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are expanded into its series up to second order in k:
v 2 k = − 1
2 +
= − 1
2 +
2
µ + 2t(1 + k2
2 + O(k4 ))
µ + 2t + tk2 + O(k4 2 )
µ + 2t(1 + k2
2 + O(k4 ))
2|k| 4t 2 + 2µt + O(k 2 ) ∼
|k|≪1
− (µ + 2t) 2
This means, that even at zero temperature the fluctuations will diverge as
the thermodynamic limit is made since the sum over all discrete k states
becomes a integral which diverges. This behavior is illustrated in figure 3.1.
That this happens is not surprising since BEC can only take place in systems
with more than one dimension. To be precise it occurs in the two dimensional
case if the gas is trapped or in the free case just at zero temperature. For
three dimensions BEC can take place also for a free gas for higher tempera-
tures than zero [3].
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1
|k|