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

Part of: Equilibrium statistical mechanics Special processes

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: [email protected]
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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

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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