= Solution
<Detailed balance> says that each equilibrium transition is balanced by its time reverse:
$$
P_{\rm eq}(1)\mathbb P_F(1\to2)
=P_{\rm eq}(2)\mathbb P_B(2\to1).
$$
With the <Boltzmann distribution> $P_{\rm eq}\propto e^{-\beta H}$ and $\Delta H=H_2-H_1$, this requires
$$
\frac{\mathbb P_F}{\mathbb P_B}=e^{-\beta\Delta H}.
$$
Parts a and b instead give $\exp[-(2\zeta/\sigma^2)\Delta H]$. Equality for every pair of endpoints yields the <fluctuation-dissipation relation for a Langevin particle>
$$
\boxed{\sigma^2=\frac{2\zeta}{\beta}=2\zeta k_BT.}
$$
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