We can apply our fundamental result, equation (8.72), to a number of known special cases. In the linear regime our equation obviously becomes,
(8.73)
The notation '< .. >0' denotes an equilibrium average over the field-free thermostatted dynamics implicit in the Liouvillean, iL0. This equation is the well-known result of time dependent linear response theory, (see §5.3).
Another special case that can be examined is the time independent nonlinear response. In this circumstance the Liouvillean iL(t) is independent of time, iL, and the propagator, UR(0,t) becomes much simpler,
(8.74)
One does not need to use time ordered exponentials. In this case the response is,
(8.75)
Again all time propagation is generated by the field-dependent thermostatted Liouvillean, iL. This equation is new. As was the case for the Kawasaki form of the nonequilibrium distribution function, explicit normalisation can be easily achieved.
Comparing equation (8.75) with the following identity that can be obtained using the equivalence of the Schrödinger and Heisenberg representations, (§8.7),
(8.76)
implies that,
(8.77)
The integral (0,t), on the right hand side of the equation can be performed yielding,
(8.78)
The correctness of this equation can easily be checked by differentiation. Furthermore it is clear that this expression is just the unnormalised form of the Kawasaki distribution function (7.25).
This equation can be used to renormalize our expression for the time independent nonlinear response. Clearly
(8.79)
is an explicitly normalised distribution function. By differentiating this distribution in time and then reintegrating we find that,
(8.80)
To simplify the notation we define the brace { }s as
(8.81)
Using this definition our renormalised expression for the response is (Evans and Morriss, 1988)
(8.82)