In mathematics, and in statistical mechanics in physics, 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 Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... It has been suggested that this article or section be merged with Markov property. ...

where P is the Markov transition matrix (transition probability), ie P_ij = P( X_t =j | X_t−1 = i ); and π_i and π_j 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:

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. It applies separately pairwise to each pair of states, so a steady-state probability current A -> B -> C -> A does not suffice.

Detailed balance is a weaker condition than requiring the transition matrix be symmetric, P_ij = P_ji. 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. The principle of indifference is a rule for assigning epistemic probabilities. ...

See also

• Gibbs sampling
• Master equation
• Atomic spectral line (deduction of the Einstein coefficients)
• Random walks on graphs
• DMP