In mathematics and statistical mechanics, a Markov process is said to show detailed balance if the transition rates between each pair of states i and j in the state space obey
P_{ij} \pi_{i} = P_{ji} \pi_{j}\,
where P is the Markov transition matrix (transition probability), ie Pij = P( Xt =j | Xt−1 = i ); and πi and πj are the equilibrium probabilities of being in states i and j, respectively.
The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and P a transition kernel:
P(s',s) \pi(s') = P(s,s') \pi(s).\,
A Markov process that satisfies the detailed balance equations is said to be a reversible Markov process or reversible Markov chain with respect to π.
Note that the detailed balance condition is stronger than that required merely for a stationary distribution. Detailed balance also implies that around any closed cycle of states, there is no net flow of probability.
\forall a,b,c, P(a,b) P(b,c) P(c,a) = P(a,c) P(c,b) P(b,a)
When a Markov process is reversible, its dynamics can be described in terms of an entropy function that act like a potential, in that the entropy of the process is always increasing, and reaches its maximum at the stationary distribution.
Detailed balance is a weaker condition than requiring the transition matrix to be symmetric, Pij = Pji. That would imply that the uniform distribution over the states would automatically be an equilibrium distribution. However, for continuous systems it may be possible to continuously transform the co-ordinates until a uniform metric is the equilibrium distribution, with a transition kernel which then is symmetric. In the discrete case it may be possible to achieve something similar, by breaking the Markov states into a degeneracy of sub-states.
Such an invariance is a supporting justification for the principle of equal a-priori probability in statistical mechanics.
