Calculation of the Stability of Phase Space Trajectories using Molecular Dynamics Simulations

Principal Investigator

Denis J. Evans

Research school of Chemistry

The Lyapunov exponents of a liquid system are a measure of the dynamical stability of a system, and can be related to transport properties of the liquid such as viscosity and thermal conductivity. To date, the behaviour of the Lyapunov spectrum as the thermodynamic limit is approached is unknown. Therefore studies into this behaviour are important.

The nonequilibrium response of a classical many body systems for constant external fields can be found using either the transient time-correlation function (TTCF) theory or Kawasaki theory. We derived the first theory for the nonlinear response in time-dependent fields, which is a generalisation of the TTCF formalism. This new theory has been tested on the colour conductivity system in an oscillatory colour field and on a system undergoing steady shear.

The basic questions addressed by this work are:

How does the maximum Lyapunov exponent behave in the thermodyamic limit? What is the behaviour of the Lyapunov spectrum of a liquid system as the thermodynamic limit is approached?

What is the general expression for the response in a time-dependent driving field? In which circumstances is it more efficient to calculate the response using this general TTCF expression, and when is it better to calculate the response using direct molecular dynamics simulation?


Debra J. Searles

Janka Petravic

Research School of Chemistry


s02 - VPP, PC

What are the results to date and the future of the work?

Preliminary results suggest that the maximum Lyapunov exponent diverges logarithmically in the thermodynamic limit. Studies on the isothermal compressibility and the thermal conductivity of a Lennard-Jones fluid at the critical point have been carried out. Enhancement of these properties has been observed.

The general response theory for nonautonomous systems requires a concept of extended phase space, with an additional phase space coordinate characterising the external field. The theory has been found to be more efficient than direct simulation in obtaining the time dependent response of phase functions for all investigated field amplitudes, both when the response has a linear component and when it is entirely nonlinear.

In future we would like to apply this method to a range of time dependent systems, such as the shear flow system with a shear rate which is a combination of a constant and oscillatory component, and oscillatory colour field of different frequencies, as well as to apply the extended phase space method to generalise the Kawasaki response formula.

What computational techniques are used?

Equilibrium and nonequilibrium molecular dynamics simulation methods are being used and are developed. Supercomputers are required to obtain statistically valid data for small systems and due to large system size requirements.


Petravic, J., Evans, D. J., Nonlinear response for time dependent external fields, Phys. Rev. Lett. 78, 1997, 1199-1202.

Evans, D.J. and Searles, D.J., Causality and response theory, In Proceedings of 1st Tohwa University International Meeting on Statistical Physics, Fukuoka, Japan, November 1995 (Ed. M. Tokuyama, Kyoto Institute for Theoretical Physics), Butsusei Kenkyu, 66(3), 452454 (1996).

Evans, D.J. and Searles, D.J., Causality, response theory and the second law of thermodynamics, Physical Review E, 53, 5808-5815 (1996).

Searles, D.J. and Evans, D.J., On the lifetimes of antisteady states, Australian Journal of Physics, 49, 39-49 (1996).

Searles, D. J., Evans, D. J. and Isbister, D. J. The number dependence of the maximum Lyapunov exponent , Physica A, accepted for publication (1996).

- Appendix A