 Detailed balance

For a Markov chain with transition matrix $P$, Detailed balance with a probability distribution $\pi$ meansThe left side is the equilibrium Probability of the directed transition $i\to j$; Detailed balance equates it to the reverse flow. Summing over $i$ gives ${\sum }_i{\pi }_i{P}_{ij}={\pi }_j{\sum }_i{P}_{ji}={\pi }_j$, so Detailed balance implies stationarity. Under the stationary initial law, repeated application of the pairwise identity equates a finite path's Probability to that of its reversal, establishing reversibility.^[ref]

For example, takeThe off-diagonal flows are both $ab/(a+b)$, which verifies Detailed balance and the Stationary distribution directly. A directed three-state cycle has a uniform Stationary distribution but no reverse transitions, so it is not reversible. Stationarity alone therefore does not imply Detailed balance, and Detailed balance alone does not imply rapid convergence.

Detailed balance is a principle used in statistical mechanics and thermodynamics that describes a specific condition of equilibrium in a system. It refers to the condition whereby, for every possible transition between states of a system, the rate of transitions in one direction is balanced by the rate of transitions in the reverse direction. This ensures that, over time, the system reaches a steady-state distribution of states.