Drag dissipation of a bead-spring pair ($D_{\mathrm{pair}}=\zeta |\dot{\mathbf{R}}-K\mathbf{R}{|}^2/2$)

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.

If the relative bead velocity $\mathbf{v}$ is independent of its separation and has a centered Maxwell-Boltzmann distribution with $⟨v_i{v}_j⟩=(2k_{B}T/m){\delta }_{ij}$, averaging the drag dissipation of a bead-spring pair gives the displayed gross loss. The cross term vanishes by the zero conditional mean. At zero velocity gradient, this remains $3n\zeta k_{B}T/m>0$, although mechanical stress power is zero. A thermal bath replenishes microscopic kinetic energy at equilibrium. Equating this gross loss to polymer stress power omits heat-bath input and cannot follow from the independently specified Maxwellian velocity distribution. The overdamped free-energy balance of a Hookean dumbbell uses a probability current velocity instead.