Solving the hydrodynamic formulation of quantum mechanics: A parallel MLS method
Corresponding Author
R. Glenn Brook
Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762-9627
Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762-9627Search for more papers by this authorPaul E. Oppenheimer
Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762-9627
Search for more papers by this authorCharles A. Weatherford
Department of Physics, Florida A&M University, Tallahassee, Florida 32307
Search for more papers by this authorIoana Banicescu
Department of Computer Science and Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762
Search for more papers by this authorJianping Zhu
Department of Mathematics and Statistics and Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762
Search for more papers by this authorCorresponding Author
R. Glenn Brook
Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762-9627
Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762-9627Search for more papers by this authorPaul E. Oppenheimer
Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762-9627
Search for more papers by this authorCharles A. Weatherford
Department of Physics, Florida A&M University, Tallahassee, Florida 32307
Search for more papers by this authorIoana Banicescu
Department of Computer Science and Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762
Search for more papers by this authorJianping Zhu
Department of Mathematics and Statistics and Engineering Research Center, Mississippi State University, Mississippi State, Mississippi 39762
Search for more papers by this authorAbstract
This article documents the first implementation of a parallel algorithm for solving the governing equations of the hydrodynamic formulation of quantum mechanics. The algorithm employs a quantum trajectory method (QTM) based on the serial algorithm introduced by Wyatt and Lopreore. The OpenMP API is employed to parallelize the code across the quantum trajectories in a shared memory environment. An outline of the parallel algorithm is provided; the analytical solution for a moving free particle is used to verify the solution obtained by the parallel algorithm. Further validation against several of the results obtained by Wyatt and Lopreore is also provided. The parallel speedups and runtimes are presented, and several performance issues are noted. Finally, the results of a preliminary accuracy study examining a moving free particle are provided. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
References
- 1Lopreore, C. L.; Wyatt, R. E. Phys Rev Lett 1999, 82(26), 5190–5193.
- 2deBroglie, L. C R Acad Sci Paris 1926, 183, 447.
- 3deBroglie, L. Nature 1926, 118, 441.
10.1038/118441b0 Google Scholar
- 4deBroglie, L. C R Acad Sci Paris 1927, 184, 273.
- 5deBroglie, L. C R Acad Sci Paris 1927, 185, 380.
- 6deBroglie, L. J de Phys 1927, 8, 225.
- 7Goldberger, M. L.; Watson, K. M. Collision Theory; Wiley: New York, 1964.
- 8Vijay, A.; Wyatt, R. E.; Billing, G. D. J Chem Phys 1999, 111, 10794.
- 9Wyatt, R. E. J Chem Phys 1999, 111, 4406–4413.
- 10Wyatt, R. E. Chem Phys Lett 1999, 313, 189–197.
- 11Mayor, F. S.; Askar, A.; Rabitz, H. A. J Chem Phys 1999, 111, 2423–2435.
- 12Dey, B. K.; Askar, A.; Rabitz, H. A. J Chem Phys 1998, 109, 8770–8782.
- 13Holland, P. Phys Rev A 1999, 60, 4326–4330.
- 14Dürr, D.; Goldstein, S.; Müuch-Berndl, K.; Zanghí, N. Phys Rev A 1999, 60, 2729–2736.
- 15Wyatt, R. E.; Kouri, D. J.; Hoffman, D. K. J Chem Phys 2000, 112, 10730–10737.
- 16Bittner, E. R.; Wyatt, R. E. J Chem Phys 2000, 113, 8888–8897.
- 17Wyatt, R. E.; Bittner, E. R. J Chem Phys 2000, 113, 8898–8907.
- 18Na, K.; Wyatt, R. E. Int J Quantum Chem 2000, 81, 206–213.
- 19Hu, X.-G.; Ho, T.-S.; Rabitz, H. Phys Rev E 2000, 61, 5967–5976.
- 20Bittner, E. R. J Chem Phys 2000, 112, 9703–9710.
- 21Bohm, D. Phys Rev 1952, 85, 166–179 (reprinted in Wheeler & Zurek, 1983).
- 22Bohm, D. Phys Rev 1952, 85, 180–193 (reprinted in Wheeler & Zurek, 1983).
- 23Dewdney, C.; Hiley, B. J. Found Phys 1982, 12, 27–48.
- 24Madelung, E. Z Phys 1926, 322–326.
