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We introduce two new thermostats, one of Langevin type and one of gradient(Brownian) type, for rigid body dynamics. We formulate rotation using thequaternion representation of angular coordinates; both thermostats preserve theunit length of quaternions. The Langevin thermostat also ensures that theconjugate angular momenta stay within the tangent space of the quaternioncoordinates, as required by the Hamiltonian dynamics of rigid bodies. We haveconstructed three geometric numerical integrators for the Langevin thermostatand one for the gradient thermostat. The numerical integrators reflect keyproperties of the thermostats themselves. Namely, they all preserve the unitlength of quaternions, automatically, without the need of a projection onto theunit sphere. The Langevin integrators also ensure that the angular momentaremain within the tangent space of the quaternion coordinates. The Langevinintegrators are quasi-symplectic and of weak order two. The numerical methodfor the gradient thermostat is of weak order one. Its construction exploitsideas of Lie-group type integrators for differential equations on manifolds. Wenumerically compare the discretization errors of the Langevin integrators, aswell as the efficiency of the gradient integrator compared to the Langevin oneswhen used in the simulation of rigid TIP4P water model with smoothly truncatedelectrostatic interactions. We observe that the gradient integrator iscomputationally less efficient than the Langevin integrators. We also comparethe relative accuracy of the Langevin integrators in evaluating various staticquantities and give recommendations as to the choice of an appropriateintegrator.

New Langevin and Gradient Thermostats for Rigid Body Dynamics