detailed balance in Markov chains

Detailed balance in Markov chains is a condition ensuring that, in a steady state, the rate of transitioning from one state to another equals the reverse rate. This means the probability flow between any two states is balanced, so the overall system remains stable over time. Mathematically, for states \(i\) and \(j\), the probability of being in state \(i\) multiplied by the transition rate to \(j\) equals the probability of being in \(j\) multiplied by the transition rate back to \(i\). This condition simplifies analysis and guarantees the system reaches equilibrium where probabilities remain constant.