The equations of motion used to generate the flux-statted dynamics in general will be of the form
The intermolecular forces are given by 
. In these
equations, 
and 
are computed by a Nosé-Hoover feedback
mechanism to keep the flux J, and the temperature
fluctuating about fixed mean values 
and 
. Specifically, we have
and
The phase variables 
and 
are are chosen so that
For example, in the case of constant colour current dynamics, 
, where 
is the charge on the ion, and
. In the case of stress-statted dynamics [Brown and Clarke [1986] , Hood et al. [1987]] , the flux to be kept constant is the xz component of the stress tensor, namely
with 
and 
given by 
and 
.
Consider an initial ensemble characterized by the distribution function
,
If we assume adiabatic incompressibility of phase space
(AI
), then the Kawasaki expression for the average of an
arbitrary phase variable, B, can be derived
[Morriss and Evans
[1985]]
Here the subscript J denotes that 
is evolved
under constant flux dynamics. The ensemble average 
is taken with respect to the distribution function 
.
By differentiating and reintegrating (
) in the usual
way, a transient time correlation function expression for the
nonequilibrium phase average is generated. Thus
in the thermodynamic limit, where 
. Since the
flux-statted propagators do not depend on the average flux 
, the
only dependence on 
in the above expression is either
explicit, or comes in indirectly through 
. By the chain rule,
. The first derivative can be evaluated by the inverse function theorem
and the second derivative is simply
So
The derivatives of () with respect to 
can be
easily evaluated around 
, and the first three are:
In comparing these results with Evans and Lynden-Bell
[1988], it should be noted that one is interested in
computing
the phase average of the force required to maintain a steady
current. This phase variable is antisymmetric with respect to a
reflection in space, whereas J is independent of position, and
so all the averages of the form 
will vanish.
Similarly, 
and all odd moments of J will
vanish. The quantity 
can be evaluated, and is
found to be 
.
Similar simplifications will also apply with the case of planar Couette flow, where one is attempting to deduce the nonlinear viscosity, defined by
where 
is the shear stress, and 
the
strain rate associated with it.
Both 
and J are antisymmetric under a reflection in
the x direction in both position and velocity space
), but
under a time reversal 
, 
is
antisymmetric, and J is symmetric. Thus the Burnett coefficients
simplify to
where 
and
with 
being the infinite frequency shear modulus [Brown and
Clarke
[1986]]
.
