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

The first law of thermodynamics, with $\Delta Q$ defined as heat lost by the particle to the bath, isThus $W-\Delta H=\Delta Q$, and part d becomes the local detailed balance identityThe logarithmic path-probability ratio is the bath's entropy production $\Delta S_{\mathrm{bath}}=\Delta Q/T$ in units of $k_B$.