Unified Memory Kernel for Polymer Dynamics: Coupling Hydrodynamic and Excluded Volume Effects in Generalized Langevin Equations

The memory kernel function (MKF) in the generalized Langevin equation (GLE) plays a crucial role in modeling polymer dynamics, yet constructing it while incorporating hydrodynamic interactions (HIs) and excluded volume (EV) effects remains a significant challenge. Here, we demonstrate that the end-to-end vector MKF, K(t), can be analytically determined by the autocorrelation function Crr(t) and force–position cross-correlation Cfr(t). Notably, under the preaveraged approximation (averaging chain conformational fluctuations), the ratio Crr(t)/Cfr(t) becomes time-independent, simplifying K(t) to a function solely dependent on Crr(t). We derive a universally applicable analytical expression for K(t) by investigating the longest relaxation time and time-dependent friction across different solvent qualities. This expression explicitly links HIs, EV, and their interplay through two key parameters (α, β), which govern subdiffusive scaling on intermediate time scales and can be directly expressed as linear functions of the Flory exponent. To validate the theory, we extract correlation functions directly from Langevin dynamics simulation trajectories. This approach not only confirms the relationship between Crr(t) and Cfr(t) but also verifies the validity of the derived MKF K(t). This framework decodes the multiscale mean-square displacement behavior by establishing direct links between memory effects and the Flory exponent.