In standard classical MD, each nuclear DoF couples to Gaussian noise that injects noise with amplitude \(\propto T\)
\[m \ddot{x} = F(x) - \gamma \dot{x} + R(t)\]
For harmonic modes at low temp (\(T \to 0)\), we get average energy \(k_BT \to 0\) via equipartition. Quantum mechanics gives ZPE \(\hbar \omega / 2\) for harmonic modes \(\implies\)energies too low in low-temp regime with stiff modes (e.g. Hbonds).
Fix: change \(k_BT\) to a more general \(\theta(T, \omega)\) so that higher freq. modes can be injected with more energy (QTB method). But energy can leak between modes classically.
Fluctuation dissipation theorem (FDT) captures this leakage, and by calculating the violation of FDT on the fly and adjusting energy injections accordingly, QTB can be fixed.
THIS MATTERS A LOT FOR WATER BECAUSE OF HBONDS (among other things). Results show that this makes water simulation much much better (e.g. densities, vaporization enthalpies, etc.). Could try using this thermostat with a NNP trained on loads of DFT calculations for better liquid simulation?