Reply to ”Comment on ’Thomson rings in a disk’ ” A. Puente Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain 7 1 R.G. Nazmitdinov 0 2 Departament de F´ısica, Universitat de les Illes Balears, n a E-07122 Palma de Mallorca, Spain and J 2 Bogoliubov Laboratory of Theoretical Physics, 2 Joint Institute for Nuclear Research, 141980 Dubna, Russia ] h c e M. Cerkaski m - Department of Theory of Structure of Matter, t a t Institute of Nuclear Physics PAN, 31-342 Cracow, Poland s . t a m K.N. Pichugin - d Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia n o (Dated: January 24, 2017) c [ Abstract 1 v We demonstrate that our model [Phys.Rev. E91, 032312 (2015] serves as a useful tool to trace 9 7 the evolution of equilibrium configurations of one-component charged particles confined in a disk. 1 6 0 Ourapproachreducessignificantlythecomputationaleffortinminimizingtheenergyofequilibrium . 1 configurations and demonstrates a remarkable agreement with the values provided by molecular 0 7 dynamics calculations. We show that the comment misrepresents our paper, and fails to provide 1 : v plausible arguments against the formation hexagonal structure for n ≥ 200 in molecular dynamics i X calculations. r a PACS numbers: 64.70.kp,64.75.Yz,02.20.Rt 1 In our recent publication [1], we have developed a semi-analytical approach that allows to determine equilibrium configurations for arbitrary, but finite, number of charged particles confined in a disk geometry. In the Comment [2] by Amore it was found that the minimum energy configuration of N = 395 charges confined in disk and interacting via the Coulomb potential has a lower energy than the result of our molecular dynamics (MD) calculations [1]. BasedonlyonthisresultAmoreconcludedthat”...the formation of a hexagonal core and valence circular rings for the centered configurations, predicted by the model of Ref.[1], is not supported by numerical evidence and the configurations obtained with this model cannot be used as a guide for the numerical calculations, as claimed by the authors. In light of this > findings, the validity of the model of Ref.[1] must be questioned, particular for N ∼ 200.” Hereafter, for the sake of convenience we refer to our model as the circular model (CM). We agree with the author that his possible global MD minimum is better than our estimate for the particular case N = 395. However, this is not enough to conclude that the CM can not help to reduce substantially the computational effort in MD or simulated annealing (SA) calculations for the following reasons. 1. From the Monte Carlo and MD calculations, even for a relatively small number of chargedparticles, itfollowsthattheamountofstableconfigurationsgrowsveryrapidly with the number of particles. Sometimes, metastable states with lower (or higher) symmetry are found with much higher probability than the true ground state. This fact was confirmed by the author who ”generated 3001 configurations ...” to get just one instance of the improved E = 110664.44 new tentative ground state, with our MIN prediction for the particle number at the boundary ring: ”... Np = 147 charges are disposed on the border of the disk, in agreement with Ref.1”. Evidently, in contrast to his claim, Amore has confirmed the usefulness of the CM. Indeed, the particle number on the boundary ring N is one of the key elements p for any calculation, since once it is defined, it is necessary to simulate less various configurations (with a number of charges N − N ). We recall that N >> N > p p p−1 N > ··· > 1, where p is a number of rings, and N is a total number of charges. p−2 Infact, externalringoccupationsareextremelywellpredictedwithsomeoccasional±1 √ mismatch by means of the expression N (N) = [2.795N2/3−3.184], where p (cid:39) [ N/2] p [1]. It is noteworthy that these expressions are obtained from the systematic CM 2 results. 