Past exam of the biological physics course of the University of Oxford / 2022 / 4 / a

Detailed balance means that at thermodynamic equilibrium each elementary transition and its reverse carry equal probability flux. For normalized state probabilities $A_{\mathrm{eq}}$ and $B_{\mathrm{eq}}$,With the printed convention $\Delta G_{AB}=G_A-G_B$, the sign is thereforeThe $G$ values are free energies per system, including internal degeneracies; for molar Gibbs free energies, replace $k_{B}T$ by $RT$. If B has lower free energy, its forward-to-reverse rate ratio exceeds one, as it should.

For constant rates, conservation gives $A=1-B$, so the master equation is $\dot{B}=k_1(1-B)-k_{2}B$. Its steady state is $B_0=k_1/(k_1+k_2)$, and separation of the decaying deviation yieldsfor the specified initial state A. The relaxation time constant is $(k_1+k_2{)}^{-1}$.