**Principal Investigator**
Denis J Evans **Project **s02

Research School of Chemistry **Machine **VP

**Co-Investigators** Debra
J Searles and Karl P Travis

Research School of Chemistry

**Calculation of the Stability of Phase Space Trajectories
using Molecular Dynamic Simulations**

We derived an expression that explains why it is so difficult to find initial microstates that will at long times, under the influence of an external dissipative field and a thermostat, lead to the Second Law violating nonequilibrium steady states. To verify the derivation, nonequilibrium molecular dynamics simulations of a system under the influence of shear were carried out. Computer simulations were carried out to verify the predictions of nonlinear response theory for classical systems subject to dissipative external fields. The Kawasaki normalisation factor, used in the Kawasaki nonlinear response expression, is shown to be unity. The influence of Lyapunov instability on the lifetimes of antisteady states was investigated using nonequilibrium molecular dynamics simulations.

**What are the basic questions addressed?**

Why, when a dissipative field is applied to a system at equilibrium and the system is thermostatted, do we overwhelmingly observe trajectories that satisfy the Second Law of Thermodynamics? Can the predictions of nonlinear response theory be verified using computer simulations? Does the Kawasaki normalisation factor differ from unity? What factors contribute to the time for which a trajectory can be reversed?

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

The expression relating the ratio of the probability of observing the Second Law violating and obeying trajectories was verified using molecular dynamics simulations of a system under the influence of shear. Proof that the Kawasaki normalisation factor is unity in an irreversible steady state relies on the fact that the distribution of values of time averaged fluxes becomes Gaussian as the length of time averaging becomes large, as expected from the central limit theorem and a Green-Kubo type expression exists for the standard deviation of the distribution. Simulations were carried out which verified that these assumptions are valid. For the first time, we were able to provide numerical evidence that the predictions of nonlinear response theory agree with the directly measured nonlinear response. The Kawasaki and the Transient Time Correlation function expressions for nonlinear response were found to be in agreement with the directly measured nonlinear response for an autonomous system of just two particles.

The lifetime of antisteady states was found to be inversely proportional to the smallest Lyapunov exponent of the steady state system and proportional to the logarithm of the trajectory error. These results were verified using nonequilibrium molecular dynamics simulations. further studies of the stability of phase space trajectories are currently being extended by investigation of the Lyapunov exponents of model systems such as hard disks.

**What computational techniques are used and why
is a supercomputer required?**

Equilibrium and nonequilibrium molecular dynamics simulations are used and are being developed. A supercomputer is required to obtain statistically valid data for small systems and due to large system size requirements.

**Publications**

*Steady states, invariant measures and response
theory*, D.J. Evans, and D.J. Searles,
Physical Review E**, 52**, 5839-5848 (1995).

*Computer simulation algorithms for molecules undergoing
planar Couette flow: An NEMD study,* K.P.
Travis, P.J. Daivis, and D.J. Evans, Journal of Chemical Physics,
**103**, 1109-1118 (1995).

*Thermostats for molecular fluids undergoing shear
flow: Application to liquid chlorine,*
K.P. Travis, P.J. Daivis, and D.J. Evans, Journal of Chemical
Physics,** 103**, 10638-10652 (1995).