The nonlinear Burnett Coefficients have already been introduced in section
. Ever since the Green-Kubo formalism for calculating the linear
transport coefficients was developed, there has been interest in a
corresponding theory for the nonlinear Burnett coefficients. The discovery
of long-time tails in the velocity autocorrelation
function by Alder and Wainwright [1970] indicated that
the hydrodynamic transport coefficients do not exist in two dimensions,
but do exist in three dimensions. By applying mode-coupling theories,
Ernst et al. [1978] showed that
the relation between stress and strain rate should be 
for hard disks and 
for hard spheres, rather than the analytic form suggested by
(). This result indicates that the nonlinear
Burnett coefficients do not exist at all, so the interest has intensified
for a numerical simulation to test the mode-coupling theories.
In a recent paper by Evans and Lynden-Bell [1988],
equilibrium fluctuation expressions for inverse Burnett
coefficients were derived for the colour
conductivity problem.
The coefficients, 
, give a Taylor series representation of
a nonlinear transport coefficient L, in terms of the
thermodynamic force F.
Thus if a thermodynamic flux J is
written in terms of the coefficient's defining constitutive
relation as 
, then the Burnett coefficients are
related by 
. In order to derive
closed form expressions for the Burnett coefficients, it was
found necessary to work in the Norton ensemble, in which the
flux J, rather than the thermodynamic force F was the
independent variable. The constitutive relation in this case is
. In the
thermodynamic limit, we may write 
, and
so the non-linear Burnett coefficients can be computed by inverting the
series.
Evans and Lynden-Bell [1988] applied constant current
dynamics to
a canonical ensemble with the currents distributed about an
average current 
. This allowed the derivation of a
transient time correlation function for the non-equilibrium
phase average <F>. It was then a simple matter to compute the
derivatives of <F> with respect to the average current 
,
as the constant current propagator commutes with the derivative
operator. However, this method appeared to be limited to colour
currents, for which an appropriate canonical distribution could
be found. In this chapter, we show that this
method can be applied to the situation of an arbitrary thermodynamic flux.
This result has been reported by Standish and Evans [1990].