We have described a consistent formalism for the nonlinear response of many-body systems to time dependent external perturbations. This theory reduces to the standard results of linear response theory in the linear regime and can be used to derive the Kawasaki form of the time-independent nonlinear response. It also is easy to show that our results lead to the transient time correlation function expressions for the time-independent nonlinear case.

If we consider equation (8.64) in the time-independent case and remember that,

(8.83)

then we can see immediately,

(8.84)

This is the standard transient time correlation function expression for the nonlinear response, (7.33).

It may be thought that we have complete freedom to move between the various forms for the nonlinear response: the Kawasaki form equation (8.78), the transient correlation function expression equation (8.84) and the new formulation developed in this chapter, equation (8.82). These various formulations can be characterised by noting the times at which the test variable B and the dissipative flux J, are evaluated. In the Kawasaki form B is evaluated at time zero, in the transient correlation approach J is evaluated at time zero, and in the new form developed in this paper, B is evaluated at time t. These manipulations are essentially trivial for the linear response.

As we have shown, these forms are all equivalent for the nonlinear response to time-independent external fields. However for the time-dependent nonlinear case only our new form equation (8.82), seems to be valid. One can develop a Kawasaki version of the nonlinear response to time-dependent fields but it is found that the resulting expression is not very useful. It, like the corresponding transient correlation form, involves convolutions of incremental propagators, Liouvilleans and phase variables which have no directly interpretable meaning. None of the operators in the convolution chains commute with one another and the resulting expressions are intractable and formal.