\sect{Brownian motion of a diatomic particle} 

Consider the 1D motion of two beads of mass $m$, attached by a very
flexible bond with spring constant $k$. 
The beads are immersed in viscous liquid with friction coefficients $\gamma_1,\gamma_2$,  
and temperature $T$. Disregard the hydrodynamic interactions
between the beads and the direct collisions of the beads.

\Dn

(1) Write down Langevin equations for the beads. Neglect accelerations.

\Dn

(2) For $\gamma_1=\gamma_2\equiv\gamma$, 
define $R=\frac{x_{1}+x_{2}}{2}$, and $r=x_{1}-x_{2}$. 
Find $\langle(r(t)-r_0)^2\rangle$ and $\langle(R(t)-R_0)^2\rangle$, 
where $r_0$ is half the initial distance between the "atoms" 
and $R_0$ is the initial location of the "molecule".

\Dn

(3) Solve the equations for $\gamma_1\neq\gamma_2$, 
and show that the same solution as in (2) is obtained by setting $\gamma_1=\gamma_2\equiv\gamma$.

\Dn

(4) Generalize the results to 3D.

