The <first law of thermodynamics>, with $\Delta Q$ defined as heat lost by the particle to the bath, is
$$
\Delta H=W-\Delta Q.
$$
Thus $W-\Delta H=\Delta Q$, and part d becomes the <local detailed balance> identity
$$
\boxed{\frac{\mathbb P_F}{\mathbb P_B}=e^{\beta\Delta Q}.}
$$
The logarithmic path-probability ratio is the bath's <entropy production> $\Delta S_{\rm bath}=\Delta Q/T$ in units of $k_B$.

Solution

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

{title2=$\Delta S_{\rm tot}$}
{wiki}

Entropy production is the sum of the entropy changes of a system and its environment. Under local detailed balance it is represented by a forward-to-reversed path-probability logarithm and has nonnegative expectation.


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