Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 1 / d

Condition on the molecule being empty. Its instantaneous incoming-charge jump rate is $f_1$, so the unconditional stationary number of arrivals per unit time isThis stationary jump traffic at a single contact is a unidirectional jump count, not a signed net current. Stationarity also gives $R_{\mathrm{out}}=f_4{p}_A^{\mathrm{st}}=R_{\mathrm{in}}$. Multiplying their difference by the charge gives zero net stationary electrical current through the molecule's single contact, even though individual arrival and departure events continue.

An $A\to B$ hop can occur only while the charge occupies $A$. Hence its stationary number per unit time isThe reverse count is $R_{B\to A}=f_3{p}_B^{\mathrm{st}}$, which equals this result. There is therefore no signed stationary current along that edge. The individual hopping traffic need not equal the contact-arrival traffic; their ratio is $R_{A\to B}/R_{\mathrm{in}}=f_2/f_4$. A charge can make several internal visits before leaving, so neither conservation nor the single-charge constraint forces those two one-way rates to coincide.