The transition from order to disorder in Voronoi Diagrams with applications
TThe transition from order to disorder in VoronoiDiagrams with applications
L. ZaninettiDepartment of Physics, via P.Giuria 1,I-10125 Turin,Italy ∗ Corresponding author: [email protected] 20, 2018
Abstract
The transition from ordered to disordered structures in Voronoi tessel-lation is obtained by perturbing the seeds that were originally identifiedwith two types of lattice in 2D and one type in 3D. The area in 2D and thevolume in 3D are modeled with the Kiang function. A new relationshipthat models the scaling of the Kiang function with a geometrical param-eter is introduced. A first application models the local structure of sub-and supercritical ammonia as function of the temperature and a secondapplication models the volumes of cosmic voids.
Keywords:
Voronoi diagrams; Monte Carlo methods; Cell-size distribution
The seeds in 1D, 2D and 3D Voronoi Diagrams are usually taken to be ran-domly distributed, the so called Poisson-Voronoi tessellations (PVT) Okabeet al. (2000). A first generalization of the PVT are the quasi random seeds, suchas the Sobol seeds Sobol (1967); Bratley, P. and Fox, B. L. (1988); Zaninetti(1992, 2009); Zaninetti and Ferraro (2015) and the eigenvalues of complex ran-dom matrices Le Caer and Ho (1990). A second major generalisation modifiesregular structures to produce non-Poissonian seeds for the Voronoi tessellation(NPVT). We select some methods, including perturbation of cubic structuresLucarini (2009), generation of seeds with controlled regularity Zhu et al. (2014),an information geometric model to simulate graphene Dodson (2015), a 3Dtopological analysis Lazar et al. (2015) and two-dimensional perturbed systemsLeipold et al. (2016).The rest of this paper is organised as follows: In section 2, we first reviewthe existing formulae which model the areas and volumes in 2D/3D PVT. Af-terwards, we introduce two models for NPVT, see Section 3. Two applicationsare discussed in Section 4. ∗ [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec igure 1: Histogram (step-diagram) of the 2D PVT reduced area distribution. The probability density function (PDF) for segments (1D) in PVT is modeledby a gamma variate H ( x ; c ) = c Γ( c ) ( cx ) c − exp( − cx ) , (1)where 0 ≤ x < ∞ , c > c ) is the gamma function with argument c,see formula (5) in Kiang (1966). In the case of 1D, c=2 which is an analyticalresult. Conversely the PDF for areas in 2D and volumes in 3D was conjecturedto follow the above gamma variate with c=4 and c=6. Later on, the ”Kiangconjecture” was refined by Ferenc and N´eda (2007) with the following PDF F N ( x ; d ) = Const × x d − exp ( − (3 d + 1) x/ , (2)where Const = √ √ d + 12 2 / d (3 d + 1) − / d Γ (3 / d + 1 / , (3)and d ( d = 1 , ,
3) represents the dimension of the considered space. This PDFallows us to fix the ”Kiang conjecture” in c=3.5 and c=5 for the 2D and 3DPVT case. A typical 2D result is reported in Figure 1 for the reduced areadistribution when the Poissonian seeds are 20000, d = 2 and c = 3 . d = 3 .
03 and c = 5 .
05. and Figure 42igure 2: Histogram of the expected number of edges in the 2D PVT.the histogram of the number of faces with average value 14.897. A comparisonshould be done with the expected value 15.535, see Table 5.5.2 in Okabe et al.(2000).
The lognormal PDF, f LN , is f LN ( x ; m, σ ) = e − σ ( ln ( xm )) xσ √ π , (4)where m is the median and σ the shape parameter, see Evans et al. (2000). Thedistribution function (DF), F LN , is F LN ( x ; m, σ ) = 12 + 12 erf (cid:32) √ − ln ( m ) + ln ( x )) σ (cid:33) , (5)where erf(x) is the error function, defined aserf( x ) = 2 √ π (cid:90) x e − t dt , (6)see Abramowitz and Stegun (1965). Two regular geometrical models are perturbed to have NPVT and a functionwhich models the transition from order to disorder is introduced.3igure 3: Histogram (step-diagram) of the 3D PVT reduced volume distribu-tion.Figure 4: Histogram of the expected number of faces in the 3D PVT.4igure 5: An example of 2D LNPVT.
