Snap, Crackle, and Pop: This is why the potential of mean force clashes with the fluctuation dissipation relation

Abstract

We analyze the non-linear generalized Langevin equation which contains a thermodynamic force. We show that even for systems in thermal equilibrium the presence of the thermodynamic force implies that the auto-correlation function of the fluctuating force becomes non-stationary. We further illustrate that a standard coarse-graining procedure that neglects this fact predicts waiting-time distributions incompatible with the original, microscopic process. We conclude that one needs to proceed with care when adding thermodynamic driving forces to the Langevin equation.

Figures (3)

• Figure 1: Snapshot of the simulated system with $500$ Lennard-Jones particles under gravity in a cuboid box
• Figure 2: Results from the MD simulations of vertically confined Lennard-Jones particles under gravity. a) Potential of mean force with two vertical lines marking the region used for the analysis of the residence times. b) Variance of the fluctuating force as a function of time. The standard deviation of the mean is included as error band in light gray. c) Autocorrelation function of the fluctuating force in the stationary limit and the memory kernel (multiplied with $\langle p_z^2\rangle$).
• Figure 3: Distribution of residence times within the region in-between the two vertical lines in \ref{['fig:md_results']}a for the original molecular-dynamics (MD) simulations as well as the coarse-grained simulations via the auxiliary-variables (AV) method