Consistent Projection of Langevin Dynamics: Preserving Thermodynamics and Kinetics in Coarse-Grained Models

TL;DR

The paper develops a projection-based coarse-graining framework for underdamped Langevin dynamics using the Zwanzig projection, deriving explicit CG drift and diffusion and proving thermodynamic consistency via a potential of mean force. It combines TI for cross-thermodynamic-state sampling with gEDMD to learn and validate CG dynamics from full-space data, enabling accurate kinetic characterization without long simulations. Through a two-dimensional Lemon Slice model, the approach demonstrates preservation of metastable states and transition timescales, with TI providing robust extrapolation to unseen temperatures. This yields a data-efficient pathway to thermodynamically and kinetically faithful coarse-grained Langevin models applicable to multi-scale systems.

Abstract

Coarse graining (CG) is an important task for efficient modeling and simulation of complex multi-scale systems, such as the conformational dynamics of biomolecules. This work presents a projection-based coarse-graining formalism for general underdamped Langevin dynamics. Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method, which was developed in the context of Koopman operator methods, can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI), a generative approach to transform samples between thermodynamic conditions, to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

Figures (11)

• Figure 1: Potential field of the Lemon Slice example.
• Figure 2: CG definition for the Lemon Slice system, in position and momentum space. Points belonging to the blue line in positional space have spatial CG component of $\xi(r, \phi) = \phi_0$, and points belonging to the blue line in the momentum space are perpendicular to $\nabla\phi$, and therefore have zero momentum CG component. Note that for $\mathbf{v} \neq 0$, the pre-image in momentum space would be more complex as the magnitude of the gradient $\nabla \phi$ depends on $r$.
• Figure 3: A) Slowest timescales $t_1$ to $t_3$ of the coarse grained Langevin equation \ref{['eq:params_effective_sde']} with the Lemon slice potential, corresponding to different values of the inverse temperature $\beta$, computed on rejection sampling data (RS, dashed lines) or on simulations of the underdamped Langevin SDE (ULD, solid lines), compared against the baseline timescales from the EDMD method (EDMD, black dash-dotted lines). B) timescales of the underdamped CG Langevin dynamics (ULD, solid lines) versus overdamped CG Langevin dynamics (OLD, dotted lines).
• Figure 4: Histogram of generated data using TI model without (middle) and with (right) importance sampling weights, compared to the one using the dataset coming from rejection sampling (left). First row is for the case of $\beta=1.50$ and the second row is for the case of $\beta=2.00$.
• Figure 5: Slowest timescales $t_1$ to $t_3$ of the Lemon slice potential corresponding to different values of the inverse temperature $\beta$, computed using data generated via simulations of the underdamped Langevin SDE (solid lines) and via the TI generative model (dash-dot-dotted lines).

Theorems & Definitions (2)

• Proposition 1
• Proposition 2