We wish to develop relations between macroscopic transport coefficients
and microscopic averages - which hopefully may be evaluated in
simulation experiments.
Let be the Hamiltonian of the given system when it is isolated.
If we apply a weak disturbing field that couples to some
property (with ) the Hamiltonian of the perturbed
system is given by
(8.1)
Linear response theory then leads to the following first order
expression for the mean temporal change of :
(8.2)
where the average is to be taken over the unperturbed
system. Assuming a constant field switched on in the distant
past we may write this as
(8.3)
Independently, the fundamental Green-Kubo relations tell us that
for any conserved quantity the appropriate transport coefficient
is given by the equilibrium average
(8.4)
Combining this with the above equation we find that
, or
(8.5)
Thus we may determine the transport coefficient either from an
equilibrium simulation using equ. 8.4, or from a non-equilibrium
simulation with applied field using equ. 8.5
Generally the second method yields better statistics but is more prone
to nonlinearity problems (large fields); also, systems responding to an
external field must be thermostated.
Example: Consider a fluid sample of ions in an electric
field
. The charge distribution is described by
the quantity
which couples to the field
according to
(8.6)
The electrical current density is defined by
(8.7)
( ... volume).
Now consider the conductivity . It may be determined in two ways:
In an equilibrium simulation, using the Green-Kubo relation
(8.8)
In a non-equilibrium simulation, using the measured response to an applied
field
:
(8.9)
Note:
In the derivation of equ. 8.5 it is sufficient but not necessary
that the external perturbation may be formulated as an additional term in
the
Hamiltonian. It is only necessary that the equations of motion
contain perturbative terms which have to fulfill certain requirements.
Specifically, the set