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Stochastic thermostats: comparison of local and global schemes
Giovanni Bussi, Michele Parrinello
TL;DR
The paper asks how stochastic rescaling relates to Langevin dynamics while retaining canonical sampling and reducing dynamical disturbance. It derives the continuous and discrete global scheme, compares it with Langevin dynamics, and finds preserved dynamics over a wide coupling range together with faster sampling.
Problem
The paper examines whether a stochastic thermostat can provide canonical sampling without the substantial dynamical disturbance associated with local Langevin coupling.
Method
The paper derives stochastic velocity rescaling as a global Langevin-like thermostat that minimizes disturbance while enforcing the same total-energy variation.
Results
The global scheme preserves dynamical properties over a wide range of coupling parameters and enables faster phase-space sampling than the local scheme.
Takeaways & Limitations
The global thermostat is a global version of Langevin dynamics that retains canonical sampling while reducing trajectory disturbance.
Abstract
from arXiv · showhide
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.
I. CONTINUOUS EQUATIONS OF MOTION
The paper derives a stochastic correction force that preserves canonical sampling while minimizing disturbance to Hamiltonian dynamics. Applied globally to total kinetic energy, this scheme is equivalent to Langevin dynamics for one degree of freedom and supports intermediate group-wise thermostats.
- I. CONTINUOUS EQUATIONS OF MOTION: The equations begin with Hamiltonian motion plus a correction force that alone changes total energy and drives thermalization.The system is defined by coordinates qi, momenta pi, masses mi, and Hamiltonian H(p,q) = K(p) + U(q), with the target canonical distribution.
- I. CONTINUOUS EQUATIONS OF MOTION: Unlike Langevin dynamics, which uses independent noises for each degree of freedom, the stochastic rescaling equation uses one noise term shared by all particles.This implements a global thermostat acting on total kinetic energy rather than independent local degrees of freedom.
- I. CONTINUOUS EQUATIONS OF MOTION: The method chooses a force proportional to each momentum, minimizing disturbance for a prescribed kinetic-energy change.The same energy variation as Langevin dynamics is retained, but the disturbance measure is minimized.
- I. CONTINUOUS EQUATIONS OF MOTION: The correction force preserves exact canonical sampling while acting as a continuous stochastic version of velocity rescaling.Its action is parallel to the momenta, so it rescales them while enforcing the stochastic total-energy increment.
- I. CONTINUOUS EQUATIONS OF MOTION: For Nf = 1, the correction force becomes exactly equivalent to the Langevin thermostat.Applying one thermostat per degree of freedom gives Langevin dynamics, while applying one thermostat to total kinetic energy gives stochastic rescaling; intermediate molecular or atomic-group schemes are possible.
II. FINITE TIMESTEP ALGORITHM
The finite-timestep implementation separates Hamiltonian integration from stochastic momentum updating using Trotter decomposition and velocity-Verlet. Its sign correction accounts for rare momentum flips and recovers the standard Langevin integrator in the single-degree-of-freedom limit.
- II. FINITE TIMESTEP ALGORITHM: The discrete algorithm uses Trotter decomposition to separate Hamiltonian integration from stochastic momentum updates, with velocity-Verlet for the Hamiltonian part.The stochastic update is integrated through the kinetic-energy evolution over a finite timestep.
- II. FINITE TIMESTEP ALGORITHM: The finite-timestep kinetic-energy update uses c = e^-2γ∆t = e^-∆t/τ, a unit-variance Gaussian R(t), and a sum of Nf − 1 squared Gaussian numbers.These quantities determine the finite-step rescaling factor.
- II. FINITE TIMESTEP ALGORITHM: The sign of the rescaling factor must account for the finite probability of momentum flips, using the same Gaussian number as the kinetic-energy update.The flip probability is extremely small for large Nf and c ≈ 1, but the sign calculation is required when the thermostat acts on only a few degrees of freedom.
- II. FINITE TIMESTEP ALGORITHM: For Nf = 1, the stochastic-rescaling update combined with velocity-Verlet is completely equivalent to the standard Langevin integration scheme.The single-degree-of-freedom limit therefore preserves the formal connection between the global and local thermostats at finite timestep.
III. EXAMPLES
The Lennard-Jones tests compare local Langevin and global stochastic-rescaling thermostats across relaxation times, measuring dynamical disturbance through diffusion and sampling efficiency through autocorrelation times.
- The global thermostat keeps the diffusion coefficient nearly independent of relaxation time, whereas short-relaxation local Langevin dynamics strongly quenches diffusion.The local limit approaches high-friction Langevin behavior, with diffusion proportional to (βmγ)^-1.
- Sampling efficiency is assessed from autocorrelation times of kinetic, potential, and total energy.For total energy, the autocorrelation time also indicates equilibration speed from an unlikely initial configuration.
- The autocorrelation estimate integrates the correlation function with a windowing function that reduces the influence of statistically noisy long-time points.Its relative accuracy is assessed using the total run length T′ = 5 × 10^4.
IV. CONCLUSIONS
The paper derives the global stochastic thermostat continuously, establishes its analogy to Langevin dynamics, and tests its dynamical and sampling properties. The global scheme preserves dynamics and samples faster than the local scheme in the reported test case.
- The global thermostat minimizes disturbance to Hamiltonian dynamics while preserving dynamical properties in the test case.
- The global scheme enables faster sampling, as measured by the autocorrelation time of the total energy.