Bit-Reversible Version of Milne’s Fourth-Order Time-Reversible Integrator for Molecular Dynamics
Ruby Valley Research Institute
Highway Contract 60, Box 601, Ruby Valley, Nevada 89833, USA
E-mail: hooverwilliam@yahoo.com
Received:
Received: 28 June 2017; accepted: 29 June 2017; published online: 17 August 2017
DOI: 10.12921/cmst.2017.0000031
Abstract:
We point out that two of Milne’s fourth-order integrators are well-suited to bit-reversible simulations. The fourth-order method improves on the accuracy of Levesque and Verlet’s algorithm and simplifies the definition of the velocity v and energy e = (q2 + v2 )/2. (We use this one-dimensional oscillator problem as an illustration throughout this paper). Milne’s integrator is particularly useful for the analysis of Lyapunov (exponential) instability in dynamical systems, including manybody molecular dynamics. We include the details necessary to the implementation of Milne’s Algorithms.
Key words:
bit-reversible molecular dynamics, chaotic dynamical systems, Lyapunov instability
References:
[1] W. E. Milne, Numerical Calculus – Approximations, Interpo-
lation, Finite Differences, Numerical Integration, and Curve
Fitting (Princeton University Press, 1949 and 2015).
[2] D. Levesque and L. Verlet, Molecular Dynamics and Time Re-
versibility, Journal of Statistical Physics 72, 519-537 (1993).
[3] L. Verlet, ‘Computer Experiments’ on Classical Fluids. I.
Thermodynamical Properties of Lennard-Jones Molecules,
Physical Review 159, 98-103 (1967).
[4] H. A. Posch and W. G. Hoover, Large-System Phase-Space
Dimensionality Loss in Stationary Heat Flows, Physica D 187,
281-293 (2004).
[5] O. Kum and Wm. G. Hoover, Time-Reversible Continuum Me-
chanics, Journal of Statistical Physics 76, 1075-1081 (1994).
[6] B. L. Holian, Wm. G. Hoover, and H. A. Posch, Resolution of
Loschmidt’s Paradox: The Origin of Irreversible Behavior in
Reversible Atomistic Dynamics, Physical Review Letters 59,
10-13 (1987).
[7] C. Grebogi, E. Ott, and J. A. Yorke, Roundoff-Induced Peri-
odicity and the Correlation Dimension of Chaotic Attractors,
Physical Review A 38, 3688-3692 (1988).
[8] C. Dellago and Wm. G. Hoover, Finite-Precision Stationary
States At and Away from Equilibrium, Physical Review E 62,
6275-6281 (2000).
[9] M. Romero-Bastida, D. Pazó, J. M. Lopéz, and M. A. Ro-
driguez, Structure of Characteristic Lyapunov Vectors in
Anharmonic Hamiltonian Lattices, Physical Review E 82,
036205 (2010).
[10] Wm. G. Hoover and Carol G. Hoover, Time-Symmetry Break-
ing in Hamiltonian Mechanics, Computational Methods in
Science and Technology 19, 77-87 (2013).
[11] Wm. G. Hoover and C. G. Hoover, The Kharagpur Lectures
(World Scientific, Singapore, 2018, in preparation).
We point out that two of Milne’s fourth-order integrators are well-suited to bit-reversible simulations. The fourth-order method improves on the accuracy of Levesque and Verlet’s algorithm and simplifies the definition of the velocity v and energy e = (q2 + v2 )/2. (We use this one-dimensional oscillator problem as an illustration throughout this paper). Milne’s integrator is particularly useful for the analysis of Lyapunov (exponential) instability in dynamical systems, including manybody molecular dynamics. We include the details necessary to the implementation of Milne’s Algorithms.
Key words:
bit-reversible molecular dynamics, chaotic dynamical systems, Lyapunov instability
References:
[1] W. E. Milne, Numerical Calculus – Approximations, Interpo-
lation, Finite Differences, Numerical Integration, and Curve
Fitting (Princeton University Press, 1949 and 2015).
[2] D. Levesque and L. Verlet, Molecular Dynamics and Time Re-
versibility, Journal of Statistical Physics 72, 519-537 (1993).
[3] L. Verlet, ‘Computer Experiments’ on Classical Fluids. I.
Thermodynamical Properties of Lennard-Jones Molecules,
Physical Review 159, 98-103 (1967).
[4] H. A. Posch and W. G. Hoover, Large-System Phase-Space
Dimensionality Loss in Stationary Heat Flows, Physica D 187,
281-293 (2004).
[5] O. Kum and Wm. G. Hoover, Time-Reversible Continuum Me-
chanics, Journal of Statistical Physics 76, 1075-1081 (1994).
[6] B. L. Holian, Wm. G. Hoover, and H. A. Posch, Resolution of
Loschmidt’s Paradox: The Origin of Irreversible Behavior in
Reversible Atomistic Dynamics, Physical Review Letters 59,
10-13 (1987).
[7] C. Grebogi, E. Ott, and J. A. Yorke, Roundoff-Induced Peri-
odicity and the Correlation Dimension of Chaotic Attractors,
Physical Review A 38, 3688-3692 (1988).
[8] C. Dellago and Wm. G. Hoover, Finite-Precision Stationary
States At and Away from Equilibrium, Physical Review E 62,
6275-6281 (2000).
[9] M. Romero-Bastida, D. Pazó, J. M. Lopéz, and M. A. Ro-
driguez, Structure of Characteristic Lyapunov Vectors in
Anharmonic Hamiltonian Lattices, Physical Review E 82,
036205 (2010).
[10] Wm. G. Hoover and Carol G. Hoover, Time-Symmetry Break-
ing in Hamiltonian Mechanics, Computational Methods in
Science and Technology 19, 77-87 (2013).
[11] Wm. G. Hoover and C. G. Hoover, The Kharagpur Lectures
(World Scientific, Singapore, 2018, in preparation).
