American Institute of Mathematical Sciences

Advanced
August  2003, 3(3): 313-342. doi: 10.3934/dcdsb.2003.3.313

Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere

Chjan C. Lim 1, , Joseph Nebus 2, and  Syed M. Assad 3,

1. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, United States
2. 

Department of Computational Science, National University of Singapore
3. 

Department of Physics, National University of Singapore

Received  November 2002 Revised  February 2003 Published  May 2003

The problem of N bodies on the surface of the sphere interacting by a logarithmic potential is examined for selected N ranging from $4$ to $40,962$, comparing the energies found by placing points at the vertices of certain polyhedrons to the lowest energies found by a Monte Carlo algorithm. The polyhedron families are generated from simple polyhedrons through two triangular face splitting operations which are used iteratively to increase the number of vertices. The closest energy of these polyhedron vertex configurations to the Monte Carlo-generated minimum energy is identified and the two energies are found to agree well. Finally the energy per particle pair is found to asymptotically approach a mean field theory limit of $- 1/2 (log(2) - 1)$, approximately $0.153426$, for both the polyhedron and the Monte Carlo-generated energies. The deterministic algorithm of generating polyhedrons is shown to be a method able to generate consistently good approximations to the extremal energy configuration for a wide range of numbers of points.
Keywords: Monte Carlo-generated minimum energy., Polyhedron families.
Mathematics Subject Classification: 74G6.
Citation: Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[1]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
[2]

Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81
[3]

Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004
[4]

Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014
[5]

Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335
[6]

Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683
[7]

Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125
[8]

Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803
[9]

Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025
[10]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[11]

Salma Souhaile, Larbi Afifi. Minimum energy compensation for discrete delayed systems with disturbances. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2489-2508. doi: 10.3934/dcdss.2020119
[12]

Adam Bobrowski, Adam Gregosiewicz, Małgorzata Murat. Functionals-preserving cosine families generated by Laplace operators in C[0,1]. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1877-1895. doi: 10.3934/dcdsb.2015.20.1877
[13]

Ugo Boscain, Thomas Chambrion, Grégoire Charlot. Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 957-990. doi: 10.3934/dcdsb.2005.5.957
[14]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
[15]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793
[16]

Maria Cameron. Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree. Networks & Heterogeneous Media, 2014, 9 (3) : 383-416. doi: 10.3934/nhm.2014.9.383
[17]

Zoltán Horváth, Yunfei Song, Tamás Terlaky. Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2997-3013. doi: 10.3934/dcds.2015.35.2997
[18]

Nguyen Thi Bach Kim. Finite algorithm for minimizing the product of two linear functions over a polyhedron. Journal of Industrial & Management Optimization, 2007, 3 (3) : 481-487. doi: 10.3934/jimo.2007.3.481
[19]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029
[20]

Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020130

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences