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.
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For a Markov chain with transition matrix , Detailed balance with a probability distribution meansThe left side is the equilibrium Probability of the directed transition ; Detailed balance equates it to the reverse flow. Summing over gives , 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]
Mixing Time Bounds via the Spectral Profile by Sharad Goel and others. Section 1.1: invariant measure and reversibility
For example, takeThe off-diagonal flows are both , 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.