Our simulations indicate that the oscillations are of nonlinear nature and that the lattice symmetry has no influence on the averages of the phase functions in equilibrium, although the phase space distribution function for different lattices is different. The Kawasaki formula is a closed expression for the evolution of the distribution function in a constant external field. Its derivation relies on the time independence of Liouville operators. The extended phase space generalisations of Liouville operators have no explicit time dependence even when the equations of motion are non-autonomous. This makes this computer simulation very time consuming.
Improved efficiency is obtained if the distribution function is renormalised at each timestep.
For the first time we have developed a time-dependent nonlinear response theory suitable for computer simulations. Our approach is based on the definition of the extended phase space in which the system becomes autonomous. In the nonlinear case the time integration of the transient time-correlation functions from the whole extended phase space are required in order to obtain the time dependent response. The theory reduces to the known limits in the cases of time-dependent linear response and time independent nonlinear response. The algorithm based on this time-dependent TTCF formalism has been tested on the simple system of two disks in a sinusoidal colour field.