On the non-stationary generalized Langevin equation

Area of Science:

• Statistical Mechanics
• Computational Biophysics
• Physical Chemistry

Background:

• Averaging observables over trajectories is standard in molecular dynamics (MD) simulations and single-molecule experiments.
• The generalized Langevin equation describes time-evolution of averages under stationary conditions.
• The behavior of averaged observables during non-stationary dynamics remains unclear.

Purpose of the Study:

• To derive the equation of motion for trajectory-averaged observables in non-stationary (non-equilibrium) dynamics.
• To characterize the non-stationary auto-correlation function for these observables.
• To develop a framework applicable to systems not at equilibrium.

Main Methods:

• Utilized time-dependent projection operator techniques.
• Derived a generalized Langevin equation with a time-dependent memory kernel and initial-condition-dependent fluctuating force.
• Established a fluctuation-dissipation-like relation between the memory kernel and the fluctuating force's autocorrelation function.

Main Results:

• The derived equation of motion mirrors the generalized Langevin equation but incorporates a time-dependent memory kernel.
• A novel relation connects the memory kernel to the fluctuating force's autocorrelation, resembling a fluctuation-dissipation relation.
• Demonstrated how projection operator choice links memory kernel expansion to simulation/experimental data for equation construction.

Conclusions:

• Successfully derived a theoretical framework for describing averaged observables in non-stationary dynamics.
• The derived equation provides a tool for analyzing non-equilibrium processes in MD simulations and experiments.
• Validated the approach using Brownian motion initialized in non-equilibrium conditions, showing consistency with direct simulations.