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For glass states, one obtains a slightly different expression than Eq. 21 when the generalized hydrodynamic approximation is applied to Eq. 13 instead of Eq. 11. As far as one is interested in the small- q regime where Ωq can be approximated by its leading-order contribution, Ωq q v0, it does not matter whether the approximation is applied to Eq. 11 or 13, and this is the case in Sec. 3 where the q range of primary interest is q 2. In Sec. 4, on the other hand, one needs to apply the approximation to Eq. 13 [see Eq. 33], since there we are also interested in larger q regime. The interested reader is referred to Ref. for details.
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