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Dam Son has kindly informed us of his recent work in which the same tail of the dynamic structure factor is derived using operator product expansions;
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A Galilean shift such as that used to arrive at Eq. (8) is strictly only valid for a uniform velocity field u (r, t) = u (t). In this case, Eq. (8) gives the Hamiltonian in the rest frame of the "walls" bounding the system in accordance with the principle of Galilean relativity [22]. However, the fact that the velocity field is (necessarily) nonuniform means that translational symmetry is broken and there is no suitable reference frame in which the system can be in equilibrium. Nonetheless, in the long-wavelength limit required to define the viscosity coefficients, Eq. (8) reduces to the Hamiltonian in the rest frame of the walls, meaning that it is suitable for defining equilibrium thermodynamic averages.
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Consider, e.g., the induced current in the α = x direction, and set q z = 0. The transverse correlator is obtained by first taking the q x → 0, followed by q y → 0 limit. Interchanging the order, one finds the longitudinal result.
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