Talk:Detailed balance

Physical Significance

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)

Entropy as Lyapunov function?

Quantling (talk · contribs) has (reasonably in my view) put a {{fact}} tag on the following assertion:

When a Markov process is reversible, its dynamics can be described in terms of an entropy function that acts like a potential, in that the entropy of the process is always increasing, and reaches its maximum at the stationary distribution.

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)

To be clear, are we talking about the entropy of the joint probability distribution of the probability vector at time zero with the probability vector at a later time t? I believe this entropy has to be non-decreasing as t increases, whether or not a stationary Markov chain shows detailed balance, or for what it is worth, whether or not it is ergodic, aperiodic, and/or has a unique stationary state. (Though, if the Markov chain is not a stationary process that changes things.) I wish I could say I knew where to find this in a citable text. Quantling (talk) 20:00, 20 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

I am aware of "detailed balance" very much as a Physics term. Searching google books for '"detailed balance" thermodynamics' brings in almost 10,000 hits. In statistical physics the assumption that equilibrium obeys detailed balance is taken as equivalent to an assumption of "microscopic reversibility" (presumably for the reasons given in the other article).

It's a fair point that we say here that if π is an equilibrium distribution, and the equation holds, then we have detailed balance; whereas the other article shows that if the equation holds, then π must be an equilibrium distribution come what may. It may be that that is not such an interesting result for physicists, as they have already assumed a context of equilibrium thermodynamics from the start, before introducing the idea of detailed balance at all.

The reversibility gives another (though weaker) indication of why π perforce has to be an equilibrium distribution -- it implies that whether the chain is run forward or backwards, the entropy is non-decreasing; therefore it must be at a local maximum. (Though this would not exclude the possibility of the system running through a chain of states, each of equal entropy).

In the next (insert indeterminate time period here) I'll try to look through some standard statistical mechanics textbooks to see what they have to say about detailed balance, to put together a confirmatory references list. But many thanks to Melcombe for pointing out the content in the other article, which I wasn't aware of and found very interesting; and for this spur to further investigation. Jheald (talk) 14:19, 23 August 2010 (UTC)

Detailed balance is a weaker condition than requiring the transition matrix to be symmetric

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)

If you start with a uniform probability distribution, and apply a symmetric transition matrix to it, then at the next tick you will have the same distribution. So this is a stationary state for the transition matrix. By inspection, it clearly also obeys the equation for detailed balance. Therefore it can indeed be said to be a stronger condition. Jheald (talk) 19:37, 18 August 2010 (UTC)

Thank you for explaining that to me. Quantling (talk) 19:47, 18 August 2010 (UTC)

The detailed balance condition is stronger than that required merely for a stationary distribution

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 $P_{ij}$ and $\pi _{i}$ change with time, then at which time(s) do we evaluate the $\pi _{i}$, $\pi _{j}$, $P_{ij}$, and $P_{ji}$ that appear in

$\pi _{i}P_{ij}=\pi _{j}P_{ji}\,$ ?

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)

Although exhibiting detailed balance does not necessarily imply that a Markov process is a stationary process, it does necessarily imply that the Markov process has a stationary distribution. My questions in this section reflect my confounding of these two "stationaries" and are thus somewhat nonsensical. Please disregard them. In the current text, I have partially reverted my edits, which had introduced stationary process where stationary distribution is appropriate. —Quantling (talk) 16:42, 27 August 2010 (UTC)

Replace this page with a redirect to Markov chain#Reversible Markov chain?

