Hamiltonian model for the Brownian motion of a rigid rotor

Abstract

Abstract We derive stochastic equations for the motion of a rigid rotor in a linear heat bath starting from a fully dynamical (Hamiltonian) description. We obtain a generalized Langevin equation for the angular velocity of the rotor in which the fluctuations in the torque arise from the initial conditions of the heat bath. The dissipation is in general nonlinear and is related to the fluctuations via a fluctuation-dissipation relation that is a natural consequence of our description. In the Langevin limit we show that our equations are equivalent to the phenomenological Euler-Langevin equations. However, whereas the latter are necessarily expressed in components referred to a body-fixed coordinate system, our representation is valid in an arbitrary system of axes and is in that sense more general. Finally, we also examine the reduction of our model to the rotational diffusion equation in the high viscosity limit. We show that this reduction can be carried out via standard methods of adiabatic elimination if our representation of the rotor equations of motion is used.