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We have extended previous kinetic results to compute the exchange correctionto the electrostatic electron susceptibility for arbitrary frequencies andwavenumbers in the low temperature limit. This has allowed us to make a generalcomparison with a much used hydrodynamic expression for exchange effects. It isfound that the hydrodynamic expression gives a useful approximation when thephase velocity is roughly a factor $2.5$ larger than the Fermi velocity. Forlow phase velocities, as for ion-acoustic waves, wave-particle interactionleads to a strong enhancement of the exchange correction and the hydrodynamicresults is smaller by an order of magnitude.

Do hydrodynamic models misestimate exchange effects? Comparison with kinetic theory for electrostatic waves