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

For a specified trajectory, the <nonconserved order-parameter dynamics> equation determines the noise realization
$$
\mathbf f_F
=\dot{\mathbf p}
+\Gamma\frac{\delta F}{\delta\mathbf p}.
$$
The forward <Onsager--Machlup path probability for Model A dynamics> is therefore
$$
P_F[\mathbf p]
=N_F\exp\left[
-\frac1{2\sigma^2}
\int_{t_1}^{t_2}dt\int d\mathbf r\,
\left|
\dot{\mathbf p}
+\Gamma\frac{\delta F}{\delta\mathbf p}
\right|^2
\right].
$$

Assume the order-parameter field is even under <time-reversal symmetry> and its free-energy functional is time-reversal invariant. The reversed path is
$$
\mathbf p^R(t)=\mathbf p(t_1+t_2-t).
$$
Its time derivative changes sign, so
$$
P_B[\mathbf p^R]
=N_B\exp\left[
-\frac1{2\sigma^2}
\int_{t_1}^{t_2}dt\int d\mathbf r\,
\left|
-\dot{\mathbf p}
+\Gamma\frac{\delta F}{\delta\mathbf p}
\right|^2
\right].
$$
For additive <Gaussian white noise>, the trajectory-to-noise Jacobian is the same in the two directions. We assume it and all path-independent normalization factors are absorbed into equal constants $N_F=N_B$. A time-reversal-odd order parameter would require the corresponding parity transformation as well.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 3 / a / Solution

{c}

For additive Gaussian white noise of variance $\sigma^2$, a Model A trajectory has path weight proportional to
$$
\exp\left[-\frac1{2\sigma^2}\int dt\,d^dr\,
\left|\dot\phi+\Gamma\frac{\delta F}{\delta\phi}\right|^2\right].
$$
Comparing a path with its time reverse yields the free-energy change and hence the thermal fluctuation-dissipation relation.


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