Fokker-planck equation for brownian motion of rotating spherical particles: PMM vol. 40, n≗ 6, 1976, pp. 1127–1131

Abstract

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time. In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors. In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].