Formal theory for smart‑sampling GLE design

Develop a formal theoretical framework to design generalized Langevin equation (GLE) thermostats that achieve "smart sampling," specifically ensuring maximal sampling efficiency at a target minimum frequency while obtaining an approximately 1/√ω decay in efficiency at higher frequencies, without degrading the sampling of slow collective modes.

Background

The authors discuss optimizing GLE thermostats to enhance sampling efficiency over broad frequency ranges. An "optimal sampling" GLE seeks uniform efficiency across modes, but a more pragmatic "smart sampling" approach targets maximal efficiency at a minimum resolvable frequency and tolerates reduced efficiency for faster modes.

They report an empirical ability to achieve a 1/√ω decay in sampling efficiency above the minimum frequency, improving over the 1/ω decay of white‑noise Langevin thermostats, but point out that a formal theoretical treatment establishing and guiding this design is currently missing.

References

Empirically, it seems to be possible to obtain a decay of $\kappa_V(\omega)\sim 1/\sqrt{\omega}$ above $\omega_\text{min}$ -- rather than the $1/\omega$ decay that would be expected for white noise (see Figure~\ref{fig:ho1d-smart}) -- but a formal treatment of this problem is not yet available.

— Path Integral Methods in Atomistic Modelling: An Introduction  (2603.28588 - Ceriotti et al., 30 Mar 2026) in Subsection "Smart sampling GLE," Chapter "Colored-noise methods"