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'm minded to leave it as it is at the moment. "Detailed balance" is the term that's heavily used in physics, and I think there is scope to add more on the idea's physical significance. Besides, if you combine everything that is already on this page with everything that is already under Reversible Markov chain, I think that gives more than sits easily as a section of the other article. Better a mention there, handing off to a more detailed and complete discussion per WP:SUMMARY. In my view. Jheald (talk) 14:46, 27 August 2010 (UTC)

There must be a lot to say about why the concept of "detailed balance" is important in its particular context in physics that wouldn't be appropriate for the Markov chain article. Besides which, my guess is that what is wanted in the physics context is actually a continuous-time process, not the discrete-time Markov chain. Melcombe (talk) 09:48, 31 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)

I am not sure this is true. Detailed balance does not in general require the transition matrix to be doubly stochastic; and a general doubly stochastic matrix need not typically obey detailed balance. And any Markov chain will obey the second law of thermodynamics (steadily degrading information). Jheald (talk) 13:57, 14 September 2010 (UTC)

I second that double stochasticity is not required by physics. Quantling (talk) 15:32, 14 September 2010 (UTC)

The source given (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.

Someone has added a clause in the article stating that the 2nd law is implied by the reversibility of physics. This is doubly wrong: it is not implied by reversibility, which is just as well, since physics is not reversible - physics possesses CPT invariance, which is not quite the same.--Michael C. Price ^talk 16:27, 14 September 2010 (UTC)

There are various issues here that probably need to be untangled.

First up, the second law of thermodynamics. In the realm of statistical thermodynamics this is generally "derived" by citing Boltzmann's H theorem. (Though one can take issues with its assumptions -- see eg Kittel [1]; also PCW Davies, p. 43 et seq [2]). The derivation we have on that page assumes a symmetrical transition matrix, and consequentially an equilibrium probability distribution that is uniform over the possible states. If we lump some of the states together, then the equilibrium probability distribution is no longer uniform, but the H-theorem still holds, the "lumping together" process having turned the symmetric transition matrix into a transition matrix which obeys detailed balance. Kubo, p. 60, shows how this changes the derivation of the H theorem. [3]

(Above I wrote that "any Markov chain will obey the second law of thermodynamics (steadily degrading information)". That's in fact clearly not correct, because a Markov chain could be constructed that tends to dump everything into one state. That would degrade the original information, but it would also reduce the total amount of information needed to specify the state of the system. So the entropy, the missing information, would in fact fall. We therefore need the dynamics to preserve the requirement that to specify the state of the system continues to need a large amount of information).

A symmetric transition matrix is doubly stochastic, because each column gives the effect of applying the matrix to a pure state, which must preserve probability and therefore sum to one; and each row is just a transpose of each column. But does being doubly stochastic imply detailed balance? I think not. Consider the deterministic transitions {2->1, 3->2, 1->3}. The transition matrix for this is doubly stochastic. But it does not obey detailed balance. (It describes a probability current flowing round a loop).

Nevertheless, there are connections between conservation of probability, and unitarity, and reversibility of dynamics. They're not quite as simple (I think) as Michael makes out. But it's something I'd need to do more reading and thinking to get to the bottom of. Nevertheless that's what we should do: to expose as clearly as we can what the connections are, and, as importantly, why they are -- what are the reasons for them; and, of course, reference them (connections and explanations), to as mainstream standard works as possible. Jheald (talk) 22:46, 14 September 2010 (UTC)

To respond to an earlier point, physics - i.e. the transition matrix - is doubly stochastic; this follows from the time development operator being unitary and can be verified quite easily. Everett supplied a proof that this double stochasticity implies the 2nd Law - the proof is not complex, just slightly tedious. But the important thing is that this demonstration did not just apply to equilibrium conditions.

Where I am confused is about whether detailed balance is defined for non-equilibrium conditions. Everett gives a definition (he says detailed balance is equivalent to double stochasticity, which requires neither reversibility nor equilibrium conditions) which applies in general, whereas this article defines detailed balance only for equilibrium conditions - where it is dependent on reversibility. A google seems to show that detailed balance has been defined elsewhere for non-equilibrium conditions; whether these definitions are synonomous with Everett's, I am not clear. --Michael C. Price ^talk 05:30, 15 September 2010 (UTC)

Detailed balance basically just means that for each pair of states, there is as much probability flow in one direction as there is in the opposite direction. So each pair of states are in dynamic equilibrium with each other. That equilibrium is central to the notion of detailed balance, so it's not a requirement I think you can just drop. On the other hand, it might be possible to stretch the definition to having just particular pairs of states in equilibrium with each other, and therefore in detailed balance with each other, while the system as whole is not at equilibrium. The Kubo book cited above is non-equilibrium statistical mechanics, so might shed more light on this.

