As a simple model for disorder we use the potential produced by a
uniform random distribution of quenched impurities. The random
external potential is then given by
(2.1)
where is the number of impurities
present in the volume
located at the fixed positions
and
is a two-body potential which describes the
particle-impurity interaction.
This model of disorder is particularly convenient for two reasons:
it can be easily treated analytically within the Bogoliubov
theory of a dilute Bose gas
can be easily implemented in a numerical simulation.
If the gas of impurities is dilute, as it is the case in the
``weak'' disorder regime which is of interest here, the
particle-impurity interaction potential can be replaced by a
pseudopotential
(2.2)
The coupling constant is defined by the -wave scattering
length of the particle-impurity collision process
(2.3)
here is the mass of the scattering particle, since the mass of
the impurity is taken to be infinite (quenched impurities). In this
case the particle-impurity reduced mass is twice as large as the
corresponding particle-particle reduced mass. This explains the
factor two difference between (1.2) and (2.3).
The important quantities which describe the statistical properties
of the random external potential are the mean value
(2.4)
and the correlation function
,
where denotes the Fourier component
(2.5)
Here
means average over disorder
configurations.
For our random external potential (2.2) the correlation
function can be rewritten as
By assuming that the impurities have a uniform distribution
the mean value of the random potential is given by
(2.6)
while
for . It can be
easily shown that the correlation function becomes
(2.7)
where
is the density of impurities and
is the concentration of impurities. Eq.
(2.7) implies that the external potential is treated as a
short correlated white noise in momentum space with amplitude
proportional to .
The independent parameters that describe the properties of
the system are the following