Langevin equation with colored noise for constant-temperature molecular dynamics simulations

We discuss the use of a Langevin equation with a colored (correlated) noise to perform constant-temperature molecular dynamics simulations. Since the equations of motion are linear in nature, it is easy to predict the response of a Hamiltonian system to such a thermostat and to tune at will the relaxation time of modes of different frequency. This allows one to optimize the time needed to thermalize the system and generate independent configurations. We show how this frequency-dependent response can be exploited to control the temperature of Car-Parrinello-like dynamics, keeping at low temperature the electronic degrees of freedom, without affecting the adiabatic separation from the vibrations of the ions.

💡 Research Summary

The paper introduces a thermostat for constant‑temperature molecular dynamics (MD) based on a Langevin equation driven by colored (time‑correlated) noise rather than the conventional white‑noise formulation. The authors begin by recalling that the standard Langevin thermostat couples each degree of freedom to a stochastic force ξ(t) with ⟨ξ(t)ξ(t′)⟩ = 2 kBT γ δ(t−t′), where γ is a constant friction coefficient. This white‑noise approach imposes the same relaxation time on all vibrational modes, which is sub‑optimal when a system contains both high‑frequency electronic degrees of freedom and low‑frequency ionic motions.

To overcome this limitation, the authors replace the delta‑correlated noise by a noise with a prescribed autocorrelation function γ(t). By construction the fluctuation‑dissipation theorem is preserved: ⟨ξ(t)ξ(t′)⟩ = kBT γ(|t−t′|). The Fourier transform γ̃(ω) then determines a frequency‑dependent effective friction γ_eff(ω) = Re