Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4.1. Hydrodynamic limit I: ξ = 0
In one dimension, there is only the backscattering contribution k → −k
1
klB
= V 2
0
2µ 2
kσ
(1 + k 2 ξ 2 ) 2C1(2kσ).
(4.13)
Due to the finite support of the speckle correlation function Cd(2k) (3.12a),
there is no backscattering at all for kσ > 1, and the inverse Boltzmann mean
free path goes to zero, at least within the scope of the Born approximation.
In dimensions d ≥ 2, there are still contributions from scattering angles
between 0 and π/2 [see insets of figure 4.3(b)], thus the inverse Boltzmann
transport length does not go abruptly to zero at kσ = 1, figure 4.3(b).
Conclusion
How strongly are the Bogoliubov excitations affected by elastic scattering?
In order to justify the description in terms of plane-wave states with a
well-defined sound velocity, the Boltzmann length lB and the localization
length lloc should be much larger than the wave length λ = 2π/k. This
is easily fulfilled for the Boltzmann length (3.51). The curves shown in
figure 4.3(b) are bounded and the scaling with V 2
0 /µ 2 guarantees lB ≫ λ.
The localization lengths are equal to or even exponentially larger than the
Boltzmann length. The only thing to keep in mind is that for large values
of kσ, forward scattering events, which produce incoherence in the phase,
may occur frequently.
4.1.3. Speed of sound
We compute the relative correction of the speed of sound Λ = Re� Σ(k,ck)µ 2
2c2k2V 2
0
from the self-energy (4.9)
Λ = − 1
2 P
� d ′ d k σ
(2π) d Cd(|k − k ′ |σ)
(k · k ′ ) 2
k 2 (k ′2 − k 2 )
. (4.14)
This formula has simplified significantly compared with the principal-value
integral (3.60).
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