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I'm pleased to see that there is now a discussion about the physical significance of detailed balance now. But I'm confused by the statement that the unitary nature of operators from quantum mechanics justifies our belief in the reversibility of all physical processes. Isn't this just replacing the assumption of detailed-balance with an assumption about a different set of matrices? It seems to me that the second-law is best justified by our experiences and experiments. —Preceding unsigned comment added by 146.186.131.40 ( talk) 23:33, 15 February 2011 (UTC)
Quantling ( talk · contribs) has (reasonably in my view) put a {{ fact}} tag on the following assertion:
The issue I see with this statement is, why does this require the Markov process to be reversible (i.e. for its stationary state to observe detailed balance) ?
At least for discrete process with a finite number of states, it's plain that entropy is clearly defined; and with each tick of the clock it cannot decrease (data processing theorem). The limiting stationary distribution will therefore have an entropy which is greater than any state previously encountered.
I suppose the problem is to show that there are no saddle-like stationary states, stable to small perturbations in some directions but not others.
Presumably this is quite an important result in the theory of markov chain monte carlo -- to establish that there really is only one stationary state, that the state-probability of the chain must converge to. Jheald ( talk) 18:33, 18 August 2010 (UTC)
I observe that the definition here is not the same as that given at Markov chain#Reversible Markov chain, since π is initially allowed to be anything and is then shown to be a stationary distribution. Both articles are short of actual citations and it would be good to have a citation for someone using the term "detailed balance" —Preceding unsigned comment added by Melcombe ( talk • contribs) 10:20, 23 August 2010
The current text says "Detailed balance is a weaker condition than requiring the transition matrix to be symmetric." However, this is false, yes? That is, a symmetric transition matrix need not represent a process with detailed balance. As such neither condition is weaker than the other, right? Quantling ( talk) 19:16, 18 August 2010 (UTC)
Thank you for explaining that to me. Quantling ( talk) 19:47, 18 August 2010 (UTC)
The current text says "The detailed balance condition is stronger than that required merely for a stationary distribution." I think what is really meant is that detailed balance is not well defined unless the process is stationary. For instance, if and change with time, then at which time(s) do we evaluate the , , , and that appear in
Can we change the text to remove the "stronger" language in favor of something that says that the concept of detailed balance makes sense only for a stationary process? Quantling ( talk) 19:44, 18 August 2010 (UTC)
This page appears to be entirely redundant with the Reversible Markov chain section of the Markov chain page. Shall we replace this page with a redirect to that section (or to an anchor associated with that section)? — Quantling ( talk) 13:46, 27 August 2010 (UTC)
I concur that there is a lot to say about detailed balance in a physics context. My understanding is that detailed balance <=> transition probability matrix is doubly stochastic => 2nd law of thermodynamics. And this applies under general non-equilibrium conditions, not, as the article implies, just to equilibrium states. -- Michael C. Price talk 11:52, 14 September 2010 (UTC)
Dear Colleguaes, Everett states that having a doubly stochastic transition matrix "amounts to a principle of detailed balancing holding" (page 29) and that this implies (the proof is supplied in appendix I) that (Shannon) entropy can't decrease. The doule stochastic matrics means just that the probability conserves (stochasticity from one side) and the equidistribution is an equilibrium ditribution (stochasticity from another side). It has no relations to the reversibility condition . What Everett really used is the balance equation. It is well known since Shannon work (1948) (cited by Everett) that entropy increases for any Markov chain with equidistibuted equilibrium and the conditional entropy with respect to equilibrium also changes monotonically in time for ANY Markov process. These statements have no relation to the detailed balance (the only benefit from the detailed balance is the simple formula for the entropy production. About unitarity and detailed balance: Stueckelberg in 1952 proved that the semi-detailed balance for non-linear Boltzmann kinetics follows from unitarity (or, what is equivalent, from the Markov microkinetics) and this is enough for the entropy growth. Semi-detailed balance, not detailed balance follows from the unitarity. Everett did not make a mistake in his proof but used unconventional terminology: detailed balance instead of balance. He proved the entropy increase for the general Markov chains, not only for the reversible ones. This is the more general statement (BTW, it was proved earlier by Shannon). I propose to delete this reference and to substitute it by more relevant references to earlier results about detailed balance and the second law.- Agor153 ( talk) 15:49, 23 September 2011 (UTC)
The article, and some of the above discussion implies that this connection is obvious, but I don't see it. If it is obvious, can we put a quick explanation in the article? If it is not sufficiently obvious can we at least cite a source? Thanks — Quantling ( talk | contribs) 12:47, 28 April 2011 (UTC)
Forgetting the complications from quantum physics, I believe the current text of the article would support the statement that detailed balance is achieved in classical physics because classical physics is time reversible. However, classical physics is deterministic, and detailed balance concerns a stochastic process. What does it mean that a deterministic process satisfies detailed balance? For instance, what are the physical interpretations of π and P in ? — Quantling ( talk | contribs) 14:26, 29 April 2011 (UTC)
The article says that detailed balance is a "sufficient conditions for the strict increase of entropy in isolated systems". However, from my understanding detailed balance holding means that entropy stays constant because it implies the system is in equilibrium. — Preceding unsigned comment added by Ryrythescienceguy ( talk • contribs) 02:31, 15 May 2022 (UTC)
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I'm pleased to see that there is now a discussion about the physical significance of detailed balance now. But I'm confused by the statement that the unitary nature of operators from quantum mechanics justifies our belief in the reversibility of all physical processes. Isn't this just replacing the assumption of detailed-balance with an assumption about a different set of matrices? It seems to me that the second-law is best justified by our experiences and experiments. —Preceding unsigned comment added by 146.186.131.40 ( talk) 23:33, 15 February 2011 (UTC)
Quantling ( talk · contribs) has (reasonably in my view) put a {{ fact}} tag on the following assertion:
The issue I see with this statement is, why does this require the Markov process to be reversible (i.e. for its stationary state to observe detailed balance) ?
