Assoc. Prof. Dennis Isbister - Staff Page - School of Physical, Environmental and Mathematical Sciences (PEMS)

Research Interests:
Liquid state physics, equilibrium and nonequilibrium statistical mechanics, NMR in solid state physics, NQR in liquids, surface physics, computer algebra processors.

Biography

Teaching

Dennis teaches in the School of Physical, Environmental and Mathematical Sciences at the University of New South Wales @ADFA.

Research

Molecular Dynamics: Theory and Simulations

Physical, chemical and biological systems are collectively characterised by their constantly changing states, hopefully a minimum number of which can be astutely chosen to be followed as time changes. For mathematicians and physicists, the optimal set of such independent variables are the positions and speeds (actually, velocities or momenta) of the molecules that make up such an ideal system that is proposed to be modelled or studied. Through a careful, usually numerical, study of the equations responsible for these changes, the so-called equations of motion, the underlying model can be solved for its evolution in time. This picture of completely determining the future possible states of a physical or biological system given only the equations of motion of the molecules and their exact initial starting values, was initially proposed by Lagrange several hundred years ago.

The basic ingredients in the formulation of our knowledge of the molecular motion has not substantially changed since Newton's days: given the force between molecules (planets) Newton derived his famous equations of motion for the changes in velocities of all molecules(planets) moving under the force law. In this context the evolution of the planetary system can be followed deterministically from the laws of gravity along with sufficient information of the initial states of the universal molecules (the astronomical bodies themselves).

Over the last twenty years advances in computer technology have seen the implementation of these principles on laptop computers take the form of actual visualisation of the pathways/trajectories that molecules (planets) would follow through solving Newton 's equations of motion. It turns out that a few hundred molecules can be collectively analyzed in a simulation box through computer simulation of molecular dynamics. Such computer simulations or experiments have shown excellent agreement with experimental data and thereby validated the technique of molecular dynamics simulation. Accompanying such a technique has been its extension to far more complicated molecules (for instance DNA) which require correspondingly far more memory and extended the limits of available machines.

One of the more exciting areas of research in the development of simulation technology has centred on the numerical aspects of the algorithms involved in the molecular dynamics mentioned above. Earlier it was thought that the correct approach capitalised on the smallness of errors generated in the numerical solution of Newton 's equations. Incidentally this view precluded the initial algorithm separately used by Verlet and Rahman, the pioneers of molecular dynamics methodology. For many years this original algorithm was considered too simple to be used without further refinement to its accuracy. However a dramatic change has taken over the simulation field in the past twelve months. As the interest for higher accuracy and longer runs (time wise) is dictated from more exotic physical systems, the previously used accurate algorithms are too expensive to accommodate on today's computers.

A typical problem comes in chaos theory and its modelling through molecular dynamics. In systems exhibiting chaos, the chaotic motions are actually manifested in the exponentially fast growth in difference between two almost identically located initial positions for the system's configuration. Depending on the complexity of the molecular system, there are a set of these exponents that measure such sensitivity to initial conditions of the whole system, these being the generic signatures of chaos, namely the Lyapunov exponents. The accurate determination and understanding of the interplay between the microscopic dynamics and these signatures of chaos underlies all physical phenomena. For example, the accurate determination of the trajectories to be followed by interplanetary probes are paramount for their success. Exceedingly small errors in their predicted trajectory/path through space over very long simulation times could be disastrous for the expected rendezvous with the flyby path of the target planet system. In the language of the chaos simulators the dynamical equations of motion are highly sensitive to initial conditions and have positive Lyapunov exponents.

Another characteristic associated with Newton 's equations of motion has been the lack of variation in the total energy of the motion (although not strictly true for space flights). For systems displaying constant energy, an equivalent but easier to use set of equations for the motion is given by Hamilton . Systems obeying Hamiltonian equations of motion mathematically and exactly conserve energy: the actual numerical results from any simulation however show finite yet exceedingly small deviations from this conservation rule due to inaccuracies in the trajectories being generated by our highly accurate algorithms that were supposed to guarantee sufficiently accurate evolution of the system where natural variables like energy are exactly conserved ( exactly here meaning to "within the precision allowed by the computer being used in the actual calculation").

Hamiltonian systems however are expected to preserve more than just their energy: a system's energy is a global concept. Yet all molecules move in such a way as to make their individual contributions to various plaited shapes constants of the motion, in addition to the energy being globally constant throughout the motion. A dot of ever-so-slightly displaced systems (with the same energy yet infinitesimally different starting conditions) moves about the original trajectory in such a way as to preserve its original size while maybe losing itsshape. This conservation of volume is reflected in the symmetry of the set of Lyapunov exponents, the so-called Conjugate Pairing Rule. The Lyapunov exponents can be ordered in pairs according to their magnitudes so that the largest and least add to zero, the second largest and the second least largest add to zero, etc...Pictorially the Lyapunov exponents resemble a horseshoe with a symmetry axis of zero. An easy-to-read proof of this expected symmetry on the Lyapunov exponents was included at the July Molecular Dynamics conference at Lyon in July by Evans, Searles and Isbister. As expected, the traditionally used algorithms often verified this symmetry of the Lyapunov spectrum, albeit from very long and expensive computer simulations. These algorithms were not designed to fully capitalise on the symmetry properties of the Lyapunov exponents. In order to reflect these symmetry characteristics of Hamiltonian flow it is necessary to use corresponding Hamiltonian/hypervolume preserving algorithms. The basic process of following the "intimate intergrowth between two nearly identical trajectories" has seen the coining of the label of symplectic 1 algorithm for use in these Hamiltonian systems.

