β Scribed by Michio Tokuyama; Irwin Oppenheim
No coin nor oath required. For personal study only.
Fluctuations of a large and heavy spherical Brownian particle (B) floating in an equilibrium fluid are rigorously studied by using two kinds of methods for asymptotic evaluation proposed by Mori; a projection operator method and a scaling method. It is shown that depending on the relative magnitudes of the mass (M) and the mass density (ps) of B to those of the bath particles (m and p). there exist three kinds of kinetic processes which are described by three types of linear stochastic equations. One is a Markou kinetic process when m/M 4 1 and pips <e 1, which leads to the Stokes law 5 = 4rqR, where 5 is the friction constant, n the shear viscosity and R the radius of B. The others are non-Markoo kinetic processes when m/M < 1 and pips = 1, and m/M * 1 and dps * 1, which are characterized by the long time t-"2 tail and the time t-l'* decay of an average velocity of B, respectively. Next the usual Brownian point heavy particle case is also discussed from our viewpoint and a Markov kinetic process with the friction constant &/(l + n/D& is obtained, where &, is the bare diffusion constant and DsE the Stokes-Einstein diffusion constant given by ksT/4qR. Finally, it is shown that the hydrodynamic process of B obeys the diffusion equation whose diffusion constant is given by Dsa and Q + L&s for a large particle and for a point particle, respectively.
π SIMILAR VOLUMES
The usefulness of non-Markov reduced equations of motion (REM) for the description of the time evolution of macrovariables is examined. We show that in general one should be very cautious when using such equations since the results may strongly depend on the addition of more variables into the REM.
In this paper a problem belonging to the moving boundary class is tackled with a 2-D application of computational fluid dynamics techniques.