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Space-resolved stress correlations and viscoelastic moduli for polydisperse systems: the faces of the stress noise

Several advances in the theory of space-resolved viscoelasticity of liquids and other amorphous systems are discussed in the present paper. In particular, considering long-time regimes of stress relaxation in liquids we obtain the generalized compressibility equation valid for systems with mass polydispersity, and derive a new relation allowing to calculate the wavevector-dependent equilibrium transverse modulus in terms of the generalized structure factors. Turning to the basic relations between the spatially-resolved relaxation moduli and the spatio-temporal correlation functions of the stress tensor, we provide their new derivation based on a conceptually simple argument that does not involve consideration of non-stationary processes. We also elucidate the relationship between the stress noise associated with the classical Newtonian dynamics and the reduced deviatoric stress coming from the Zwanzig-Mori projection operator formalism. The general relations between the stress noise and the tensor of relaxation moduli are discussed as well.