Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T03:56:48.780Z Has data issue: false hasContentIssue false
  • English
  • Français

Long-range orientational order of random near-lattice hard sphere and hard disk processes

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  16 July 2020

Alexisz Tamás Gaál
Alexisz Tamás Gaál*
Affiliation:
Courant Institute of Mathematical Sciences
*
*Postal address: 251 Mercer Street, New York, NY 10012, USA. E-mail: gaal@cims.nyu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that a point process of hard spheres exhibits long-range orientational order. This process is designed to be a random perturbation of a three-dimensional lattice that satisfies a specific rigidity property; examples include the FCC and HCP lattices. We also define two-dimensional near-lattice processes by local geometry-dependent hard disk conditions. Earlier results about the existence of long-range orientational order carry over, and we obtain the existence of infinite-volume measures on two-dimensional point configurations that turn out to follow the orientation of a fixed triangular lattice arbitrarily closely.

Keywords

spontaneous symmetry breakinghard spherehard diskrigidity estimate

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B21: Continuum models (systems of particles, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 559 - 577
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aristoff, D. (2014). Percolation of hard disks. J. Appl. Prob. 51 (1), 235–246.10.1239/jap/1395771426CrossRefGoogle Scholar
Aumann, S. (2015). Spontaneous breaking of rotational symmetry with arbitrary defects and a rigidity estimate. J. Statist. Phys. 160, 168–208.10.1007/s10955-015-1234-9CrossRefGoogle Scholar
Bauerschmidt, R., Conache, D., Heydenreich, M., Merkl, F. and Rolles, S. W. W. (2019). Dislocation lines in three-dimensional solids at low temperature. Ann. H. Poincaré 20 (9), 3019–3057.10.1007/s00023-019-00829-9CrossRefGoogle Scholar
Dereudre, D., Drouilhet, R. and Georgii, H.-O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Prob. Theory Relat. Fields 153 (3–4), 643–670.10.1007/s00440-011-0356-5CrossRefGoogle Scholar
Friesecke, G., James, R. and Müller, S. (2002). A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55 (11), 1461–1506.10.1002/cpa.10048CrossRefGoogle Scholar
Gaál, A. T. (2013). Spontaneous breaking of rotational symmetry in a probabilistic hard disk model in Statistical Mechanics. Master’s thesis, Institute of Mathematics, University of Munich. Available at https://math.nyu.edu/~gaal/thesis/GaalMASoSe2013.pdf.Google Scholar
Gaál, A. T. (2014). Long-range order in a hard disk model in statistical mechanics. Electron. Commun. Prob. 19 (9), 1–9.10.1214/ECP.v19-3047CrossRefGoogle Scholar
Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter, Berlin.10.1515/9783110850147CrossRefGoogle Scholar
Georgii, H.-O. and Zessin, H. (1993). Large deviations and the maximum entropy principle for marked point random fields. Prob. Theory Relat. Fields 96, 177–204.10.1007/BF01192132CrossRefGoogle Scholar
Giulian, A. and Theil, F. (2019). Long range order in atomistic models for solids. Available at arXiv:1907.07923.Google Scholar
Heydenreich, M., Merkl, F. and Rolles, S. W. W. (2014). Spontaneous breaking of rotational symmetry in the presence of defects. Electron. J. Prob. 19 (111), 1–17.10.1214/EJP.v19-2971CrossRefGoogle Scholar
Hofer-Temmel, C. (2019). Disagreement percolation for the hard-sphere model. Electron. J. Prob. 24, 91, 1–22.Google Scholar
Lazzaroni, G., Palombaro, M. and Schlömerkemper, A. (2017). Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires. Discrete Contin. Dyn. Syst. S 10, 119–139.10.3934/dcdss.2017007CrossRefGoogle Scholar
Magazinov, A. (2018). On percolation of two-dimensional hard disks. Commun. Math. Phys. 364 (1), 1–43.10.1007/s00220-018-3193-xCrossRefGoogle Scholar
Merkl, F. and Rolles, S. W. W. (2009). Spontaneous breaking of continuous rotational symmetry in two dimensions. Electron. J. Prob. 14, 57, 1705–1726.10.1214/EJP.v14-671CrossRefGoogle Scholar
Richthammer, T. (2007). Translation-invariance of two-dimensional Gibbsian point processes. Commun. Math. Phys. 274, 81–122.10.1007/s00220-007-0274-7CrossRefGoogle Scholar
Richthammer, T. (2016). Lower bound on the mean square displacement of particles in the hard disk model. Commun. Math. Phys. 345, 1077–1099.10.1007/s00220-016-2584-0CrossRefGoogle Scholar