Consider an equilibrium ensemble of systems, characterised by a distribution function, f₀, subject at t=0, to an external time dependent field F_e(t). We assume that the equilibrium system (t<0), has evolved under the influence of the Gaussian isokinetic Liouvillean iL₀. This Liouvillean has no explicit time dependence. The equilibrium distribution could be the canonical or the isokinetic distribution. These assumptions are summarised by the equation,

(8.52)

The equations of motion for the system can be written as,

(8.53)

Provided that the temperature can be obtained from the expression, 3Nk_BT∕2 = Σp _i ²∕2m, the term αp _i represents the Gaussian thermostat. α is chosen so that Σp _i ²∕2m is a constant of the motion.

(8.54)

The terms C,D couple the external field F_e(t) to the system. The adiabatic, unthermostatted equations of motion need not be derivable from a Hamiltonian (i.e. C,D do not have to be perfect differentials). We assume that the AIΓ holds,

(8.55)

The dissipative flux is defined in the usual way,

(8.56)

where,

(8.57)

The response of an arbitrary phase variable B(Γ) can obviously be written as,

(8.58)

In this equation iL(t) is the p-Liouvillean for the field-dependent Gaussian thermostatted dynamics, t>0. If we use the Dyson decomposition of the field-dependent p-propagator in terms of the equilibrium thermostatted propagator we find that,

(8.59)

By successive integrations we unroll U_R0 propagator onto the distribution function.

(8.60)

However U^† _R0 is the equilibrium f-propagator and by equation (8.10.1) it has no effect on the equilibrium distribution f₀.

(8.61)

We can now unroll the Liouvilleans to attack the distribution function rather than the phase variables. The result is,

(8.62)

From equation (8.52) it is obvious that it is only the operation of the field-dependent Liouvillean which needs to be considered. Provided AIΓ is satisfied, we know from (7.29, et. seq.) that,

(8.63)

For either the canonical or Gaussian isokinetic ensembles therefore,

(8.64)

Thus far the derivation has followed the same procedures used for the time dependent linear response and time independent nonlinear response. The operation of U_R(s,t) on B however, presents certain difficulties. No simple meaning can be attached to U_R(s,t) B. We can now use the Composition and the Inverse theorems to break up the incremental p-propagator U_R(s,t). Using equations (8.18),

(8.65)

Substituting this result into (8.64) we find

(8.66)

Using the Inverse theorem (8.3.1), and integrating by parts we find,

(8.67)

where after unrolling U_R ^-1(0,s) we attack B with U_R(0,t) giving B(t). As it stands the exponential in this equation has the right time ordering of a p-propagator but the argument of the exponential contains an f-Liouvillean. We obviously have some choices here. We choose to use (8.48) to rewrite the exponential in terms of a p-propagator. This equation gives

(8.68)

where

(8.69)

α(Γ,s) is the Gaussian isokinetic multiplier required to maintain a fixed kinetic energy. Substituting these results into equation (8.67), using the fact that,

(8.70)

gives,

(8.71)

or,

(8.72)

This equation is the fundamental result of this chapter. It must be remembered that all time evolution is governed by the field-dependent thermostatted equations of motion implicit in the Liouvillean, iL(t).