Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 3 / e

For $f>0$ set $\tau =ft$ and $y(\tau )=p_0(t)$. The master equation becomesUsing $p_A=1-y-p_B$ and $p_B=(y^{\prime }+y)/2$ givesInitially the molecule occupies $B$, so $y(0)=0$ and $y^{\prime }(0)=2$. The characteristic roots are $-2,-3$; the constant particular solution is $1/3$. ConsequentlywhenceIt has the required initial slope $2f$, tends to the stationary uncharged probability $1/3$, and remains nonnegative. As a further check, $p_B=1/6-e^{-2ft}/2+4e^{-3ft}/3$ and $p_A=1/2-e^{-2ft}/2$ are nonnegative and sum with $p(t)$ to one. If $f=0$, no jump occurs and $p(t)=0$, also obtained directly from the boxed expression.