- 25Goldstein, S. Phys Today 1998, 42–46.
- 26Goldstein, S. Phys Today 1998, 38–42.
- 27Haroche, S. Phys Today 1998, 36–42.
- 28Bohm, D.; Hiley, B. J. The Undivided Universe; Routeledge: London, 1993.
- 29Cushing, J. T.; Fine, A.; Goldstein, S. Bohmian Mechanics and Quantum Theory: An Appraisal; Kluwer: Dordrecht, 1996.
10.1007/978-94-015-8715-0 Google Scholar
- 30Holland, P. R. The Quantum Theory of Motion; Cambridge: New York, 1993.
10.1017/CBO9780511622687 Google Scholar
- 31Grib, A. A.; Rodrigues, Jr., W. A. Nonlocality in Quantum Physics; Kluwer: New York, 1999.
10.1007/978-1-4615-4687-0 Google Scholar
- 32Bamzai, A. S.; Deb, B. M. Rev Mod Phys 1981, 53, 95.
- 33Bamzai, A. S.; Deb, B. M. Rev Mod Phys 1981, 53, 593.
- 34Ghosh, S. K.; Deb, B. M. Phys Rep 1982, 92, 1.
- 35Deb, B. M.; Chattaraj, P. K. Chem Phys Lett 1988, 148, 550.
- 36Chattaraj, P. K.; Koneru, S. R.; Deb, B. M. J Comput Phys 1987, 72, 504.
- 37Deb, B. M.; Chattaraj, P. K. Phys Rev A 1989, 39, 1696.
- 38Deb, B. M.; Ghosh, S. K. J Chem Phys 1982, 77, 342.
- 39Deb, B. M.; Chattaraj, P. K. Phys Rev A 1988, 37, 4030.
- 40Weiner, J. H.; Partom, Y. Phys Rev 1969, 187, 1134.
- 41Weiner, J. H.; Askar, A. J Chem Phys 1971, 54, 1108.
- 42Askar, A.; Weiner, J. H. Am J Phys 1971, 39, 1230.
- 43Meirmanov, A. M.; Pukhnachov, V. V.; Shmarev, S. I. Evolution Equations and Lagrangian Coordinates; Walter de Gruyter: New York, 1997.
10.1515/9783110874440 Google Scholar
- 44McWeeny, R. Methods of Molecular Quantum Mechnics, 2nd ed.; Academic: New York, 1989.
- 45Wyatt, R. E. Moving least squares computer code.
- 46Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P. Comput Methods Appl Mech Eng 1996, 139, 3–47.
- 47Liszka, T. J.; Duarte, C. A. M.; Tworzydlo, W. W. Comput Methods Appl Mech Engrg 1996, 139, 263–288.
- 48Beissel, S.; Belytschko, T. Comput Methods Appl Mech Eng 1996, 139, 49–74.
- 49Liu, W. K.; Chen, Y.; Uras, R. A.; Chang, C. T. Comput Methods Appl Mech Eng 1996, 139, 91–157.
- 50Beylkin, G.; Keiser, J. M.; Vozovoi, L. J Comp Phys 1998, 147, 362–387.
- 51Liu, W. K.; Belytschko, T.; Zhang, Y. F. Int J Numer Meth Eng 1984, 20, 1581–1597.
- 52Liszka, T. Int J Numer Meth Eng 1984, 20, 1599–1612.
- 53Lancaster, P.; Salkauskas, K. Math Comp 1981, 37, 141–158.
- 54Hennessy, J. L.; Patterson, D. A. Computer Organization & Design: The Hardware/Software Interface; Morgan Kaufmann: San Francisco, CA, 1994.
- 55Sima, D.; Fountain, T.; Kacsuk, P. Advanced Computer Architectures: A Design Space Approach; Addison-Wesley: New York, 1997.
- 56Heinrich, J. Origin and Onyx2 Theory of Operations Manual; Silicon Graphics, Inc.: Mountain View, CA, 1997.
- 57 OpenMP Architecture Review Board. Openmp: Simple, portable, scalable smp programming. URL: http://www.openmp.org/.
- 58 OpenMP Architecture Review Board. OpenMP Fortran Application Program Interface, Version 1.1. OpenMP Architecture Review Board, November 1999. URL: http://www.openmp.org/specs/.
- 59Eckart, C. Phys Rev, 2nd Series 1930, 35(11), 1303–1309.
