nLab detailed balance

Idea

Detailed balance is a strong form of equilibrium for Markov processes? and dynamical systems, stronger than stationarity and independent from ergodicity.

It can be roughly interpreted as a situation where for every two regions (of the state space, etc.) $A$ and $B$, the amount of mass (or probability, etc.) flowing from $A$ to $B$ equals the amount of mass flowing from $B$ to $A$.

Definition

For discrete Markov chains

A finite-state stationary Markov chain on $X$ with stationary measure $p$ and transition matrix? $k$ is said to satisfy detailed balance if and only if

$$p(x)\,k(y|x) \;=\; p(y)\,k(x|y) .$$

Equivalently, if for each $x,y\in X$, the probability of the following two-state trajectories are equal:

$$P(x,y) \;=\; P(y,x) .$$

Note that this is strictly stronger than stationarity. For example, consider a cyclic permutation $k$ on the set $\{a,b,c,d\}$ given by $a\mapsto b, b\mapsto c$, etc. This admits the uniform measure as stationary measure (call it $p$), however we have that

$$p(a)\cdot k(b|a) \;=\; \frac{1}{4} \cdot 1 \;=\; \frac{1}{4}$$

and

$$p(b)\cdot k(a|b) \;=\; \frac{1}{4} \cdot 0 \;=\; 0 .$$

Indeed, the “mass” flows in the direction $a\to b$, but not $b\to a$ (at least, not in a single step). Still, the system is stationary: the amount of mass that leaves $a$ (and goes to $b$) is equal to the amount of mass that enters $a$. It just does not come from $b$.

General case

More generally, given a measure-preserving Markov kernel $k:(X,p)\to(X,p)$, we say that the resulting process satisfies detailed balance if and only if for all measurable subsets $A$ and $B$ of $X$,

$$\int_A k(B|x)\,p(d x) \;=\; \int_B k(A|x)\,p(d x) .$$

In other words, $k$ is its own Bayesian inverse.

The same definition can be given for a class (“semigroup”) of measure-preserving Markov kernels indexed by an arbitrary monoid.

Properties

• A Markov process satisfies detailed balance if and only if it is time-reversible.

Examples

• Every idempotent Markov chain? satisfies detailed balance.

(…)

Related entries

• probability theory, categorical probability
• invariant measure, invariant set
• equilibrium, ergodicity,
• Markov process?, Markov kernel, Markov category
• time-reversal symmetry, time-reversible stochastic process

References

• Wikipedia, Ergodic process

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

category: probability