At least for discrete process with a finite number of states, it's plain that entropy is clearly defined; and with each tick of the clock it cannot decrease (data processing theorem). The limiting stationary distribution will therefore have an entropy which is greater than any state previously encountered.
I suppose the problem is to show that there are no saddle-like stationary states, stable to small perturbations in some directions but not others.
Presumably this is quite an important result in the theory of markov chain monte carlo -- to establish that there really is only one stationary state, that the state-probability of the chain must converge to. Jheald ( talk) 18:33, 18 August 2010 (UTC)
I observe that the definition here is not the same as that given at Markov chain#Reversible Markov chain, since π is initially allowed to be anything and is then shown to be a stationary distribution. Both articles are short of actual citations and it would be good to have a citation for someone using the term "detailed balance" —Preceding unsigned comment added by Melcombe ( talk • contribs) 10:20, 23 August 2010
The current text says "Detailed balance is a weaker condition than requiring the transition matrix to be symmetric." However, this is false, yes? That is, a symmetric transition matrix need not represent a process with detailed balance. As such neither condition is weaker than the other, right? Quantling ( talk) 19:16, 18 August 2010 (UTC)
Thank you for explaining that to me. Quantling ( talk) 19:47, 18 August 2010 (UTC)
The current text says "The detailed balance condition is stronger than that required merely for a stationary distribution." I think what is really meant is that detailed balance is not well defined unless the process is stationary. For instance, if and change with time, then at which time(s) do we evaluate the , , , and that appear in
Can we change the text to remove the "stronger" language in favor of something that says that the concept of detailed balance makes sense only for a stationary process? Quantling ( talk) 19:44, 18 August 2010 (UTC)
This page appears to be entirely redundant with the Reversible Markov chain section of the Markov chain page. Shall we replace this page with a redirect to that section (or to an anchor associated with that section)? — Quantling ( talk) 13:46, 27 August 2010 (UTC)
I concur that there is a lot to say about detailed balance in a physics context. My understanding is that detailed balance <=> transition probability matrix is doubly stochastic => 2nd law of thermodynamics. And this applies under general non-equilibrium conditions, not, as the article implies, just to equilibrium states. -- Michael C. Price talk 11:52, 14 September 2010 (UTC)
Dear Colleguaes, Everett states that having a doubly stochastic transition matrix "amounts to a principle of detailed balancing holding" (page 29) and that this implies (the proof is supplied in appendix I) that (Shannon) entropy can't decrease. The doule stochastic matrics means just that the probability conserves (stochasticity from one side) and the equidistribution is an equilibrium ditribution (stochasticity from another side). It has no relations to the reversibility condition . What Everett really used is the balance equation. It is well known since Shannon work (1948) (cited by Everett) that entropy increases for any Markov chain with equidistibuted equilibrium and the conditional entropy with respect to equilibrium also changes monotonically in time for ANY Markov process. These statements have no relation to the detailed balance (the only benefit from the detailed balance is the simple formula for the entropy production. About unitarity and detailed balance: Stueckelberg in 1952 proved that the semi-detailed balance for non-linear Boltzmann kinetics follows from unitarity (or, what is equivalent, from the Markov microkinetics) and this is enough for the entropy growth. Semi-detailed balance, not detailed balance follows from the unitarity. Everett did not make a mistake in his proof but used unconventional terminology: detailed balance instead of balance. He proved the entropy increase for the general Markov chains, not only for the reversible ones. This is the more general statement (BTW, it was proved earlier by Shannon). I propose to delete this reference and to substitute it by more relevant references to earlier results about detailed balance and the second law.- Agor153 ( talk) 15:49, 23 September 2011 (UTC)
The article, and some of the above discussion implies that this connection is obvious, but I don't see it. If it is obvious, can we put a quick explanation in the article? If it is not sufficiently obvious can we at least cite a source? Thanks — Quantling ( talk | contribs) 12:47, 28 April 2011 (UTC)
Forgetting the complications from quantum physics, I believe the current text of the article would support the statement that detailed balance is achieved in classical physics because classical physics is time reversible. However, classical physics is deterministic, and detailed balance concerns a stochastic process. What does it mean that a deterministic process satisfies detailed balance? For instance, what are the physical interpretations of π and P in ? — Quantling ( talk | contribs) 14:26, 29 April 2011 (UTC)
The article says that detailed balance is a "sufficient conditions for the strict increase of entropy in isolated systems". However, from my understanding detailed balance holding means that entropy stays constant because it implies the system is in equilibrium. — Preceding unsigned comment added by Ryrythescienceguy ( talk • contribs) 02:31, 15 May 2022 (UTC)