Detailed balance

From Wikipedia, the free encyclopedia

Jump to: navigation, search

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

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:

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, 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.

[edit] See also

Gibbs sampling

Master equation

Atomic spectral line (deduction of the Einstein coefficients)

Random walks on graphs

DMP

Retrieved from "http://en.wikipedia.org/wiki/Detailed_balance"

Categories: Probability theory | Stochastic processes | Statistical mechanics

Views

Article

Discussion

Edit this page

History

Personal tools

Log in / create account

Navigation

Main Page

Contents

Featured content

Current events

Random article

Interaction

About Wikipedia

Community portal

Recent changes

Contact Wikipedia

Donate to Wikipedia

Help

Search

Toolbox

What links here

Related changes

Upload file

Special pages

Printable version

Permanent link

Cite this page

This page was last modified on 9 March 2008, at 00:02.

All text is available under the terms of the GNU Free Documentation License. (See Copyrights for details.)
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a U.S. registered 501(c)(3) tax-deductible nonprofit charity.

Privacy policy

About Wikipedia

Disclaimers