To use all functions of this page, please activate cookies in your browser.
my.chemeurope.com
With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.
 My watch list
 My saved searches
 My saved topics
 My newsletter
Detailed balanceIn 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 Additional recommended knowledgewhere 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 steadystate 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 coordinates 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 substates. Such an invariance is a supporting justification for the principle of equal apriori probability in statistical mechanics. See also

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Detailed_balance". A list of authors is available in Wikipedia. 