Stochastic thermostats: comparison of local and global schemes

We show that a recently introduced stochastic thermostat [J. Chem. Phys. 126 (2007) 014101] can be considered as a global version of the Langevin thermostat. We compare the global scheme and the local one (Langevin) from a formal point of view and through practical calculations on a model Lennard-Jones liquid. At variance with the local scheme, the global thermostat preserves the dynamical properties for a wide range of coupling parameters, and allows for a faster sampling of the phase-space.

💡 Research Summary

The paper provides a thorough comparison between a recently introduced stochastic thermostat that operates globally and the traditional Langevin thermostat, which acts locally on each degree of freedom. After reviewing the most common isothermal molecular‑dynamics schemes—Nosé‑Hoover, Nosé‑Hoover chains, the Berendsen weak‑coupling method, and Langevin dynamics—the authors focus on the fundamental difference between local and global temperature control. Langevin dynamics adds a friction term γ p_i and a stochastic term proportional to √(2 γ k_BT m_i) dW_i to each particle’s momentum. While this guarantees sampling of the canonical ensemble, a large γ strongly damps the physical trajectories, reducing diffusion and corrupting dynamical observables.

The authors derive the energy balance for Langevin dynamics using the Itô calculus, obtaining an expression in which the total energy change depends on a single stochastic term rather than N_f independent noises. They then pose the problem of designing a correction force ˜g_i that reproduces exactly the same total‑energy variation as Langevin but minimizes the disturbance of the underlying Hamiltonian flow. By requiring the disturbance measure Σ_i (˜g_i dt)² to be minimal for a fixed kinetic‑energy increment, they find that ˜g_i must be proportional to the instantaneous momentum p_i, i.e. ˜g_i = λ(t) p_i. The scalar λ(t) contains both deterministic and stochastic contributions, A(K) and B(K), which are determined by matching the Langevin energy change. This leads to the compact global thermostat equation (7):

 ˜g_i dt = ½τ