The Smoluchowski Equation for an Elastic Dumbbell

For the dumbbell, as shown in Fig. 6.1, we assume that the two beads have an identical mass m and radius a. Their position vectors are indicated by Ri and R2, respectively. Then the configuration vector between the two 

As seen in Chapter 3, the force constant on the harmonic spring is of entropic origin. The spring forces on bead 1 and bead 2, denoted by Fi and F2, respectively, are given by 

Let (Ri, R2)dRidR2 represent statistically the number of dumbbells that will be found within the configuration range from Ri to Ri + ctRi for bead 1 and from R2 to R2 + dR2 for bead 2. For a homogeneous flow with no concentration gradient, the configurational distribution of the dumbbells is expected to be independent of the locations of the dumbbells and we can write 

As defined above, tp(R, t)d L is the probability that a randomly selected dumbbell from the fluid system will be in the configuration range R to R+ dR, and is normalized as  

Then the average value of any dynamic quantity M(R) can be defined by 

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