Ruelle, David. “A generalized detailed balance relation.” Journal of Statistical Physics 164, no. 3 (2016): 463-471.
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Given a system M in a thermal bath we obtain a generalized detailed balance relation for the ratio r = π τ ( K → J ) / π τ ( J →K ) of the transition probabilities M:J→KM:J→K and M:K→JM:K→J in time ττ. We assume an active bath, containing solute molecules in metastable states. These molecules may react with M and the transition J→KJ→K occurs through different channels αα involving different reactions with the bath. We find that r=∑pαrα, where pα is the probability that channel α occurs, and rα depends on the amount of heat (more precisely enthalpy) released to the bath in channel αα.