Balance, Detailed Balance, Super Detailed Balance

Introduction

The important concepts with lots of applications in many fields, usually suffer from the the ambiguity induced by presenting the same subject with different languages. In my experience, probability theory is a great example for this phenomena. It is a so useful concept that its applications ranges multiple discipline in physics, biology, chemistry, ecology, computer science, financial market, industrial and urban planning, and etc. And because of this reason different people have introduced these concepts with different language that has lead to many ambiguities in diversity in notation, that is a nightmare for someone trying to learn them for the first time.

The notion of balance, detailed balance, and as discussed by some papers, super-detailed balance, is also suffering from this wide variety of different expositions. Of course, one way to prevent such confusions is to use the proper mathematical language to communicate the ideas. The problem is that the new students are not usually fluent in the mathematics language, and they are often exposed to the hand-wavy expositions on the web. In this document I’ll try to give a clean exposition of these notions that is understandable by young undergraduate students from different fields.

Please note that this document assumes familiarity with Markov Chains. So please see probability-1 for more details on the background.

Balance

Let $(S,P,{\pi }^{(0)})$ be a Markov chain, where $S$ is the underlying space, $P$ is the transition probabilities, and ${\pi }^{(0)}$ is some probability measure on $S$ (interpreted as a row vector). The probability measure evolves as

$${\pi }^{(n)}={\pi }^{(n-1)}P.$$

The stationary probability measure (stationary probability distribution) is a distribution $\pi$ that is invariant under $P$. I.e.

$$\pi =\pi P.$$

In components this translates to

$${\pi }_i=\sum _j{\pi }_j{P}_{ij}.$$

This is what often referred to as the balance condition. But what is the reason behind this specific terminology? The terminology becomes more tangible by translating the probability measure of state $i$, i.e. ${\pi }^{(n)}(i)$ as some mass in this state.

To be more specific, assume we have unit amount of mass, and we want to put that mass in different states, i.e. the elements of $S$, according to their probability measure. So the amount of mass in state $i\in S$ is ${\pi }^{(n)}(i)$. Then the equation

$${\pi }^{(n+1)}={\pi }^{(n)}P,$$

translates to how much mass is moving between each state. Let $P_{ij}$ denote the fraction of mass in $i$ that will move to the state $j$ at each step. Observe that since ${\sum }_j{P}_{ij}=1$ for all $i$, it implies that at each step, all the mass comes out of state $i$ and different fractions of it gets distributed between the neighbors of $i$ (according to the fractions $P_{ij}$). With this terminology, at each step (denote by Markov update), all the incoming mass to $i$ is

$$\text{In}(i)=\sum _j{P}_{ji}{\pi }_j^{(n)},$$

and the outgoing mass from state $i$ will be

$$\text{Out}(i)=\sum _j{P}_{ij}{\pi }_i^{(n)}={\pi }_i^{(n)}.$$

So the following conditions are equivalents

$$\text{In}(i)=\text{Out}(i)\text{for all i}⟺{\pi }^{(n+1)}={\pi }^{(n)}P.$$

So the balance condition for a probability distribution $\pi$ is the condition that with each Markov update, the incoming and outgoing mass from each node is the same.

Detailed Balance

Let $m_{i\to j}$ denote the amount of mass that flows from the node $i$ to $j$ at each Markov update. As discussed above, we have

$$m_{i\to j}={\pi }_i{P}_{ij}.$$

So the equation for $\text{In}(i)$ can be written as

$$\text{In}(i)=\sum _j{P}_{ji}{\pi }_j^{(n)}=\sum _j{m}_{j\to i},$$

i.e. as the sum of all the masses coming to this state through different channels. Similarly the outgoing sum can be written as

$$\text{Out}(i)=\sum _j{P}_{ij}{\pi }_i^{(n)}=\sum _j{m}_{j\to i}.$$

Of course one necessary condition for $\text{In}(i)=\text{Out}(i)$ to hold is to set

$$m_{i\to j}=m_{j\to i},\text{or equivalently},{\pi }_i{P}_{ij}={\pi }_j{P}_{ji}.$$

This is what referred to as detailed balance.

Super-Detailed Balance

In chemistry and physics literature, you might encounter the notion of super detailed balance and you might have asked “WTF? (Where’s The Fine-print?)”. It might sound very complicated, but the idea is still simple and easy to understand. Remember $m_{i\to j}$? Now assume that there are many finer micro channels in the channel from $i$ to $j$ and similarly in the channel from $j$ to $i$. So we can write

$$m_{i\to j}=\sum _r{m}_{i\to j}^{(r)},m_{j\to i}=\sum _r{m}_{j\to i}^{(r)}.$$

Note that we are assuming there are $r$ micro channels from $i$ to $j$ and same as $j$ to $i$. In order for detailed balance to hold, we need to have $m_{i\to j}=m_{j\to i}$. However, one sufficient condition for this to hold is to demand

$$m_{i\to j}^{\text{(r)}}=m_{j\to i}^{(r)}\text{for all r}.$$

This is called super-detailed balance.

In short, if there are multiple ways to go from state $i$ to state $j$, and the same number of ways to go from $j$ to $i$, in order for detail balance to hold, we ask the flow of mass in pairs of micro-channels to be the same.

So in summary we have

$$\text{super-detailed balance}⟹\text{detailed balance}⟹\text{balance}.$$

When this happens in reality?

There are multiple scenarios in physics and biology that we make intermediate choices to make an specific transition. Some times a group of these transitions all lead to the same transition, and this can be modeled as having fine transition channels between states.