2. In our publication [1], in order to obtain our estimate of the MD ground state E , MD we generated only 100 configurations with the boundary ring N = 147 charges, p=9 where the internal charges were randomly distributed. As a result, we have obtained E = 110667.6 > E = 110665.1 > E = 110664.44. Note, however, that the CM MD MIN disagreementbetweentheauthor’snewresultandourmodelpredictionE isstillless CM than 3×10−3% (as we stated in our paper it is 2×10−3%). Moreover, the occupations for the external (approximately circular) shells are quite accurately predicted within CM for any N. In the case of N = 395 we have obtained (147,65,50,40), while the analysis of the Amore’s MD ground state yields (147,66,51,40). This comparison suggests that the effectivity of the CM prediction might be improved if the second ring, the nearest neigbor to the boundary one, should be taken into account. 3. To prove the usefulness of this idea we consider initial configurations characterized by external occupations: N = 147 (Set 1); N = 147, N = 65 (Set 2); N = 147, 9 9 8 9 N = 66(Set3). Inallcaseswehavegenerated2000configurations, whereN particles 8 9 were initially set on the boundary at R = 1, and for two other sets N particles 1 8 have been distributed at R = 0.96. That value was chosen to take into account 2 monopole oscillations around the equilibrium configuration. The remaining particles were distributed randomly. For the Set 1 (Fig.1, top panel) we found the lowest state E = 110664.52 > E = MD MIN 110664.44, that occurs just once. In the middle panel (Set 2) we use two boundary shells N = 147, N = 65, predicted by the CM partition, and obtain slightly lower 9 8 state. However, the ground state is not reached yet. The systematic analysis of the CM results leads us to conclusion that the second shell occupation is fitted by the formula N (N) = [1.351N2/3 − 6.566] that yields p−1 N = 66. Considering the initial configuration with N = 147, N = 66 (Set 3) 8 9 8 with randomly distributed internal charges (Fig.1, bottom panel), we obtain that the ground state E occurs three times (0.15%). In other words, with this initialization MIN it appears once every 666 generated configurations. Note that Amore has generated 3001 configurations to obtain just one realization of the possible ground state. 3 3 Set 1 (147) 2 ) % ( p 1 0 10 Set 2 (147, 65) ) % ( 5 p 0 10 Set 3 (147, 66) ) % ( 5 p 0 5 0 5 0 5 0 5 0 4. 5. 5. 6. 6. 7. 7. 8. 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Energy FIG. 1: (Color online) Histograms for energy states of N = 395 charges in the disk geometry obtained by means of the MD method for different initialization procedures. 4. We recall that for infinite systems the hexagonal lattice has the lowest energy of all two-dimensional Wigner Bravais crystals [3]. Evidently, the decrease of system size places primary emphasis upon system boundaries (see, for example, discussion in Refs.