An alternative approach to nonlinear response theory was pioneered by Yamada and Kawasaki (1967). Rather than developing power series expansions about F_e=0 they derived a closed expression for the perturbed distribution function. The power of their method was demonstrated in a series of papers in which Kawasaki first predicted the non-analyticity of the shear viscosity with respect to strain rate (Kawasaki and Gunton, 1973, and Yamada and Kawasaki, 1975). This work predates the first observation of these effects in computer simulations. The simplest application of the Kawasaki method is to consider the adiabatic response of a canonical ensemble of N-particle systems to a steady applied field F_e.

The Liouville equation for this system is

(7.8)

The Liouvillean appearing in this equation is the field dependent Liouvillean defined by the equations of motion, (7.1). Equation (7.8) has the formal solution,

. (7.9)

For simplicity we take the initial distribution function f(0), to be canonical, so that f(t) becomes

(7.10)

The adiabatic distribution function propagator is the Hermitian conjugate of the phase variable propagator, so in this case exp[-iLt] is the negative-time phase variable propagator, (exp[-iL(-t)]). It operates on the phase variable in the numerator, moving time backwards in the presence of the applied field. This implies that

(7.11)

Formally the f-propagator leaves the denominator invariant since it is not a phase variable. The phasedependence of the denominator has been integrated out. However since the distribution function must be normalised, we can obviously also write,

(7.12)

This equation is an explicitly normalised version of (7.11) and we will have more to say concerning the relations between the so-called bare Kawasaki form, (7.11), and the renormalized Kawasaki form, (7.12), for the distribution function in §7.7. In Kawasaki’s original papers he referred only to the bare Kawasaki form, (7.11).

Using the equations of motion (7.1) one can write the time derivative of H₀ as the product of a phase variable J(Γ) and the magnitude of the perturbing external field, F_e.

(7.13)

For the specific case of planar Couette flow, we saw in §6.2 that Ḣ₀^ad is the product of the strain rate, the shear stress and the system volume, -γP_xyV and thus in the absence of a thermostat we can write,

(7.14)

The bare form for the perturbed distribution function at time t is then

(7.15)

It is important to remember that the generation of P_xy(-s) from P_xy(0) is controlled by the field-dependent equations of motion.

A major problem with this approach is that in an adiabatic system the applied field will cause the system to heat up. This process continues indefinitely and a steady state can never be reached. What is surprising is that when the effects of a thermostat are included, the formal expression for the N-particle distribution function remains unaltered, the only difference being that thermostatted, field-dependent dynamics must be used to generate H₀(-t) from H₀(0). This is the next result we shall derive.

Consider an isokinetic ensemble of N-particle systems subject to an applied field. We will assume field dependent, Gaussian isokinetic equations of motion, (5.3.1). The f-Liouvillean therefore contains an extra thermostatting term. It is convenient to write the Liouville equation in operator form

(7.16)

The operator iL is the f-Liouvillean, and iL is the p-Liouvillean. The term Λ, is

(7.17)

is the phase space compression factor (§3.3). The formal solution of the Liouville equation is given by

(7.18)

In the thermostatted case the p-propagator is no longer the Hermitian conjugate of the f-propagator.

We will use the Dyson decomposition derived §3.6, to relate thermostatted p- and f-propagators. We assume that the both p-Liouvilleans have no explicit time dependence. We make a crucial observation, namely that the phase space compression factor Λ, is a phase variable rather than an operator. Taking the reference Liouvillean, to be the adjoint of iL* we find

(7.19)

Repeated application of the Dyson decomposition to exp[-iLs-Λs] on the right hand side gives

(7.20)

In deriving the second line of this equation we use the fact that for any phase variable B, exp[-iLs]B=B(-s)exp[-iLs]. Substituting (7.20) into (7.18) and choosing, f(0)=f_T(0)=δ(K-K₀)exp(-βΦ)∕Z(β), we obtain

(7.21)

If we change variables in the integral of the phase space compression factor and calculate Φ(-t) from its value at time zero we obtain,

(7.22)

We know that for the isokinetic distribution, β=3N∕2K (see §5.2). Since under the isokinetic equations of motion, K is a constant of the motion, we can prove from (5.3.1), that,

(7.23)

If AIΓ is satisfied the dissipative flux J is defined by equation (7.13). Substituting (7.23) into (7.22) we find that the bare form of the thermostatted Kawasaki distribution function can be written as,

(7.24)

Formally this equation is identical to the adiabatic response (7.15). This is in spite of the fact that the thermostat changes the equations of motion. The adiabatic and thermostatted forms are identical because the changes caused by the thermostat to the dissipation (Ḣ₀), are exactly cancelled by the changes caused by the thermostat to the form of the Liouville equation. This observation was first made by Morriss and Evans (1985). Clearly one can renormalize the thermostatted form of the Kawasaki distribution function giving (7.25), as the renormalized form of the isokinetic Kawasaki distribution function,

(7.25)

As we will see, the renormalized Kawasaki distribution function is very useful for deriving relations between steady state fluctuations and derivatives of steady state phase averages. However, it is not useful for computing nonequilibrium averages themselves. This is because it involves averaging exponentials of integrals which are extensive. We will now turn to an alternative approach to this problem.