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Still many things to add: discussion on existence/uniqueness; phase transitions; alternative definition for translation invariant states using the thermodynamic formalism, etc. One should also make it more readable ;) . --YVelenik 16:14, 5 November 2005 (UTC)
The way the article is written down at the moment (esp. the more introductory parts), it looks as though Gibbs measures and Markov random fields are the same thing. But that's not the case. The concept of Gibbs random field is more general, as it does not require the Markov property (in its usual meaning), thus allowing for (suitable) infinite range interactions in the potential. What replaces the Markov property are the DLR equations (which are stated in the lattice section). This should be stressed, maybe after the Hammersley-Clifford theorem (which is restricted to finite collections of random variables). Also notice that to write that the probability of the state to be x is given by the Boltzmann factor makes only sense for finite collections of random variables. Infinite collections are very important in applications to Statistical Physics and Probability Theory, as it is only for those that a given potential can give rise to several states (several solutions to the DLR equations), i.e. that first-order phase transitions can occur.-- 129.194.8.73 ( talk) 08:52, 15 March 2009 (UTC)
Rewrote the top matter completely. I think it is important to distinguish Gibbs measures, which are a tool for studying infinite systems, from Gibbs distributions, which apply only to finite systems. The previous version of the article conflated the two. Only the latter topic should be merged with canonical ensemble or partition function; the former deserves its own article. I plan to add more on phase transitions, symmetries, and pure states (extreme points in the space of Gibbs measures) when I have the time, but maybe others will also contribute. Eigenbra ( talk) 20:18, 14 August 2014 (UTC)
In the second paragraph:
any probability measure that satisfies a Markov property
What does this mean? I thought a Markov property applied to stochastic processes, not to measures. 178.38.78.134 ( talk) 01:34, 22 January 2015 (UTC)
![]() | This article is rated Start-class on Wikipedia's
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Still many things to add: discussion on existence/uniqueness; phase transitions; alternative definition for translation invariant states using the thermodynamic formalism, etc. One should also make it more readable ;) . --YVelenik 16:14, 5 November 2005 (UTC)
The way the article is written down at the moment (esp. the more introductory parts), it looks as though Gibbs measures and Markov random fields are the same thing. But that's not the case. The concept of Gibbs random field is more general, as it does not require the Markov property (in its usual meaning), thus allowing for (suitable) infinite range interactions in the potential. What replaces the Markov property are the DLR equations (which are stated in the lattice section). This should be stressed, maybe after the Hammersley-Clifford theorem (which is restricted to finite collections of random variables). Also notice that to write that the probability of the state to be x is given by the Boltzmann factor makes only sense for finite collections of random variables. Infinite collections are very important in applications to Statistical Physics and Probability Theory, as it is only for those that a given potential can give rise to several states (several solutions to the DLR equations), i.e. that first-order phase transitions can occur.-- 129.194.8.73 ( talk) 08:52, 15 March 2009 (UTC)
Rewrote the top matter completely. I think it is important to distinguish Gibbs measures, which are a tool for studying infinite systems, from Gibbs distributions, which apply only to finite systems. The previous version of the article conflated the two. Only the latter topic should be merged with canonical ensemble or partition function; the former deserves its own article. I plan to add more on phase transitions, symmetries, and pure states (extreme points in the space of Gibbs measures) when I have the time, but maybe others will also contribute. Eigenbra ( talk) 20:18, 14 August 2014 (UTC)
In the second paragraph:
any probability measure that satisfies a Markov property
What does this mean? I thought a Markov property applied to stochastic processes, not to measures. 178.38.78.134 ( talk) 01:34, 22 January 2015 (UTC)