[4, 5]). Therefore, one needs to understand how the Wigner crystallization may settle down, in a particular, in a disk geometry as a function of the number of interacting charged particles. In Supplemental Material we compare our results 4 correspondingtotheMDandthesemi-analyticalapproachfor161 ≤ N ≤ 260charges. These results demonstrate a remarkable agreement between two approaches and make it clear how the centered hexagonal lattice (CHL) evolves with the increase of charge particle number. Therefore, we strongly believe that the results obtained by means of our method can be successfully used to feed SA or MD calculations with sensible initial configurations, reducing significantly the amount of scanning, normally needed to visit the global energy minima. 5. The systematic manifestation of the CHL with the increase of particle number N ≥ 200 in our CM and MD results can be interpreted as the onset of the hexagonal crystallization in the disk geometry. There is no, however, any manifestation of a phase transition, typical for infinite systems. In a finite system a crossover takes place from the CHL to ring localization with the approaching to the disk boundary. This ring organization is clearly seen at the boundary in Fig.2 presented by the author (two clear rings). In our paper we have compared the MD configuration with the prediction of the CM for the CHL at N = 395. In fact, in our MD calculations the clean CHL takes place at N = 381 with the configuration 143,64,49,39,30,19,18,12,6,1 and the minimal energy E = 102764.53. The valence configurations with N = 49, N = 64,N = 143 form well 7 8 9 defined ring structure. The increase of particle number disintegrates slowly the CHL in the disk geometry, while the hexagonal lattice still exists. Nevertheless, with each new shell, as soon as a new particle appears at the centre it gives rise to the CHL again. Since we deal with a finite system, restrictedbythecirculargeometry,theboundaryaffectstheplainsymmetricalconfigurations giving rise to defects. In conclusion, we disagree with the main outcome of the author’s Comment formulated in his Summary. In order to argue against our model and corresponding conclusions, it is required a systematic thorough analysis of the system with increasing particle number but not only one particular case. In fact, we have demonstrated that the CM predictions for external rings (N ,N ) enable to us to reduce substantially scanning efforts needed to p p−1 reach the ground state in the MD. 5 Acknowledgments M.C. and K.P. are grateful for the warm hospitality at JINR. This work was supported in part by Bogoliubov-Infeld program of BLTP and Russian Foundation for Basic Research. [1] M. Cerkaski, R.G. Nazmitdinov, and A.Puente, Phys. Rev. E91, 032312 (2015). [2] P. Amore, arXiv:1607.08785. [3] L. Bonsall and A.A. Maradudin, Phys. Rev. B15, 1959 (1977). [4] H. Saarikoski, S.M. Reimann, A. Harju, and M. Manninen, Rev. Mod. Phys. 82, 2785 (2010). [5] J.L. Birman, R.G. Nazmitdinov, and V.I. Yukalov, Phys. Rep. 526, 1 (2013). 