To have more flexible seeds we introduce the adjustable non Poissonian seeds(LNPVT), which can be computed both in 2D and 3D following an algorithmintroduced in Zaninetti and Ferraro (2015). The algorithm is now outlined:1. The process starts by inserting the seeds on a 2D/3D regular Cartesiangrid with equal distance δ between one point and the following one.2. A random radius is generated according to the half Gaussian , HN ( x ),which is defined in the interval [0 , ∞ ] HN ( x ; s ) = 2 s (2 π ) / exp( − x s ) 0 < x < ∞ . (7)A random direction is chosen in 2D/3D and the two/three Cartesian co-ordinates of the generated radius are evaluated. These two/three smallCartesian components are added to the regular 2D/3D grid which repre-sent the seeds. To have small corrections, we express s in δ units.3. We now have N seeds and we can eliminate a given number of seeds , N hole ,according to the rule N hole = N × p hole . This elimination of seeds willallow a more disordered distribution of areas/volumes; otherwise specified p hole = 0.Figure 5 reports an example of 2D tessellation from LNPVT in which s = 0 . s = 0 . p hole = 0 . delta/ s = 0 . We can model the order disorder transition introducing the following functionfor c of Kiang as in equation (1) c ( s ; c min , c max , a ) = c max − ( c max − c min ) (cid:0) − e − a s (cid:1) , (8)where s is the scale of the half Gaussian, see equation (7), c max and c min arethe maximum and minimum value for c of Kiang and a is a scale to be foundfrom the fitting procedure. A typical 2D example for c as function of s isreported in Figure 8 for the LNPVT case where c min = 3 .
22 (the minimum), c max = 65 .
33 (the maximum) and a =2.11 and in Figure 9 for the LNPVT casewhen c min = 3 . c max = 83 .
06 and a = 1.9.A 3D example for c as function of s is reported in Figure 10 for the LNPVTcase when c min = 3 . c max = 454 .
12 and a =4.25. Two applications are presented. 6igure 7: An example of 2D TNPVT.Figure 8: The Kiang parameter c as function of s for 2D LNPVT.7igure 9: The Kiang parameter c as function of s for 2D TNPVT.Figure 10: The Kiang parameter c as function of s for 3D LNPVT.8igure 11: The Kiang parameter c as function of T. The local structure of sub- and supercritical ammonia with 250
K < T < K has been extensively analyzed by Idrissi et al. (2011) and Figure 11 reports the c of Kiang as function of the temperature for Figure 5 in Idrissi et al. (2011).These values for c of Kiang are obtained from Figure 5 in Idrissi et al. (2011)and we report as an example the deduction of the first couple: σ V (˚ A )= 4.33, T ( K ) = 250 . σ = . . = 0 .
017 which means c = 3356. Figure 12 reportsboth the chemical and theoretical data: data with stars as deduced from Figure5 in Idrissi et al. (2011) and full line derived coupling equations (8) and (9).To set up the theoretical data from equation (8) the following relationshiphas been used T = 512 × s . K . (9)
The catalog of the Baryon Oscillation Spectroscopic Survey (BOSS), see Maoet al. (2017), reports the volume in units of
M pc /h of 1228 cosmic voids where h = H /
100 and the Hubble constant, H , is expressed in km s − Mpc − . Thenumerical analysis gives c = 0 .