Some talk about detailed balance in the nuclear physics/particle physics can be seen in this section by Blatt and Weisskopf [4], showing why its not the same as "reciprocity" (ie physical time-reversal symmetry, more specifically (I think) CPT symmetry) which follows from unitarity (?). On the other hand, here's a book which says that time-reversal symmetry does imply detailed balance [5] (I suspect it may not be being as careful, but would need to look more closely than just a first glance). This by Sachs also may be helpful [6].

Again, need to do more thinking on this, and see the full chain of logic set out more systematically. But unfortunately, currently I haven't (yet) been able give the time to get to the bottom of it all really thoroughly. Jheald (talk) 08:19, 15 September 2010 (UTC)

(OT) A couple of refs to show how the concept is presented in books on queuing theory [7], [8]. Just parking them here, to make sure we can later check we're picking up the points made. Jheald (talk) 08:24, 15 September 2010 (UTC)

I am unclear about the relationship between unitarity and CPT invariance (both of which are accepted as applying to all of physics), but time symmetry does imply double stochasticity. However perhaps that is better detailed at those appropriate articles. It seems that detailed balance is a more specific concept than I thought, and less dependent on double stochasticity than Everett indicated - although I could well be wrong here - in which case this is not the appropriate article to mention his derivation (or restatement) of the 2nd Law which is only dependent on double stochasticity.

Thanks for the Blatt and Weisskop link - and how annoying that equation 2.18 is not visible. Argh! Many questions left unanswered. --Michael C. Price ^talk 21:38, 15 September 2010 (UTC)

This is just to note that, as well as the Markov chain article, there is related amterial in Balance equation. Melcombe (talk) 08:36, 15 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 $\pi _{i}P_{ij}=\pi _{j}P_{ji}$. 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)

How does a unitary time development operator imply detailed balance?

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)

Perhaps I was too hasty. Here's a good discussion:[9], which I haven't had time to digest yet. Be back later.-- cheers, Michael C. Price ^talk 13:12, 28 April 2011 (UTC)

Okay, here is the logic:

Hamiltonian is Hermitian => time development is unitary => transition matrix is doubly stochastic => detailed balance => 2nd law of thermodynamics. -- cheers, Michael C. Price ^talk 18:42, 28 April 2011 (UTC)

Still not sure that doubly stochastic => detailed balance (per my previous comments above). Though haven't time to look at your link yet. Jheald (talk) 19:26, 28 April 2011 (UTC)

I'm not sure either - see previous section discussion. -- cheers, Michael C. Price ^talk 20:20, 28 April 2011 (UTC)

The part I don't get is "time development is unitary => transition matrix is doubly stochastic". If I am thinking about this correctly ... the unitary matrix has complex entries that transition quantum coefficients to quantum coefficients at the next time step. The transition matrix has real entries that transition probabilities (somehow related to the absolute value squared of quantum coefficients?) to probabilities applicable at the next time step. How does the former being unitary make the latter be doubly stochastic? —Quantling (talk | contribs) 12:27, 29 April 2011 (UTC)

Possibly I'm failing to distinguish between the transition matrix and the probability transition matrix? To be explicit let's take the transition probability matrix elements as T_ij=|<j|U|i>|², where U is the unitary time development operator, then (sum_i) T_ij = (sum_j) T_ij = 1, which is the definition of doubly stochastic. Does that clarify my language? -- cheers, Michael C. Price ^talk 13:47, 29 April 2011 (UTC)

