Transient subtraction: A control variate method for computing transport coefficients

TL;DR

This work tackles the high variance in transport-coefficient estimators arising from Green–Kubo and nonequilibrium frameworks by introducing a transient subtraction technique. The method couples a transiently perturbed trajectory with a sensibly chosen equilibrium trajectory to serve as a control variate, thereby reducing variance while preserving the correct linear response. The authors develop a formal mathematical framework for perturbations, construct optimal initial-condition couplings via deterministic maps, and provide rigorous bias and variance analyses, including Lyapunov-based functional estimates and Poisson-equation considerations. They validate the approach numerically for Langevin dynamics and Lennard–Jones fluids, demonstrating significant variance reductions with a manageable computational overhead, and discuss extensions to TTCF and broader coupling strategies.

Abstract

In molecular dynamics, transport coefficients measure the sensitivity of the invariant probability measure of the stochastic dynamics at hand with respect to some perturbation. They are typically computed using either the linear response of nonequilibrium dynamics, or the Green--Kubo formula. The estimators for both approaches have large variances, which motivates the study of variance reduction techniques for computing transport coefficients. We present an alternative approach, called the \emph{transient subtraction technique} (inspired by early work by Ciccotti and Jaccucci in 1975), which amounts to simulating a transient dynamics started off equilibrium and relaxing towards the equilibrium state, from which we subtract a sensibly coupled equilibrium trajectory, resulting in an estimator with smaller variance. We present the mathematical formulation of the transient subtraction technique, give error estimates on the bias and variance of the associated estimator, and demonstrate the relevance of the method through numerical illustrations for various systems.

Figures (4)

• Figure 1: Illustration of coupling measure on initial conditions.
• Figure 2: Bias \ref{['eq:bias_1d-lang']} as a function of $\eta$ for the first and second-order maps, with overlayed reference lines.
• Figure 3: Trajectories for the computation of the mobility of a Lennard--Jones fluid with colored drift and associated error bars. The top graphs correspond to the instantaneous response (normalized by $\eta$) as a function of time as transient trajectory relaxes, while the bottom graphs show the integrated response over time. The dashed line corresponds to the reference value $\rho = 0.122$ obtained in meier2004blassel2024.
• Figure 4: Trajectories for the computation of the shear viscosity of a Lennard--Jones fluid and associated error bars. The top graphs correspond to the instantaneous response (normalized by $\eta$) as a function of time as transient trajectory relaxes, while the bottom graphs show the integrated response over time. The dashed line corresponds to the reference value $U_1 = 0.322$ found in blassel2024.

Theorems & Definitions (16)

• Remark 2.1: Relationship with Green-Kubo formulas
• Remark 3.1: Tangent dynamics
• Lemma 3.2
• proof
• Remark 3.3
• Proposition 3.4: Finite $\eta$ bias
• Remark 3.5
• Remark 3.6: Well-posedness of PDEs
• proof : Proof of \ref{['prop:gen_subtraction']}
• Corollary 3.7