Equations of Internal Motion for the Molecules Hydrodynamic and Brownian Forces

11 Equations of Internal Motion for the Molecules Hydrodynamic and Brownian Forces (DPL, Sect. 175) 

Next we replace the double bracket in the first term on the right side by [[(r - u2)(f - uj)]] , and add appropriate compensating terms here and elsewhere we use the notation uj (r , t) = [[f ]] for the average velocity of bead V. Then Eq. (11.1) becomes  

The terms on the left side can be differentiated by parts, and Eq. (10.6) can then be used to show that two of the terms sum to zero. Thus, the left side of Eq. (11.2) becomes  

When Eq. (11.2) is then divided by P, we get the equation of motion for the beads  

We refer to the terms on the left as acceleration terms, and on the right we have a sum of four forces. There are two forces Ft and Fj, whose sum is well determined by comparing Eqs. (11.2) and (11.4). However, the separation into two terms is arbitrary. The symbols F and Ft respectively are referred to here as the averaged Brownian force and the averaged hydrodynamic force on bead v of molecule a. We choose to define these forces, in the most natural way, as follows  

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