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A disconnected detailed-balance chain does not converge to the full reference law

Physical scope

An exact three-state finite-rate calculation showing why the connectedness condition in the convergence theorem is needed; not empirical data.

Example

Let $S = {1, 2, 3}$ , with $q_{12} = q_{21} = 1$ second $^{- 1}$ and all other off-diagonal rates zero. Let $π = (1 / 3, 1 / 3, 1 / 3)$ and $p (0) = (3 / 8, 1 / 8, 1 / 2)$ .

Facts & Assumptions

Given: The finite detailed-balance rate model.

[F1]

Finite detailed-balance rate and transition models: $π$ satisfies detailed balance; the rate graph has one edge ${1, 2}$ and isolated state $3$ .

[F2]

Relative-entropy decrease and convergence for a finite detailed-balance chain: convergence to $π$ is proved when the full undirected rate graph is connected.

Verification

Given: The master equation on the three stated states.

1.1givenF1algebra

The third probability is constant and $p_{1} + p_{2} = 1 / 2$ . The difference $d = p_{1} - p_{2}$ satisfies $\dot{d} = - 2 d$ with $d (0) = 1 / 4$ , so $d (t) = e^{- 2 t} / 4$ , $p_{1} (t) = 1 / 4 + e^{- 2 t} / 8$ , and $p_{2} (t) = 1 / 4 - e^{- 2 t} / 8$ .

2.1F1F2step 1.1algebra∎

Hence $p (t) \to (1 / 4, 1 / 4, 1 / 2) \neq π$ . The relative entropy tends to $\frac{1}{2} \log (9 / 8) > 0$ , confirming that detailed balance without connectedness does not force convergence to the full reference law.

Depends on

Dependency tree · 2 levels

Sources

C. Jarzynski, Introduction to Nonequilibrium Statistical Physics, lecture notes, Relative Entropy and Relaxation to Equilibrium