With constant positive <kinetic coefficient> $\Gamma$ and <mobility> $M$, two independent relaxation channels give
$$
\dot p_j=-\partial_iW_{ij}-\Gamma\mu_j+f_j,
\qquad W_{ij}=-M\partial_i\mu_j+N_{ij},
\qquad\mu_j=\frac{\delta F}{\delta p_j}.
$$
The deterministic mobility operator is $\Gamma-M\nabla^2$. Each channel separately satisfies <microscopic reversibility> when its noise obeys the corresponding <Model A fluctuation-dissipation relation> or <fluctuation-dissipation relation for a conserved flux>.

Mixed conserved and nonconserved order-parameter dynamics

Set the noise strengths to the <model A fluctuation-dissipation relation> and the <fluctuation-dissipation relation for a conserved flux>:
$$
\sigma^2=2\Gamma k_BT,\qquad\sigma_N^2=2Mk_BT.
$$
The preceding logarithmic ratio becomes
$$
\log\frac{P_F}{P_B}=-\frac1{k_BT}\int[\mu_j\dot p_j+\mu_j\partial_iW_{ij}+W_{ij}\partial_i\mu_j].
$$
The last two terms combine as $\partial_i(\mu_jW_{ij})$. They integrate to zero under the periodic <boundary conditions>, by <integration by parts>. The remaining term is $F_2-F_1$ by the <functional chain rule>, hence
$$
\boxed{\frac{P_F}{P_B}=e^{-(F_2-F_1)/(k_BT)}.}
$$
After including the equilibrium initial weights, the forward and reversed histories have equal probability. Thus <microscopic reversibility> is satisfied by the two channels together, with each channel's own thermal noise strength.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 344 / 2 / e / Solution

{title2=$\sigma_N^2=2Mk_BT$}

For the flux $W_{ij}=-M\partial_i\mu_j+N_{ij}$ of a <vector order parameter>, <microscopic reversibility> requires independent <Gaussian white noise> components with covariance
$$
\langle N_{ij}(\mathbf r,t)N_{kl}(\mathbf r',t')\rangle
=2Mk_BT\,\delta_{ik}\delta_{jl}\delta(\mathbf r-\mathbf r')\delta(t-t').
$$
The current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_BT)^{-1}\int W_{ij}\partial_i\mu_j$ to its logarithm; <integration by parts> converts this to the free-energy loss from the conserved dynamics.

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 Incoming links: Fluctuation-dissipation relation for a conserved flux