Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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1<br />
kls<br />
µ 2<br />
V 2<br />
0<br />
4. Disorder—Results and Limiting Cases<br />
2.0<br />
1.5<br />
1.0<br />
0.5<br />
d = 1<br />
d = 2<br />
d = 3<br />
0.0<br />
0.0 0.5 1.0 1.5<br />
kσ<br />
2.0 2.5<br />
(a)<br />
1<br />
klB<br />
µ 2<br />
V 2<br />
0<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
d = 1<br />
d = 2<br />
d = 3<br />
0.0<br />
0.0 0.5 1.0 1.5 2.0<br />
kσ<br />
2.5 3.0 3.5<br />
Figure 4.3.: (a) inverse mean free path (4.11) and (b) inverse Boltzmann transport length<br />
(3.51) in the hydrodynamic limit ξ = 0 for a speckle disorder (3.12). The insets show<br />
schematically the angular dependence <strong>of</strong> elastic scattering.<br />
Equivalently, this result is obtained from (3.50) with the elastic envelope<br />
function A 2 = cos 2 (θ) from section 2.4. This result is a function <strong>of</strong> kσ and<br />
is shown in figure 4.3(a).<br />
For small values <strong>of</strong> kσ, the correlator Cd can be approximated by Cd(0).<br />
Then, the d-dimensional angular average <strong>of</strong> cos 2 (θ) gives 1/d. This result is<br />
due to the “concentration <strong>of</strong> measure phenomenon” [125]: With increasing<br />
dimension, the measure <strong>of</strong> the sphere is concentrated at the equator, where<br />
cos θ vanishes. Altogether we find<br />
1<br />
kls<br />
≈<br />
2 V0 4µ 2(kσ)d Cd(0) Sd<br />
(2π) d−1<br />
1<br />
d<br />
(b)<br />
(4.12)<br />
Thus, the inverse mean free path, measured in units <strong>of</strong> the wave length,<br />
scales like (kσ) d .<br />
For large values <strong>of</strong> kσ, the potential allows practically only forward scattering.<br />
That means, cos 2 (θ) can be evaluated as 1 and the angular integral<br />
(4.11) over the correlator Cd(2kσ sin(θ/2)) ≈ Cd(kσθ) scales like (kσ) −(d−1) ,<br />
leading to a linear scaling (kls) −1 ∝ kσ.<br />
Boltzmann transport length and localization length<br />
By adding a factor 1−cos θ under the angular integral (4.11), we obtain the<br />
inverse Boltzmann transport length (3.51), see figure 4.3(b). At large kσ,<br />
where forward scattering dominates, this suppresses 1/(klB) compared with<br />
1/(kls), see insets <strong>of</strong> figure 4.3.<br />
86