= Hookean dumbbell model
{c}
{title2=$U(\mathbf R)=H|\mathbf R|^2/2$}
Represent a dilute <polymer> by two beads connected by a spring obeying <Hooke's law>, with stiffness $H>0$ and bead drag coefficient $\zeta>0$. Neglect bead inertia and <hydrodynamic interactions>. For component-first <velocity gradient> $K_{ij}=\partial_j u_i$, the configuration <Fokker-Planck probability current> is
$$
\mathbf J=K\mathbf R\,\psi-\frac2\zeta(H\mathbf R\psi+k_BT\nabla_{\mathbf R}\psi),\qquad\partial_t\psi=-\nabla_{\mathbf R}\cdot\mathbf J.
$$
The factor two comes from relative motion of both beads. At rest its normalized <probability density> is $\psi_{\rm eq}=(H/(2\pi k_BT))^{3/2}e^{-H R^2/(2k_BT)}$. The <conformation tensor> $C=\langle\mathbf R\mathbf R^T\rangle$ has dimensions of length squared; $(H/(k_BT))C$ is its dimensionless version. The <Hookean dumbbell in planar extension> is a particular imposed flow in this model.
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