A generalized detailed balance relation

Given a system $M$ in a thermal bath we obtain a generalized detailed balance relation for the ratio $r=π_τ(K\to J)/π_τ(J\to K)$ of the transition probabilities $M:J\to K$ and $M:K\to J$ in time $τ$. We assume an active bath, containing solute molecules in metastable states. These molecules may react with $M$ and the transition $J\to K$ occurs through different channels $α$ involving different reactions with the bath. We find that $r=\sum p^αr^α$, where $p^α$ is the probability that channel $α$ occurs, and $r^α$ depends on the amount of heat (more precisely enthalpy) released to the bath in channel $α$.

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