploring the distinction between experimental resonant modes and theoretical eigenmodes: From vibrating plates to laser cavities","volume":"89","author":"Tuan","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Chakraverty, S. (2009). Vibration of Plates, CRC Press.","DOI":"10.1201\/9781420053968"},{"key":"ref_22","unstructured":"Ventsel, E., and Krauthammer, T. (2004). Thin Plates and Shells, Marcel Dekker."},{"key":"ref_23","unstructured":"Rayleigh, J.W.S. (1945). Theory of Sound, Dover."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1088\/0959-5309\/51\/5\/312","article-title":"Vibrations of free square plates: Part I. Normal vibrating modes","volume":"51","author":"Waller","year":"1939","journal-title":"Proc. Phys. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1002\/j.1538-7305.1951.tb01366.x","article-title":"Prediction and entropy of printed English","volume":"30","author":"Shannon","year":"1951","journal-title":"Bell Syst. Tech. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1119\/1.17904","article-title":"Entangled quantum systems and the Schmidt decomposition","volume":"63","author":"Ekert","year":"1995","journal-title":"Am. J. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"127903","DOI":"10.1103\/PhysRevLett.92.127903","article-title":"Analysis and interpretation of high transverse entanglement in optical parametric down conversion","volume":"92","author":"Law","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5304","DOI":"10.1103\/PhysRevLett.84.5304","article-title":"Continuous frequency entanglement: Effective finite Hilbert space and entropy control","volume":"84","author":"Law","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1080\/00107514.2013.878554","article-title":"Schmidt modes and entanglement","volume":"55","author":"Fedorov","year":"2014","journal-title":"Contemp. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1023\/A:1023269922178","article-title":"Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle","volume":"43","year":"1998","journal-title":"Appl. Math."},{"key":"ref_32","unstructured":"Leissa, A.W. (1993). Vibration of Plates, Acoustical Society of America."},{"key":"ref_33","unstructured":"Van Vlack, L.H. (1980). Elements of Material Science and Engineering, Addison-Wesley."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2963","DOI":"10.1063\/1.1710033","article-title":"Maximum-emission principle and phase locking in multimode lasers","volume":"38","author":"Tang","year":"1967","journal-title":"J. Appl. Phys."},{"key":"ref_35","unstructured":"Haken, H. (2006). Information and Self-Organization: A Macroscopic Approach to Complex Systems, Springer."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1103\/PhysRevA.35.1360","article-title":"Quantum localization and the rate of exploration of phase space","volume":"35","author":"Heller","year":"1987","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1103\/PhysRevE.60.114","article-title":"Phase transitions in open quantum systems","volume":"60","author":"Jung","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_38","unstructured":"Albeverio, S., Gesztesy, F., H\u00f8egh-Krohn, R., and Holden, H. (2012). Solvable Models in Quantum Mechanics, Springer."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"062712","DOI":"10.1103\/PhysRevA.66.062712","article-title":"Green functions for generalized point interactions in one dimension: A scattering approach","volume":"66","author":"Schmidt","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1228","DOI":"10.1098\/rsta.2010.0330","article-title":"Non-relativistic and relativistic scattering by short-range potentials","volume":"369","author":"Arnbak","year":"2011","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"064301","DOI":"10.1103\/PhysRevC.63.064301","article-title":"Local realizations of contact interactions in two-and three-body problems","volume":"63","author":"Kruppa","year":"2001","journal-title":"Phys. Rev. C"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Demkov, Y., and Ostrovskii, V.N. (1989). Zero-Range Potentials and Their Applications in Atomic Physics, Springer.","DOI":"10.1007\/978-1-4684-5451-2"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Doniach, S., and Sondheimer, E. (1998). Green\u2019s Functions for Solid State Physicists, World Scientific.","DOI":"10.1142\/p067"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1103\/PhysRevD.19.639","article-title":"Quark confinement in the infinite-momentum frame","volume":"19","author":"Thorn","year":"1979","journal-title":"Phys. Rev. D"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1103\/PhysRevLett.64.1855","article-title":"Wave chaos in singular quantum billiard","volume":"64","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4357","DOI":"10.1103\/PhysRevE.50.4357","article-title":"Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer","volume":"50","author":"Shigehara","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0375-9601(96)00640-8","article-title":"Point interactions in two and three dimensions as models of small scatterers","volume":"222","author":"Exner","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"036206","DOI":"10.1103\/PhysRevE.63.036206","article-title":"Singular statistics","volume":"63","author":"Bogomolny","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"134013","DOI":"10.1103\/PhysRevLett.91.134103","article-title":"Intermediate wave function statistics","volume":"91","author":"Berkolaiko","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"123021","DOI":"10.1088\/1367-2630\/12\/12\/123021","article-title":"Singular statistics revised","volume":"12","author":"Tudorovskiy","year":"2010","journal-title":"New J. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1007\/s00220-012-1556-2","article-title":"Statistics of wave functions for a point scatterer on the torus","volume":"316","author":"Rudnick","year":"2012","journal-title":"Commun. Math. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"3341","DOI":"10.1103\/PhysRevE.52.3341","article-title":"Range of spectral correlations in pseudointegrable systems: Gaussian-orthogonal-ensemble statistics in a rectangular membrane with a point scatterer","volume":"52","author":"Weaver","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1103\/PhysRevE.54.1321","article-title":"Wave chaos in quantum billiards with a small but finite-size scatterer","volume":"54","author":"Shigehara","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_54","first-page":"29","article-title":"Analytic solution to field distribution in two-dimensional inhomogeneous waveguides","volume":"8","author":"Grica","year":"2015","journal-title":"J. Electromagn. Waves Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/215\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:10:54Z","timestamp":1760134254000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/215"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,29]]},"references-count":54,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["e24020215"],"URL":"https:\/\/doi.org\/10.3390\/e24020215","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,1,29]]}}}