The two time correlation functions in equations (
) to ()
are averaged over an ensemble of flux-statted trajectories. To calculate
this, one would first need to generate the distribution 
, using
molecular dynamics simulation or Monte Carlo methods. Once an initial
phase space configuration 
was produced with probability
, then its evolution under the flux-statted
equations of motion () needs to be followed. If
we wish to follow these trajectories for 
time steps, then we require
timesteps to average over 
trajectories. By
contrast, the Green-Kubo expressions for the linear transport coefficients
involve correlation functions whose propagators are independent of the
initial state of the trajectory. We can therefore form the average as
with 
being the timestep. This clearly requires only 
timesteps, and so is more efficient by roughly a factor of 
.
We shall see in this section that the ensemble averages in equations
() to () can be calculated from a single equilibrium
trajectory, with the consequent improvement in efficiency. Write the
flux-statted propagator explicitly as 
:
Now use the Dyson equation () to expand 
in terms of
:
For 
being the flux-statted Liouvillean with Nosé-Hoover feedback
mechanism, the difference in operators is contained only in the equation
of motion for 
:
Now 
is just another intensive phase variable, so we may write the series
() as
Using the result of Appendix , the higher terms will vanish
in the thermodynamic limit, and so we may write
Substituting this into () reveals
Thus the time correlation functions of (...) are expressed
in terms of an average over a single trajectory, provided that the
flux-statting propagator generates 
. This is the case for the
Nosé-Hoover
feedback mechanism discussed, for the case 
.
