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

Write the probability column vector as $\mathbf{p}=(p_0,p_A,p_B{)}^T$, with $0$ denoting the uncharged state. The arrows in the original figure give transitions $0\to A$ at rate $f_1$, $A\to B$ at rate $f_2$, $B\to 0$ at rate $f_3$, and $B\to A$ at rate $f_4$. Exclusion of double occupation means injection occurs only from state $0$. Thus the master equation isFor example, ${\dot{p}}_B=f_2{p}_A-(f_3+f_4)p_B$. Every allowed jump adds to one probability and subtracts from another, which verifies the zero column sums.