Nonlinear Dynamical Friction from the Doppler-Shifted Equilibrium Memory Kernel

We present a statistical mechanics framework for modeling equilibrium friction coefficients using the Generalized Langevin Equation (GLE). We show that the kernel, obtained via the Fluctuation-Dissipation Theorem (FDT) from the stochastic force autocorrelation measured in a thermal equilibrium state, is sufficient to model the dynamics of the system in a Non-Equilibrium Steady State (NESS). This approach provides a computationally efficient path to modeling complex equilibrium friction problems. We apply this framework to the canonical problem of test particle drag in a uniform plasma. The GLE formalism is shown to naturally capture non-Markovian phenomena through the moments of the kernel, including an effective mass renormalization and oscillatory relaxation. We demonstrate that the standard Chandrasekhar stopping power formula arises naturally as the Markovian limit of this equilibrium memory kernel. These theoretical predictions are quantitatively validated by direct Particle-in-Cell simulations, which confirm the predicted oscillatory structure of the memory kernel. This work thus establishes a practical method for predicting equilibrium friction properties from first-principles equilibrium simulations.