# On the "generalized Generalized Langevin Equation"

## TL;DR

This paper extends the generalized Langevin equation to non-stationary, non-equilibrium systems using time-dependent projection operators, enabling the derivation of equations of motion from molecular dynamics data.

## Contribution

It introduces a formalism for deriving non-stationary Langevin equations with time-dependent kernels, linking them to simulation data and initial conditions.

## Key findings

- Derived a non-stationary Langevin equation with a time-dependent memory kernel.
- Established a fluctuation-dissipation-like relation for non-equilibrium conditions.
- Validated the approach with Brownian motion simulations.

## Abstract

In molecular dynamics simulations and single molecule experiments, observables are usually measured along dynamic trajectories and then averaged over an ensemble ("bundle") of trajectories. Under stationary conditions, the time-evolution of such averages is described by the generalized Langevin equation. In contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable has. We employ the formalism of time-dependent projection operator techniques to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. The equation is similar in structure to the generalized Langevin equation, but exhibits a time-dependent memory kernel as well as a fluctuating force that implicitly depends on the initial conditions of the process. We also derive a relation between this memory kernel and the autocorrelation function of the fluctuating force that has a structure similar to a fluctuation-dissipation relation. In addition, we show how the choice of the projection operator allows to relate the Taylor expansion of the memory kernel to data that is accessible in MD simulations and experiments, thus allowing to construct the equation of motion. As a numerical example, the procedure is applied to Brownian motion initialized in non-equilibrium conditions, and is shown to be consistent with direct measurements from simulations.