Abstract
We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.
Highlights
In molecular simulations, it is often desirable to fix the temperature of the simulated system, ensuring the system samples from the NVT ensemble (Gibbs measure)
The Langevin thermostat ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies
While here we present only results for one set of thermostat parameters (γ = 5 ps−1 and Γ = 10 ps−1 for the Langevin thermostat and υ = 4 fs and Υ = 1 fs for the gradient thermostat), we have tested a wide range of parameter values
Summary
It is often desirable to fix the temperature of the simulated system, ensuring the system samples from the NVT ensemble (Gibbs measure). We first propose a new Langevin thermostat for rigid body dynamics This thermostat improves upon that in Ref. 5, as it preserves the unit length of quaternions and keeps the angular momenta conjugate to the quaternion coordinates on the tangent space. We propose a new gradient (Brownian) thermostat for rigid body dynamics and construct a 1st-order geometric integrator for it. Both the gradient thermostat and the numerical scheme preserve the unit length of quaternions. We perform numerical comparison of the proposed Langevin and gradient thermostats and of the derived numerical integrators These tests demonstrate that the Langevin thermostat integrated by Langevin A or Langevin C is a powerful approach for computing NVT ensemble averages involving rigid bodies.
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