 Drag dissipation of a bead-spring pair

When the midpoint follows a linear velocity field, the two beads' slip velocities are $\pm (\dot{\mathbf{R}}-K\mathbf{R})/2$. Summing their individual drag powers $\zeta |{\mathbf{v}}_{\mathrm{slip}}{|}^2$ proves the displayed expression. Using a probability current velocity in place of the instantaneous relative velocity defines the corresponding overdamped dissipative functional; adding independent thermal velocities is a different stochastic description.