Using these symplectic algorithms to solve equations of motion has seen immediate success in the traditional and more mundane applications of molecular dynamics to simulate ordinary properties of fluids like pressure, isothermal compressibility and heat capacities. However the exciting use of these genuinely new and time saving measures is yet to come: one of these involves the determination of the signatures of chaos for which they were originally optimised. This area is not yet fully appreciated by other simulators at large.

We propose to firstly investigate their use in simple Hamiltonian dynamics to calculate the complete set of Lyapunov exponents for model systems of fluids at equilibrium. Does the conjugate pairing rule benefit significantly from the symplectic algorithms approach? Does the energy remain constant as well as the preservation of the multi volume aspects of microscopic dynamics? If the theoretical predictions of Marsden are correct and this is answered negatively, then is there a compromise algorithm that ensures energy conservation as well as stretching and folding nearby trajectories equally?

We have already anticipated their true worth by generalising their use to chaotic systems other than Hamiltonian, principally in the simulation of modelling shear flow in fluids between two hard walls. Unfortunately such dissipative systems are definitely nonHamiltonian and therefore it is not entirely obvious that the symplectic algorithms can be extended to such systems. The structure of the Lyapunov horseshoe can be fully investigated only after the justification of the extended algorithms for these systems is done in a theoretical and rigorous manner.

Some Recent Results : The Hooke's Law Pendulum

Consider the Hooke's law Pendulum shown below with mass 1 and spring constant 4, in a gravitational field of unit strength and with minimum spring potential energy at a length of unity. The Hamiltonian for this system is

H(x,y,p l ,p y ) = f (r) + mgh + p 2 /2
&n bsp; = 2(r-1) 2 + y + (p 2 l + p 2 y )/2 .

where the pendulum length is r = Ö x 2 + y 2 and the y coordinate represents the height h. This simple system is chaotic. Just how the simple pendulum becomes chaotic is the subject of this study. William G. Hoover discusses the Hookean pendulum on p303 of his book, Computational Statistical Mechanics,Elsevier, 1991. We investigated the effect of changing the initial conditions in phase space on the same energy surface, E=1.

Grants and Awards

Select Publications

Manchanda, R., Sood, R., Grey, D., Isbister, D.J., 2008, Transport and recombination of electrons in a high pressure proportional counter using different gas mixtures, Nuclear Instruments and Methods in Physics Research, 595(3), 605-615.

Pan, G., Ely, J., McCabe, C., Isbister, D. J., 2005, Operator splitting algorithm for isokinetic SLLOD molecular dynamics, Journal of Chemical Physics, 122(9).

Woodward, C., Campion, M., Isbister, D.J., 2002, Kinetics of a two-dimensional lattice gas mixture in a color field, Journal of Chemical Physics, 116, 2983-2990.

Zhang, F., Isbister, D. J., Evans, D., 2001, Nonequilibrium Molecular Dynamics Studies of Heat Flow in One-Dimensional Systems, International Journal of Thermophysics, 22, 135-147.

Ratanapisit, J., Isbister, D.J., Ely, J. 2001, Transport properties of fluids: symplectic integrators and their usefulness, Fluid Phase Equilibria, 351-361.

Isbister, D.J., Zhang, F., Evans, D., 2001, Multiple nonequilibrium steady states for one-dimensional heat flow, Physical Review E, 64, 0211021-0211025.

Hoover, W.M., Hoover, C., Isbister, D.J., 2001, Chaos, ergodic convergence, and fractal instability for a thermostated canonical harmonic oscillator, Physical Review E, 63, 1-5.

Zhang, F., Isbister, D. J., Evans, D. J ., 2000, Non-equilibrium molecular dynamics simulations of heat flow in one-dimensional lattices, Physical Review E, 61(4), 3541-3546.

Publication Summary (1993-1998)

B. C. Sanctuary, D.J. Isbister, X. Wang, L. Lu, X. Yang, Series Expansion for Composite Pulses in NMR, J. Mag. Reson., 96, 229-236, (1992).

L.N Shakhmuratova, D.J. Isbister and D.H. Chaplin, Oscillatory free-induction decay signals in the angular distribution of nuclear radiation in ferromagnetic materials under strong inhomogeneous broadening, Hyperfine Interactions, 80, 1167-1172, (1993).

W.D. Hutchison, N. Yazidjoglou, D.J. Isbister and D.H. Chaplin, L. Shakhmuratova, Pulsed NMRON oscillatory free-induction decay signals in the angular distribution of gamma radiation from plated Co Fe , Hyperfine Interactions, 80, 1173-1178, (1993).

