[Paper Review] Position-dependent memory kernel in generalized Langevin equations: theory and numerical estimation

Hadrien Vroylandt, Pierre Monmarché|arXiv (Cornell University)|Jan 7, 2022

Theoretical and Computational Physics50 references45 citations

TL;DR

This paper rigorously derives generalized Langevin equations (GLEs) with position-dependent memory kernels for non-linear, coarse-grained dynamics in equilibrium Hamiltonian systems. It establishes a position-dependent fluctuation-dissipation theorem and introduces a numerical Volterra-based method to estimate the kernel from all-atom simulations, demonstrating that position-dependent kernels are essential for accurate modeling beyond standard assumptions.

ABSTRACT

Generalized Langevin equations with non-linear forces and position-dependent linear friction memory kernels, such as commonly used to describe the effective dynamics of coarse-grained variables in molecular dynamics, are rigorously derived within the Mori-Zwanzig formalism. A fluctuation-dissipation theorem relating the properties of the noise to the memory kernel is shown. The derivation also yields Volterra-type equations for the kernel, which can be used for a numerical parametrization of the model from all-atom simulations.

Motivation & Objective

• To rigorously derive generalized Langevin equations (GLEs) with non-linear forces and position-dependent linear friction kernels within the Mori-Zwanzig formalism.
• To establish a position-dependent fluctuation-dissipation theorem (FDT) linking the noise covariance to the memory kernel.
• To develop a numerical scheme based on Volterra-type integro-differential equations for estimating the memory kernel from all-atom simulation data.
• To demonstrate that standard GLEs with time-only kernels fail to capture dynamics accurately when the system exhibits position-dependent friction.
• To validate the model using numerical experiments on a 1D system, comparing splines, minimal, and linear GLEs.

Proposed method

• Derive the GLE with position-dependent kernel using the Mori-Zwanzig formalism, projecting the dynamics onto a set of collective variables.
• Formulate the memory kernel as satisfying Volterra-type integro-differential equations derived from the projection formalism.
• Implement a numerical inverse Volterra method to estimate the kernel from time-series data of all-atom simulations.
• Use the fluctuation-dissipation theorem to relate the noise covariance to the memory kernel, ensuring consistency with equilibrium statistical mechanics.
• Apply the method to a 1D model system with a potential of mean force, comparing results across different GLE formulations (splines, minimal, linear).
• Validate the noise model by comparing its correlation function to the kernel's constant part and assess non-Gaussian tails in noise histograms.

Experimental results

Research questions

• RQ1 Can generalized Langevin equations with non-linear forces and position-dependent friction kernels be rigorously derived from the Mori-Zwanzig formalism?
• RQ2 Does a position-dependent fluctuation-dissipation theorem hold, relating the noise covariance to the memory kernel in such systems?
• RQ3 Can the memory kernel be accurately estimated from all-atom simulation data using Volterra-type equations?
• RQ4 How do different GLE formulations (splines, minimal, linear) compare in reconstructing the true dynamics and noise statistics?
• RQ5 To what extent does the position dependence of the noise variance affect the validity of the FDT and the overall model accuracy?

Key findings

• The memory kernel in the GLE must depend on position to accurately describe non-linear, coarse-grained dynamics, contradicting the standard assumption of time-only dependence.
• A position-dependent fluctuation-dissipation theorem is rigorously derived, linking the noise covariance to the memory kernel in a way consistent with equilibrium statistical mechanics.
• The numerical estimation method based on Volterra equations successfully recovers the memory kernel from all-atom trajectories, with the splines GLE showing the best agreement with reference data.
• The noise in the minimal GLE and splines GLE exhibits non-Gaussian tails, indicating that a simple Gaussian noise model is insufficient for accurate dynamics.
• The memory kernel of the minimal GLE lacks a constant coefficient, making it incompatible with the standard FDT, while the splines and linear GLEs show good agreement between noise auto-correlation and kernel coefficient.
• The variance of the noise strongly depends on position, as confirmed by binning the noise data, highlighting the necessity of position-dependent noise modeling.