Detailed balance Totally Explained

Everything about Detailed Balance totally explained

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 ť

P_

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:
ť

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. It applies separately pairwise to each pair of states, so a steady-state probability current A -> B -> C -> A doesn't 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.

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