The Arnold cat map and elongational flow

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Title: The Arnold cat map and elongational flow
Author(s): Todd, B. D.; Hunt, T. H.
Abstract: For many years the simulation of elongational (or extensional) flows by molecular dynamics simulation was deemed impossible due to the limitation that the simulation must cease once the length of the simulation box in the contracting dimension equals twice the interaction potential radius. The problem was always that a nonequilibrium steady-state could never be achieved for molecular systems, since their relaxation times are vastly greater than the time taken for this minimum extension to be reached. This limitation was removed in 1998 when it was shown [1-3] how to remove this time constraint by the use of novel periodic boundary conditions, first devised by Kraynik and Reinelt [4] in their studies of foams. However, the derivation of Kraynik and Reinelt is algebraically involved and is not all that easy to follow. In this talk we show how to vastly simplify the derivation by use of dynamical systems theory. In particular, we show that the famous 'Cat Map', devised by Arnold [5], can be used as periodic boundary conditions for molecular dynamics simulations of planar elongational flow. Using the formalism of dynamical systems theory one can compute the required parameters of these periodic boundary conditions in several lines of algebra, compared to the pages of algebra involved in the original derivation. We also present some simulation data for linear polymer melts undergoing planar elongation and demonstrate how useful these periodic boundary conditions are in the study of polymer rheology. 1. Todd, B. D. and Daivis, P. J., 1998, Phys. Rev. Lett., 81, 1118. 2. Todd, B. D. and Daivis, P. J., 1999, Comput. Phys. Commun., 117, 191. 3. Baranyai, A. and Cummings, P. T., 1999, J. Chem. Phys., 110, 42. 4. Kraynik, A. M. and Reinelt, D. A., 1992, Int. J. Multiphase Flow. 18, 1045. 5. Arnold, V. I. and Avez, A., 1968, Ergodic problems of classical mechanics (Benjamin, New York).
Publication type: Conference poster
Research centre: Swinburne University of Technology. Faculty of Information and Communication Technologies. Centre for Molecular Simulation
Source: Paper presented at the AMSI Workshop 'Foundations and methodologies of mathematical physics and modern developments in lie theory, quantum theory and statistical mechanics', Coolangatta, Queensland, Australia, 30 November-04 December 2004
Publication year: 2004
Publisher: University of Queensland
Publisher URL: http://www.maths.uq.edu.au/cmp/Workshops/Abstracts_2004.html