= Free-energy balance of a Hookean dumbbell
{title2=$\dot{\mathcal F}=nHC:K-\mathcal D$}
For constant number density $n$, constant <temperature>, a homogeneous incompressible <linear velocity field> and decaying configuration tails, define
$$
\mathcal F=nk_BT\int\psi\log\frac\psi{\psi_{\rm eq}}\,d\mathbf R,\qquad
\mathcal D=\frac{2n}{\zeta}\int\psi|H\mathbf R+k_BT\nabla\log\psi|^2\,d\mathbf R\ge0.
$$
This <free energy> is a <relative entropy> times $nk_BT$. Differentiate it using the configuration <Fokker-Planck probability current>. One <integration by parts> gives $\dot{\mathcal F}=n\int\psi(H\mathbf R+k_BT\nabla\log\psi)\cdot\mathbf V\,d\mathbf R$. Substituting the <probability current velocity> gives the displayed balance: the diffusive affine contribution is $-nk_BT\operatorname{tr}K=0$. The polymer <stress tensor> $\sigma_p=n(HC-k_BTI)$ has mechanical power $\sigma_p:K=nHC:K$. Its instantaneous power need not be nonnegative, because it includes changes of stored <free energy>. The nonnegative quantity is $\mathcal D$, and equality of stress power and dissipation requires a stationary <free energy>.
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