One obvious way to improve on the Rouse chain, is by introducing hydrodynamic interactions between the beads. The chain so defined is called the Zimm chain . The equations describing hydrodynamic interactions have been obtained in section 5.4.

Let us now derive the equations of motion of the beads in a Zimm chain. Let
be the probability density of finding
particles
near
at time *t*.
The time development of
obeys

where is the flux of particles

The first term results from the forces felt by all the beads. On the Smoluchowski time scale , these forces make the beads move with constant velocities . This amounts to saying that the forces are exactly balanced by the hydrodynamic forces acting on the beads

(7.3) |

Multiplying this velocity by , we obtain the systematic part of the flux of particle

Combining Eqs. (7.1) and (7.2) we find

(7.4) |

where is the inverse of . In the stationary state the right hand side must be zero, and . From this it follows that

(7.5) |

which is a generalization of the Einstein equation .

The equation of motion of the probability density, i.e. the Smoluchowski
equation , now reads

(7.6) |

The Langevin equations corresponding to this Smoluchowski equation are

The reader can easily check that these reduce to the equations of motion of the Rouse chain when hydrodynamic interactions are neglected.

Let us note at this point that for the approximation of the mobility tensor
we make use of

(7.9) |

which greatly simplifies Eq. (7.7).