02 for the distribution of the reduced volumeof cosmic voids and Figure 13 reports the distribution function (DF) of thelognormal, see equation (5) with parameters as in Table 1.These results for the cosmic voids can be simulated analyzing the distributionof the reduced volumes in 3D tessellation from LNPVT, see Figure 14, wherewe have 195112 original seeds, s = 10 and p hole = 0 . D is9igure 12: The Kiang parameter c as function of T.Figure 13: Empirical DF of reduced voids distribution for BOSS (dotted line)and lognormal DF (full line). 10igure 14: Empirical DF of reduced volume distribution for 3D LNPVT.the maximum distance between theoretical and observed DF , and P KS , is thesignificance level, in the K-S test.Table 1: Lognormal parameters for reduced volumes of cosmic voids and rela-tive LNPVT simulation.case parameters D P KS astronomical observations m = 0 . σ = 1.182 0.049 0.005LNPVT simulation m = 0 . σ =1.523 0.055 0.003 We perturbed the 2D seeds identified by a Cartesian and a triangular lattice,as well the 3D seeds of a Cartesian lattice. The probability to have some holesin the resulting 2D/3D seeds is introduced. The area and volume distributionare modeled by the one parameter Kiang’s PDF. The transition from order todisorder is parametrised by a geometrical variable, which regulates the strengthof the perturbation of the ordered seeds. Two applications are presented: oneto the local structure of sub- and supercritical ammonia and the second one tothe volumes of cosmic voids.
References
Abramowitz, M. and Stegun, I. A. (1965).
Handbook of Mathematical Functions ith Formulas, Graphs, and Mathematical Tables . Dover, New York.Bratley, P. and Fox, B. L. (1988). Implementing Sobol’s quasirandom sequencegenerator. ACM Trans. Math. Softw. , 14(1):88–100.Dodson, C. T. J. (2015). A Model for Gaussian Perturbations of Graphene.
Journal of Statistical Physics , 161:933–941.Evans, M., Hastings, N., and Peacock, B. (2000).
Statistical Distributions - thirdedition . John Wiley & Sons Inc, New York.Ferenc, J.-S. and N´eda, Z. (2007). On the size distribution of Poisson Voronoicells.
Phys. A , 385:518–526.Idrissi, A., Vyalov, I., Kiselev, M., Fedorov, M. V., and Jedlovszky, P.(2011). Heterogeneity of the local structure in sub-and supercritical am-monia: A voronoi polyhedra analysis.
The Journal of Physical Chemistry B ,115(31):9646–9652.Kiang, T. (1966). Random Fragmentation in Two and Three Dimensions.
Z.Astrophys. , 64:433–439.Lazar, E. A., Han, J., and Srolovitz, D. J. (2015). Topological framework forlocal structure analysis in condensed matter.
Proceedings of the NationalAcademy of Science , 112:E5769–E5776.Le Caer, G. and Ho, J. S. (1990). The voronoi tessellation generated fromeigenvalues of complex random matrices.
Journal of Physics A: Mathematicaland General , 23(14):3279.Leipold, H., Lazar, E. A., Brakke, K. A., and Srolovitz, D. J. (2016). Statis-tical topology of perturbed two-dimensional lattices.
Journal of StatisticalMechanics: Theory and Experiment , 4:043103.Lucarini, V. (2009). Three-dimensional random voronoi tessellations: Fromcubic crystal lattices to poisson point processes.
Journal of Statistical Physics ,134(1):185.Mao, Q., Berlind, A. A., Scherrer, R. J., and et al. (2017). A Cosmic VoidCatalog of SDSS DR12 BOSS Galaxies.
ApJ , 835:161.Okabe, A., Boots, B., Sugihara, K., and Chiu, S. (2000).
Spatial tessellations.Concepts and Applications of Voronoi diagrams, 2nd ed.
Wiley, Chichester,New York.Sobol, I. M. (1967). The distribution of points in a cube and the approximateevaluation of integrals.
USSR Comp. Math. Math. Phys. , 7(4):86–112.Zaninetti, L. (1992). The Voronoi tessellation generated from different distri-butions of seeds.
Phys. Lett. A , 165:143–147.Zaninetti, L. (2009). Poissonian and non-Poissonian Voronoi diagrams withapplication to the aggregation of molecules.
Phys. Lett. A , 373:3223–3229.Zaninetti, L. and Ferraro, M. (2015). On Non-Poissonian Voronoi Tessellations.
Applied Physics Research , 7:108–124.12hu, H. X., Zhang, P., Balint, D., Thorpe, S. M., Elliott, J. A., Windle, A. H.,and Lin, J. (2014). The effects of regularity on the geometrical properties ofVoronoi tessellations.