T.J. McKenna, D.J. Isbister, S.J. Campbell and D.H. Chaplin, Ferromagnetic domain size of Tb, J. Mag. and Magnet. Materials, 125, 319-322, (1993).

D.J. Isbister, Quantum Mechanics and Maple: The reduced rotation matrix elements

, 3-j and 6-j symbols, Maple Technical Newsletters, 1, 55-63, (1995).

J. Petravic, D.J. Isbister and G.P. Morriss, Correlation dimension of the sheared hard disc Lorentz gas, J. Stat. Phys., 76, 1045-1063, (1994).

S.Z. Ageev, D.J. Isbister and B.C. Sanctuary, Composite pulses in Nuclear Quadruple Resonance, Mol. Phys., 83, 193-210. (1994).

D.J. Isbister, M.S. Krishnan and B.C. Sanctuary, Use of Computer Algebra for the Study of Quadruple Spin Systems, Mol. Phys., 86, 1517-1535, (1995).

L.N Shakhmuratova, D.H. Chaplin, W.D. Hutchison, D.J. Isbister and N. Yazidjoglou, The Oscillatory Free Induction Decay in g detected Pulsed NMRON, Proc.Ampere Conference on Mag.Res. and Related Phenomena, 1 , 376-377, (1994).

L.N Shakhmuratova, D.J. Isbister and D.H. Chaplin, Oscillatory Free Induction Decay signal in the angular distribution of g radiation from Oriented Nuclei in Ferromagnetic Materials, Bull. Acad. Sci Russia, 58, 60-63, (1994).

J. Petravic and D.J. Isbister, Pressure Tensor of the hard disk Lorentz gas, Phys. Rev., E 51, 4309-4318, (1995).

D.H. Chaplin, L.N Shakhmuratova, W.D. Hutchison, and D.J. Isbister, Oscillatory free induction decay in the angular distribution of g radiation from oriented nuclei-the extension of the theorem on coherent transients for the higher rank statistical tensors, Hyper. inter, 1C, 605-608 (1995).

D.H. Chaplin, L.N Shakhmuratova, W.D. Hutchison, and D.J. Isbister, Two-pulse stimulated echo in the angular distribution of g radiation from oriented nuclei., Hyper. inter, 1C, 6557-560 (1995).

D.H. Chaplin, L.N Shakhmuratova, W.D. Hutchison, and D.J. Isbister, Oscillatory free induction decay in NMRON: The extension of the theorem on coherent transients for the higher rank statistical tensors, Bull. Mag. Res, 17 , 230-231 (1995).

G.P. Morriss, C.P. Dettmann and D.J. Isbister, The Field Dependence of Lyapunov Exponents for NonEquilibrium Systems, Phys. Rev . 54, 4748-4754, (1996).

L.N Shakhmuratova, W.D. Hutchison, D.J. Isbister and D.H. Chaplin, Observation of a new coherent transient in NMR: two pulse nutational stimulated echo in the angular distribution of g radiation from oriented nuclei, Magnetic reson. and related phenomena, 28th Ampere Congress, 205-206, (1996).

L.N Shakhmuratova, D.K. Fowler, D.J. Isbister and D.H. Chaplin, Single pulse nuclear spin echo from multidomain ferromagnets, Magnetic reson. and related phenomena, 28th Ampere Congress, 499-500, (1996).

L.N Shakhmuratova, D.K. Fowler, D.J. Isbister and D.H. Chaplin, The formation of multiple spin echoes in ferromagnets: theory and experiment, Magnetic reson. and related phenomena, 28th Ampere Congress, 501-502, (1996).

L.N Shakhmuratova, W.D. Hutchison, D.J. Isbister and D.H. Chaplin, Observation of a new coherent transient in NMR: nutational two pulse stimulated echo in the angular distribution of g radiation from oriented nuclei, Applied Magnetic Resonance, 12, 103-117 (1997).

D.J. Searles, D.J. Evans and D.J. Isbister, The number dependence of the Maximum Lyapunov Exponent, Physica A240, 96-104 (1997).

D.J. Isbister, D.J. Searles and D.J. Evans, Symplectic properties of algorithms and simulation methods, Physica A240, 105-114 (1997).

D.J. Searles , D.J. Isbister, and D.J. Evans, NonEquilibrium Molecular Dynamics Integrators Using Maple, Mathematics and Computers in Simulation, 00-15, (1997).

D.J. Searles, D.J. Evans and D.J. Isbister, The Conjugate Pairing Rule for Non-Hamiltonian Systems, accepted for Chaos, 00-21 (1998).

Talks presented at Recent Conferences:

Complexity and NonEquilibrium Molecular Dynamics CECAM, Lyons, France (1996)

Statistical Mechanics and Mathematical Physics University of NSW, Kensington (1997)

Probabilistic and Thermodynamic aspects of Nonlinear Dynamics Brussells, (July, 1998)

School of Physical, Environmental and Mathematical Sciences

Associate Professor Dennis Isbister