Hydrodynamic Fluctuations and Stokes' Law Friction

Abstract

The fri ctional forc e on a Brownian motion particle can be expressed by means of the time· correlation of the flu ctuating force on the particle . We s how that this method, applied to a spherical particle in a vi scous in compressibl e fluid , leads to Stokes ' Law. The c al culation is based on the theory of hydrodynamic flu ctuations due to Landau and Lifs hitz, and on a hydrodynamic theorem due to Faxe n. The subj ect of thi s article is th e connecti on be tween two diffe rent me th ods for calculating th e fri cti onal force on a Brownia n moti on particle. One method is based on th e s tati s ti cal mech ani cal th eory of irre­ ve r sible processes, and involv es evalua ti on of a certain time-correlati on formula. The other me thod is based on macroscopi c hydrodynami cs, and requires solution of th e N avi er- S tokes equ a ti on. F or simplicity we r estri c t the di scussion to the s pecial case of a spheri cal p article of radiu s a moving slowly through a vi scous in compressible fluid. The vi scosity coeffi cient is T/ . In the hydrodynamic theor y, th e fri c ti onal force F on a sphere movin g with constant velocity v is giv e n by Stokes' law,