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# The Smoluchowski equation

We shall now derive the equivalent of the Fokker-Planck equation, but this time applicable at the Smoluchowski timescale.

Suppose we are given a distribution of particles which were at position at time t=0. We assume that the particles are at every instant of time in thermal equilibrium with respect to their velocities. A flux will exist, given by

 (4.29)

where D is the diffusion constant , occurring in , and is the friction coefficient  on the Smoluchowski timescale. At equilibrium, the flux must be zero and the distribution be equal to

 (4.30)

Using this in Eq. (4.29) while setting , leads to the Einstein equation (4.13).

Introducing Eq. (4.29) into the equation of particle conservation

 (4.31)

we get the Smoluchowski equation

 = (4.32) = (4.33)

In the remaining part of this section we shall substantiate the above derivation. First we define the particle distribution on the Smoluchowski timescale by

 = (4.34) = (4.35)

Averaging the Fokker-Planck equation  over the initial velocities and integrating over , we find the continuity equation Eq. (4.31), with

 = = (4.36)

where the second step serves to define the velocity at time t at position , given that the particle was originally at .

We next derive an equation describing the time development of the velocity . To this end we multiply the Fokker-Planck equation by , average over the initial velocities, and integrate over , obtaining

 (4.37)

Using the continuity equation  and rearranging we find

 (4.38)

In a strongly damped system the integral on the right hand side yields the velocity fluctuation at position , multiplied by the probability to find the particle at position , which is GkT1/m. On the left hand side we recognize the acceleration of the particle at (see section 5.1). Eq. (4.38) may then be written as

 (4.39)

In a strongly damped system the average particle velocity is almost constant. We therefore put the left hand side of Eq. (4.39) equal to zero and solve for . Introducing the result into Eq. (4.36) we find the flux Eq. (4.29) and next the Smoluchowski equation. In Eq. (4.39) is called the Brownian force.

In appendix B we shall present an alternative derivation of the Smoluchowski equation. In appendix C we will demonstrate that the Langevin equations (4.27), (4.28) are equivalent to the Smoluchowski equation.

Next: Appendix A Up: Stochastic processes Previous: The Smoluchowski time scale
W.J. Briels