The Smoluchowski time scale

From Eq. (4.8) we see that the particle loses its memory of its initial velocity after a time span $\tau \approx 1/\xi$. Using equipartition  its initial velocity may be put equal to $\sqrt{3kT/m}$. The distance it travels, divided by its diameter then is

$$\frac{1}{a}\frac{\sqrt{3kT/m}}{\xi }=\frac{\sqrt{3mkT}}{6\pi \eta a^{2}} \approx 10^{-5}$$ (4.23)

where the value 10^-5 refers to normal colloidal particles. We see that the particles have hardly moved at the time possible velocity gradients have relaxed to equilibrium. When we are interested in timescales on which particle configurations change, like in dynamic light scattering experiments , we may restrict our attention to the space coordinates, and average over the velocities.

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

$$\vec{0}=-\xi \vec{v}-\frac{1}{m}\vec{\nabla}\Phi +\vec{F}$$ (4.24)

where $\Phi (\vec{r})$ is the external potential . Rewriting this, and using the fluctuation-dissipation theorem  we get

  

$$\frac{d\vec{r}}{dt} -\frac{1}{\xi m}\vec{\nabla}\Phi +\vec{f}$$ = (4.25)

$$\langle \vec{f}(t)\cdot \vec{f}(t^{\prime })\rangle 6\frac{kT}{m\xi } \delta (t-t^{\prime })=6D\delta (t-t^{\prime }).$$ = (4.26)

These equations are correct when D is independent of the position of the particle. However it is clear that the random term now refers to a completely different timescale than in the original Langevin equation ; nevertheless $\vec{f}(t)$ is obtained here from $\vec{F}(t)$ by dividing the latter by $\xi$, meaning that the ''derivation'' given here must be wrong. The flaw occurs when the instantaneous acceleration is put equal to zero in Eq. (4.24). In fact it is the average acceleration which should be put equal to zero.

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:

  

$$\frac{d\vec{r}}{dt} -\frac{1}{m\xi }\vec{\nabla}\Phi +\vec{\nabla}D+\vec{f }$$ = (4.27)

$$\langle \vec{f}(t)\cdot \vec{f}(t^{\prime })\rangle 6D\delta (t-t^{\prime }).$$ = (4.28)

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