Document Zbl 1376.82017

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Stat. Phys.</a> <a href="/?q=in%3A368124" title="Articles in this Issue">167, No. 6, 1586-1592 (2017)</a>. </div> <div class="abstract">Summary: We show that minimizers of the Heitmann-Radin energy [<span class="zbmathjax-textit">R. C. Heitmann</span> and <span class="zbmathjax-textit">C. Radin</span>, “The ground states for sticky discs”, J. Stat. Phys. 22, No. 3, 281–287 (1980)] are unique if and only if the particle number $N$ belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in [the authors, “Crystallization in two dimensions and a discrete Gauss-Bonnet theorem”, J. Nonlinear Sci. (to appear), <a href="https://doi.org/10.1007/s00332-017-9401-6">doi:10.1007/s00332-017-9401-6</a>].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6791707">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A82B20" title="MSC2020">82B20</a> </td> <td class="space"> Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A82B21" title="MSC2020">82B21</a> </td> <td class="space"> Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acrystallization">crystallization</a>; <a href="/?q=ut%3AWulff+shape">Wulff shape</a>; <a href="/?q=ut%3AHeitmann-Radin+potential">Heitmann-Radin potential</a>; <a href="/?q=ut%3Adiscrete+differential+geometry">discrete differential geometry</a>; <a href="/?q=ut%3Aenergy+minimization">energy minimization</a></div> <div class="software"> <h3>Software:</h3><a href="/software/7248">OEIS</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1376.82017" data-ciurl="/ci/06791707" data-biburl="/bibtex/06791707.bib" data-amsurl="/amsrefs/06791707.bib" data-xmlurl="/xml/06791707.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06791707.pdf" title="Zbl 1376.82017 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10955-017-1781-3" aria-label="DOI for “Classification of particle numbers with unique Heitmann-Radin minimizer”" title="10.1007/s10955-017-1781-3">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1701.07231"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="mathoverflow"> <h3>Online Encyclopedia of Integer Sequences:</h3> <a href="https://oeis.org/A1399">a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.</a><br> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial. Differ. Equ. 44(1-2), 81-100 (2012) · <a href="/1379.74002" class="nowrap">Zbl 1379.74002</a></td> </tr><tr> <td>[2]</td> <td class="space">Davoli, E., Piovano, P., Stefanelli, U.: Sharp \[N^{3/4}\] N3/4 law for the minimizers of the edge-isoperimetric problem on the triangular lattice. J. Nonlinear Sci. 27(2), 627-660 (2017) · <a href="/1383.82063" class="nowrap">Zbl 1383.82063</a> · <a href="https://doi.org/10.1007/s00332-016-9346-1" class="nowrap">doi:10.1007/s00332-016-9346-1</a></td> </tr><tr> <td>[3]</td> <td class="space">De Luca, L., Friesecke, G.: Crystallization in two dimensions and a discrete Gauss-Bonnet Theorem, preprint. arXiv:1605.00034 (2016) · <a href="/1382.82047" class="nowrap">Zbl 1382.82047</a></td> </tr><tr> <td>[4]</td> <td class="space">Dobrushin, R.L., Kotecky, R., Shlosman, S.B.: The Wulff Construction: A Global Shape from Local Interactions. AMS, Providence (1992) · <a href="/0917.60103" class="nowrap">Zbl 0917.60103</a></td> </tr><tr> <td>[5]</td> <td class="space">Fonseca, I., Müller, S.: A uniqueness proof of the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119, 125-136 (1991) · <a href="/0752.49019" class="nowrap">Zbl 0752.49019</a> · <a href="https://doi.org/10.1017/S0308210500028365" class="nowrap">doi:10.1017/S0308210500028365</a></td> </tr><tr> <td>[6]</td> <td class="space">Heitmann, R.C., Radin, C.: The ground states for sticky discs. J. Stat. Phys. 22(3), 281-287 (1980) · <a href="https://doi.org/10.1007/BF01014644" class="nowrap">doi:10.1007/BF01014644</a></td> </tr><tr> <td>[7]</td> <td class="space">Mermin, N.D.: Crystalline order in two dimensions. Phys. Rev. 176(1), 250-254 (1968) · <a href="https://doi.org/10.1103/PhysRev.176.250" class="nowrap">doi:10.1103/PhysRev.176.250</a></td> </tr><tr> <td>[8]</td> <td class="space">Polyá, G., Szegó, G.: Problems and Theorems in Analysis I. Springer, Berlin (1991) · <a href="/0236.00003" class="nowrap">Zbl 0236.00003</a></td> </tr><tr> <td>[9]</td> <td class="space">Richthammer, T.: Translation-invariance of two-dimensional Gibbsian point processes. Commun. Math. Phys. 274(1), 81-122 (2007) · <a href="/1132.82002" class="nowrap">Zbl 1132.82002</a> · <a href="https://doi.org/10.1007/s00220-007-0274-7" class="nowrap">doi:10.1007/s00220-007-0274-7</a></td> </tr><tr> <td>[10]</td> <td class="space">Schmidt, B.: Ground states of the 2D sticky disc model: fine properties and \[N^{3/4}\] N3/4 law for the deviation from the asymptotic Wulff shape. J. Stat. Phys. 153(4), 727-738 (2013) · <a href="/1292.82027" class="nowrap">Zbl 1292.82027</a> · <a href="https://doi.org/10.1007/s10955-013-0852-3" class="nowrap">doi:10.1007/s10955-013-0852-3</a></td> </tr><tr> <td>[11]</td> <td class="space">Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. In: Differential Geometry (Proc. Sympos. Pure. Math., Vol. XXVII), Part 1, pp. 419-427. AMS, Providence (1975) · <a href="/0317.49054" class="nowrap">Zbl 0317.49054</a></td> </tr><tr> <td>[12]</td> <td class="space">The On-line Encyclopedia of Integer Sequences. www.oeis.org · <a href="/1274.11001" class="nowrap">Zbl 1274.11001</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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