Detailed balance

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 P_ij = P( X_t =j | X_t−1 = i ); and $\pi _{i}$ and $\pi _{j}$ are the equilibrium probabilities of being in states i and j, respectively.

The definition carries over straightforwardly to continuous variables, where $\pi$ 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 $\pi$ .

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 it's minimum at the stationary distribution.

Detailed balance is a weaker condition than requiring the transition matrix to 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.

See also