From Eq. (4.8) we see that the particle loses its memory of its
initial velocity after a time span
.
Using equipartition its initial velocity may be put equal to
.
The distance it travels, divided by its diameter then is

(4.23) |

where the value 10

A very crude way to arrive at the equations of motion is the following. On
the timescale referred to above, and which is called the Smoluchowski time
scale , the average velocity is zero, or constant
in case we apply an external force. Eq. (4.2) then becomes

where is the external potential . Rewriting this, and using the fluctuation-dissipation theorem we get

These equations are correct when

In the next section we shall derive the Smoluchowski equation, governing the
time development of the distribution of particles on the Smoluchowski
timescale. In Appendix C we shall then show that the correct Langevin
equations which lead to the Smoluchowski equation are:

In our treatment of the Rouse chain in Chapter 6 the diffusion constant will be independent of , in which case Eqs. (4.27) and (4.28) are equivalent to Eqs. (4.25) and (4.26).