  We show the linear response theory of spatial-scale-dependent relaxation
moduli for overdamped Brownian particle systems. We employ the Irving-Kirkwood
stress tensor field as the microscopic stress tensor field. We show that the
scale-dependent relaxation modulus tensor, which characterizes the response of
the stress tensor field to the applied velocity gradient field, can be
expressed by using the correlation function of the Irving-Kirkwood stress
tensor field. The spatial Fourier transform of the relaxation modulus tensor
gives the wavenumber-dependent relaxation modulus. For isotropic and
homogeneous systems, the relaxation modulus tensor has only two independent
components. The transverse and longitudinal deformation modes give the
wavenumber-dependent shear relaxation modulus and the wavenumber-dependent bulk
relaxation modulus. As simple examples, we derive the explicit expressions for
the relaxation moduli for two simple models the non-interacting Brownian
particles and the harmonic dumbbell model.