6 Supplemental Material The MD and Circular Model results These tables summarize our results corresponding to the minimum energy equilibrium configurations under disk confinement. E , E are the total MD (our best estimate) and MD CM the Circular Model energies. Results for 161 ≤ n ≤ 180 n E Configuration E Configuration MD CM 161 17323.885 [79,33,22,15,9,3] 17324.571 [79,33,22,15,9,3] 162 17548.672 [79,33,23,15,9,3] 17549.325 [79,33,23,15,9,3] 163 17775.066 [80,33,23,15,9,3] 17775.717 [80,33,23,15,9,3] 164 18002.880 [80,33,23,16,9,3] 18003.488 [80,33,23,16,9,3] 165 18232.249 [80,34,23,16,9,3] 18232.837 [80,34,23,16,9,3] 166 18463.081 [81,34,23,16,9,3] 18463.668 [81,34,23,16,9,3] 167 18695.335 [81,34,24,16,9,3] 18695.901 [81,34,24,16,9,3] 168 18929.198 [81,34,24,16,10,3] 18929.751 [81,34,24,16,10,3] 169 19164.419 [81,34,24,16,10,4] 19164.969 [81,34,24,16,10,4] 170 19400.980 [82,34,24,16,10,4] 19401.528 [82,34,24,16,10,4] 171 19639.241 [82,35,24,16,10,4] 19639.768 [82,35,24,16,10,4] 172 19878.944 [83,35,24,16,10,4] 19879.461 [83,34,24,17,10,4] 173 20120.029 [83,35,24,17,10,4] 20120.507 [83,35,24,17,10,4] 174 20362.731 [83,35,25,17,10,4] 20363.180 [83,35,25,17,10,4] 175 20606.871 [84,35,25,17,10,4] 20607.318 [84,35,25,17,10,4] 176 20852.540 [84,35,25,17,11,4] 20852.999 [84,36,25,17,10,4] 177 21099.703 [84,36,25,17,11,4] 21100.164 [84,36,25,17,11,4] 178 21348.301 [85,36,25,17,11,4] 21348.762 [85,36,25,17,11,4] 179 21598.311 [85,36,25,18,11,4] 21598.791 [85,36,25,17,11,5] 180 21849.924 [85,36,25,18,11,5] 21850.334 [85,36,25,18,11,5] 7 Results for 181 ≤ n ≤ 207 n E Configuration E Configuration MD CM 181 22102.961 [86,36,25,18,11,5] 22103.368 [86,36,25,18,11,5] 182 22357.440 [86,36,26,18,11,5] 22357.815 [86,36,26,18,11,5] 183 22613.517 [86,37,26,18,11,5] 22613.878 [86,37,26,18,11,5] 184 22871.031 [87,37,26,18,11,5] 22871.391 [87,37,26,18,11,5] 185 23129.918 [87,36,26,18,11,6,1] 23130.442 [87,37,26,18,12,5] 186 23390.285 [87,37,26,18,11,6,1] 23391.044 [87,37,26,19,12,5] 187 23652.188 [87,37,26,18,12,6,1] 23652.947 [87,37,26,18,12,6,1] 188 23915.459 [88,37,26,18,12,6,1] 23916.215 [88,37,26,18,12,6,1] 189 24180.381 [88,37,27,18,12,6,1] 24181.062 [88,37,26,19,12,6,1] 190 24446.798 [89,37,27,18,12,6,1] 24447.458 [88,37,27,19,12,6,1] 191 24714.561 [89,37,27,19,12,6,1] 24715.189 [89,37,27,19,12,6,1] 192 24983.853 [89,38,27,19,12,6,1] 24984.461 [89,38,27,19,12,6,1] 193 25254.755 [90,38,27,19,12,6,1] 25255.362 [90,38,27,19,12,6,1] 194 25527.119 [90,38,27,19,13,6,1] 25527.697 [90,38,27,19,13,6,1] 195 25801.014 [90,38,27,20,13,6,1] 25801.595 [90,38,28,19,13,6,1] 196 26076.378 [91,38,27,20,13,6,1] 26076.964 [91,38,28,19,13,6,1] 197 26353.122 [91,39,27,20,13,6,1] 26353.672 [91,38,27,20,13,7,1] 198 26631.365 [91,39,28,20,13,6,1] 26631.870 [91,39,27,20,13,7,1] 199 26911.103 [91,39,28,20,13,7,1] 26911.559 [91,39,28,20,13,7,1] 200 27192.287 [92,39,28,20,13,7,1] 27192.741 [92,39,28,20,13,7,1] 201 27475.149 [92,40,28,20,13,7,1] 27475.591 [92,40,28,20,13,7,1] 202 27759.495 [92,39,28,21,14,7,1] 27759.953 [93,40,28,20,13,7,1] 203 28045.151 [93,39,28,21,14,7,1] 28045.669 [93,40,29,20,13,7,1] 204 28332.320 [93,40,28,21,14,7,1] 28332.900 [93,40,29,20,14,7,1] 205 28621.069 [93,40,29,21,14,7,1] 28621.647 [93,40,29,21,14,7,1] 206 28911.236 [94,40,29,21,14,7,1] 28911.813 [94,40,29,21,14,7,1] 207 29203.054 [94,41,29,21,14,7,1] 29203.620 [94,41,29,21,14,7,1] 8 Results for 208 ≤ n ≤ 234 n E Configuration E Configuration MD CM 208 29496.341 [94,40,29,21,14,8,2] 29496.944 [94,41,29,21,14,8,1] 209 29790.985 [95,40,29,21,14,8,2] 29791.618 [95,41,29,21,14,8,1] 210 30087.107 [95,41,29,21,14,8,2] 30087.834 [95,41,30,21,14,8,1] 211 30384.840 [95,41,30,21,14,8,2] 30385.659 [95,41,30,22,14,8,1] 212 30684.005 [96,41,30,21,14,8,2] 30684.828 [96,41,30,22,14,8,1] 213 30984.546 [96,41,30,22,14,8,2] 