I would say that "doubly stochastic" means that the transition matrix is symmetric. That is, the requirement is T_ij = T_ji for all i and j. Is that what you mean? Is this requirement achieved? —Quantling (talk | contribs) 14:16, 29 April 2011 (UTC)

"Doubly stochastic" is defined at stochastic matrix. The definition does not require T to be symmetrical. Jheald (talk) 15:27, 29 April 2011 (UTC)

Yes, the definition of doubly stochastic is that the rows and columns each sum to one. -- cheers, Michael C. Price ^talk 15:34, 29 April 2011 (UTC)

Thank you for the clarification. I have removed the confusing reference to doubly stochastic in this article.

My understanding is that there are $(n-1)^{2}$ degrees of freedom for doubly stochastic matrices, but $n(n+1)/2-1$ degrees of freedom for stochastic matrices that can be put into detailed balance with the right π. The former is bigger for $n>4$, so for the physics case I think we're going to need something stronger than doubly stochastic to lead us to detailed balance. —Quantling (talk | contribs) 16:16, 29 April 2011 (UTC)

Classical physics and detailed balance

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 $\pi _{i}P_{ij}=\pi _{j}P_{ji}$? —Quantling (talk | contribs) 14:26, 29 April 2011 (UTC)

Deterministic dynamics means that an exact initial state of the system is mapped forward to an exact final state. But we can also use deterministic dynamics to map an initial probability distribution over initial states forward to a final probability distribution over final states. In such a case π_i is simply the probability weight attached to state i. For a strictly deterministic system, the mapping matrix P would have to be made up entirely of ones and zeroes, with only a single 1 in each row. An example of detailed balance can be constructed for such a set-up, but it's a bit freakish: it would require all the states to be paired, and the probability to be ping-ponged back and forth on each iteration between each state and its pair. Deterministic yes, but perhaps physically not very generally applicable.

However thinking we might be able to apply determinism in this way, with these exactly specified physical states, isn't usually very physical either -- for example, it implies specifying the position and velocity of every particle with perfect precision -- i.e. infinite information. Instead, suppose we can only specify the position and (classical) velocity of each particle to a finite precision (this is called moving to a "coarse grained" description of the system) -- ie being able to specify only that each particle is somewhere in a small n dimensional cube in the bigger space. For simplicity, suppose we were only playing with two particles A and B. This "tick" of our discrete clock, our "state of the system" is that they are each in different little cubes. Next tick, the "state of the system" we predict is that both particles happen to have moved into the same little cube. The tick after, where will they be? Well, they might have missed each other completely, in which case we might be able to say pretty accurately where they would end up. But there's also a chance of a close collision -- with the extent of the deflction that each suffers depending on just how close the collision is. But in our coarse-grained description, we can't predict just how close the collision is -- because our description doesn't define the starting positions well enough to say. So the best we can do is give a probability distribution over which pairs of cells the two particles might jointly end up in next tick. So in terms of our matrix P, we are no longer mapping 1 state -> 1 state: we now have to map 1 state -> many states, with a probability weighting over the different final states we might end up in. Also, there may be more than one previous state that would end up in each of the coarse-grained final states. So in fact, when acting on coarse-grained states our matrix P will typically be a many -> many mapping. So, classically, this is how deterministic dynamics can still nevertheless give rise to mixing behaviour; and how evolving forward from even quite a relatively well-defined initial state can still lead relentlessly towards a rather diffuse final equilibrium probability distribution.

Hope that makes at least some kind of sense, and helps. Jheald (talk) 20:50, 29 April 2011 (UTC)

Okay, and I assume that this is somehow done in a way that satisfies detailed balance. Can you or someone else summarize this in a sentence or three for the article so that the section on the physical significance of detailed balance is useful to someone who doesn't already know the answers? —Quantling (talk | contribs) 12:58, 2 May 2011 (UTC)

Detailed balance and entropy production

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)