Brownian Motion of Polyatomic Molecules: The Coupling of Rotational and Translational Motions

Abstract

The coupling of rotational and translation Brownian motions is examined from several points of view. The first is a phenomenological theory based upon generalized Langevin equations of motion and a Markoff integral equation. Next, a more detailed statistical-mechanical theory is fashioned after the pattern of Kirkwood's theory for nonequilibrium processes in monatomic liquids. Both schemes lead to a generalized Fokker—Planck—Chandrasekhar equation for the singlet-distribution function. This equation includes terms that account for separate rotational and translational motions as well as two mutually symmetric contributions which are descriptive of their coupling. The friction tensors associated with the uncoupled components of these motions are found to be proportional to the autocorrelations of the environmental force and torque which act upon a given molecule. The frictional coupling is related to the cross correlation of force and torque. From the principle of microreversibility it is possible to establish a reciprocal relationship between the two coupling tensors. A third approach is to derive the generalized Fokker—Planck equation from the Boltzmann equation for a dilute solution of rotating molecules. This has been done for the model of perfectly rough spheres and also for ``loaded spherocylinders.'' In both cases explicit formulas are obtained for the various friction tensors. The last section of the paper is devoted to the application of these theories to problems of diffusion.