30985.505 [96,41,30,22,15,8,1] 214 31286.636 [96,41,30,22,13,9,3] 31287.674 [96,41,30,22,15,8,2] 215 31590.285 [97,41,30,22,13,9,3] 31591.327 [97,41,30,22,15,8,2] 216 31895.274 [97,41,30,22,14,9,3] 31896.396 [97,42,30,22,15,8,2] 217 32201.810 [97,41,30,22,15,9,3] 32202.828 [97,41,30,22,15,9,3] 218 32509.860 [97,42,30,22,15,9,3] 32510.843 [97,42,30,22,15,9,3] 219 32819.357 [98,42,30,22,15,9,3] 32820.338 [98,42,30,22,15,9,3] 220 33130.391 [98,42,31,22,15,9,3] 33131.367 [98,42,31,22,15,9,3] 221 33443.063 [98,42,31,23,15,9,3] 33444.031 [98,42,31,23,15,9,3] 222 33757.067 [99,42,31,23,15,9,3] 33758.034 [99,42,31,23,15,9,3] 223 34072.594 [99,42,31,23,16,9,3] 34073.531 [99,42,31,23,16,9,3] 224 34389.617 [99,43,31,23,16,9,3] 34390.523 [99,43,31,23,16,9,3] 225 34708.151 [100,43,31,23,16,9,3] 34709.055 [100,43,31,23,16,9,3] 226 35028.160 [100,43,31,23,15,10,4] 35029.106 [100,43,31,23,16,9,4] 227 35349.639 [100,43,31,23,16,10,4] 35350.486 [100,43,31,23,16,10,4] 228 35672.675 [101,43,31,23,16,10,4] 35673.518 [101,43,31,23,16,10,4] 229 35997.082 [101,43,32,23,16,10,4] 35997.893 [101,43,32,23,16,10,4] 230 36323.076 [101,44,32,23,16,10,4] 36323.861 [101,44,32,23,16,10,4] 231 36650.589 [101,44,32,24,16,10,4] 36651.395 [101,44,32,24,16,10,4] 232 36979.503 [102,44,32,24,16,10,4] 36980.306 [102,44,32,24,16,10,4] 233 37309.955 [102,44,32,24,17,10,4] 37310.715 [102,44,32,24,17,10,4] 234 37642.049 [102,44,33,24,17,10,4] 37642.792 [102,44,33,24,17,10,4] 9 Results for 235 ≤ n ≤ 260 n E Configuration E Configuration MD CM 235 37975.484 [103,44,33,24,17,10,4] 37976.225 [103,44,33,24,17,10,4] 236 38310.462 [103,44,33,24,17,11,4] 38311.186 [103,45,33,24,17,10,4] 237 38646.933 [103,45,33,24,17,11,4] 38647.705 [103,45,33,24,17,11,4] 238 38984.913 [104,45,33,24,17,11,4] 38985.683 [104,45,33,24,17,11,4] 239 39324.318 [104,45,33,25,16,11,5] 39325.033 [104,45,33,24,17,11,5] 240 39665.177 [104,45,33,25,14,12,6,1] 39665.976 [104,45,33,25,17,11,5] 241 40007.632 [104,45,33,25,15,12,6,1] 40008.473 [105,45,33,25,17,11,5] 242 40351.460 [105,45,33,25,15,12,6,1] 40352.337 [105,45,33,25,18,11,5] 243 40696.877 [105,45,33,25,16,12,6,1] 40697.757 [105,46,33,25,18,11,5] 244 41043.791 [105,46,33,25,16,12,6,1] 41044.696 [105,46,34,25,18,11,5] 245 41392.174 [106,46,33,25,16,12,6,1] 41393.091 [106,46,34,25,18,11,5] 246 41741.996 [106,46,34,25,16,12,6,1] 41743.132 [106,46,34,25,18,12,5] 247 42093.362 [106,46,34,25,17,12,6,1] 42094.661 [106,46,34,25,18,11,6,1] 248 42446.278 [107,46,34,25,17,12,6,1] 42447.440 [106,46,34,25,18,12,6,1] 249 42800.557 [107,46,34,25,18,12,6,1] 42801.689 [107,46,34,25,18,12,6,1] 250 43156.448 [107,46,34,26,18,12,6,1] 43157.543 [107,46,34,26,18,12,6,1] 251 43513.864 [107,47,34,26,18,12,6,1] 43514.922 [107,47,34,26,18,12,6,1] 252 43872.683 [108,47,34,26,18,12,6,1] 43873.737 [108,47,34,26,18,12,6,1] 253 44233.025 [108,47,35,26,18,12,6,1] 44234.016 [108,47,34,26,19,12,6,1] 254 44594.889 [108,47,35,26,19,12,6,1] 44595.833 [108,47,35,26,19,12,6,1] 255 44958.250 [109,47,35,26,19,12,6,1] 44959.191 [109,47,35,26,19,12,6,1] 256 45323.172 [109,47,35,26,19,13,6,1] 45324.102 [109,48,35,26,19,12,6,1] 257 45689.578 [109,48,35,26,19,13,6,1] 45690.514 [109,48,35,26,19,13,6,1] 258 46057.509 [110,48,35,26,19,13,6,1] 46058.440 [110,48,35,26,19,13,6,1] 259 46426.854 [110,48,35,27,19,13,6,1] 46427.711 [110,48,35,26,19,13,7,1] 260 46797.668 [110,48,35,27,19,13,7,1] 46798.489 [110,48,35,27,19,13,7,1] 10