A condition for a Markov process with stationary distribution π such that for every pair of states i,j the probability fluxes balance: π_i P_{ij} = π_j P_{ji}, implying time-reversal symmetry (reversibility).

Principle

Detailed balance states that, under the invariant measure π, the one-step transition rate from i to j equals that from j to i, so microscopic transitions cancel in equilibrium and the process is statistically indistinguishable from its time-reverse.

Demonstration

For a random walk on an undirected graph with transition probabilities P_{ij}=1/deg(i) to neighbors, the stationary measure π_i ∝ deg(i) satisfies π_i P_{ij} = π_j P_{ji}, showing detailed balance and reversibility.

Misapplication

Assuming that every stationary distribution implies detailed balance; stationarity (πP=π) does not require pairwise flux equality and many stationary chains are non-reversible and have net probability currents.

Consequence

If detailed balance holds the chain is reversible, spectral analysis simplifies (self-adjointness in weighted L^2(π)), equilibrium expectations obey time symmetry, and convergence diagnostics and sampling schemes (e.g., MCMC) can exploit reversibility.

Reversal

Breaking detailed balance yields irreversible dynamics with nonzero circulation in state space and asymmetric fluxes π_i P_{ij} ≠ π_j P_{ji}, which can enhance mixing or produce sustained currents absent in reversible chains.

Boundary

Applies to Markov chains and Markov processes with well-defined transition kernels and an invariant measure; not applicable to non-Markovian processes, processes lacking an invariant measure, or cases where transitions are not pairwise comparable.

Semantic Tension

Often conflated with stationarity: stationarity demands global balance of inflow and outflow for each state, while detailed balance is a stronger, pairwise condition that implies stationarity but is not implied by it.

Synthesis

Detailed balance requires equality of pairwise probability fluxes under the invariant measure, making the Markov process reversible; it is a stricter symmetry condition than mere stationarity and has strong spectral and sampling consequences.