In this section, a derivation of an expression for phase averages
as they evolve after an external field is switched on is given. At the
time t=0, the distribution function is given by the equilibrium
distribution, 
.
Consider the impact of a steady external field, 
, on the
distribution. A discussion for the case where the external field
has an arbitrary time dependence is much more complex [Evans
and Morriss
[1988]]
, and will not be given
here. The external field 
is switched on at t=0, and
changes the dynamics to
where 
and 
are phase variables determined by the
system under study.
In this derivation we will take the thermostatting to be Nosé-Hoover,
so the equilibrium distribution will be 
. The
same argument can also be applied to the Gaussian thermostat, or to any other
type, with identical results.
In writing these equations, we are assuming that the momenta are
measured with respect to the local streaming velocity of the
fluid, hence the term `peculiar momenta'. At low Reynolds number,
this presents no major difficulties, but in the turbulent flow
regime new methods have to be applied [Evans and Morriss
[1986]] .
We assume that the external force 
is properly conservative,
so that in the absence of thermostatting 
,
Liouville's theorem holds. This assumption is known as adiabatic
incompressibility of phase space, or AI
for short. We can
of course pursue the theory without invoking this assumption, but
it has proven unnecessary to do so. With AI
, we can compute
the phase space compressibility
from (
):
The time evolution of 
from its initial state
can now be computed using Morriss's
lemma, [Morriss and Evans
[1985]]
which states:
The proof of this uses Dyson's equation [see Evans and Standish [1990]] :
where 
.
The time evolution is given by
The exponent 
can be expressed in terms of its time
derivative:
Upon substituting this, we get
Now we can write
where 
is the rate of change of internal energy
with the thermostatting turned off. This is related to
dissipative flux 
through
Substituting this into () gives
This is the Kawasaki distribution function [Yamada and Kawasaki [1967], Morriss and Evans
[1985]]
, which describes the phase space
distribution that has evolved under the influence of a
dissipative force 
, and thermostatting has been used to
guarantee a steady state. According to the Schrödinger picture,
(), we can evaluate Kawasaki phase averages for
a typical phase variable B:
where 
means average over phase space
distribution 
. The time evolution implicit in
() is generated by the full, field
dependent, thermostatted equations of motion ().
In the case of the Gaussian thermostat, only equations ()
through () differ. The corresponding equations are
The time derivative of 
is similarly related to the adiabatic heating of
the system:
Upon substituting this into (), one obtains the
Kawasaki distribution:
The phase average () is difficult to
work with owing to the extensive nature of the argument to the
exponential. We can cast this phase average into an easier form
to work with by differentiating ()
with respect to time:
Now the average dissipation flux at t=0 is zero, so we get the Transient Time Correlation Function
This relation is exact, regardless of the magnitude of the
external field 
.
