For a specified field trajectory, the required normalized noise is
$$
\boldsymbol\Lambda_F
=\frac{\dot{\mathbf p}
+\Gamma\,\delta F/\delta\mathbf p-\Gamma\mathbf Y}
{\sqrt{2k_BT\Gamma}},
$$
and time reversal changes only $\dot{\mathbf p}$ to $-\dot{\mathbf p}$. Assuming equal additive-noise Jacobians, the difference of the two <Onsager--Machlup path probability for Model A dynamics>[Onsager--Machlup actions] gives
$$
\log\frac{\mathbb P_F}{\mathbb P_B}
=-\beta\int_{t_1}^{t_2}dt\int d\mathbf r\,
\dot{\mathbf p}\mathbin\cdot
\left(\frac{\delta F}{\delta\mathbf p}-\mathbf Y\right).
$$
The <functional chain rule> identifies the first term as $-\beta\Delta F$, so
$$
\boxed{\frac{\mathbb P_F[\mathbf p]}{\mathbb P_B[\mathbf p]}
=\exp\left[
-\beta\Delta F
+\beta\int_{t_1}^{t_2}dt\int d\mathbf r\,
\mathbf Y\mathbin\cdot\dot{\mathbf p}
\right].}
$$
The forcing performs generalized work $W_Y=\int\mathbf Y\cdot\dot{\mathbf p}$, and $W_Y-\Delta F$ is the heat dissipated into the bath. The formula is therefore the field-theory form of <local detailed balance> and quantifies nonequilibrium entropy production.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / g / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-344/3/g/solution