A Generalized Detailed Balance Relation

Abstract

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