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A positive balance function $\Phi$ expresses each active route's aggregate service as this ratio. With independent Poisson arrivals and exponential document sizes, detailed balance yields stationary weights $\Phi (n){\prod }_r{\rho }_r^{n_r}$, provided their sum is finite.

The Barker acceptance probability accepts a proposal using the displayed ratio. If $a=\pi (x)q(x,y)$ and $b=\pi (y)q(y,x)$, the accepted flux is $ab/(a+b)$, symmetric in the two states. This proves detailed balance; define acceptance as zero when $a=b=0$. Rejected proposals contribute to the diagonal transition probability. Replacing $\pi$ by an unnormalized target leaves the ratio unchanged. For a common proposal, $t/(1+t)\le \min (1,t)$ shows that its acceptance probability never exceeds the Metropolis–Hastings acceptance probability. The name refers to A. A. Barker's work on Monte Carlo methods for a plasma; the initials are retained without an unverified expansion.^[ref]

On a finite graph with symmetric edge-swap rates, the stirring representation of symmetric exclusion transports the occupations by a random permutation. An independent identically distributed Bernoulli distribution occupation field is invariant under every deterministic permutation, and hence under this random permutation when it is independent of the initial field. Therefore the displayed product measure is invariant, including densities zero and one. Equivalently, an edge swap preserves the number of particles and has the same reverse rate, giving detailed balance.

For nested regression models, append one coefficient $u$ drawn from density $g_k$ while retaining the variance $v$ and existing coefficients $a$. The dimension-matching map $(a,v,u)\mapsto (a,u,v)$ has absolute Jacobian one. With joint model/parameter target $t_k$, birth-selection probability $b_k$, and reverse death-selection probability $d_{k+1}$, the displayed acceptance probability and its reciprocal death rule enforce detailed balance. Include normalized parameter-prior densities and a model-order prior in $t_k$: constants that depend on model order cannot be dropped from a cross-model ratio.

A Boltzmann machine is a stochastic neural network that learns a probability distribution specified by pairwise energies. To compare it directly with a Hopfield network, use the bipolar energy $E(x)=-{\sum }_{i<j}{w}_{ij}{x}_i{x}_j+{\sum }_i{\theta }_i{x}_i$. Here $\beta >0$ controls the preference for low energy, and the partition function $Z={\sum }_x{e}^{-\beta E(x)}$ normalizes all states. In a computational model, $\beta$ is an algorithmic inverse temperature; interpreting it as $1/(k_{B}T)$ additionally requires physical energy units.

A random single-unit Gibbs sampling update has conditional probabilityIndeed, the two possible energies differ by $-2h_k$; their normalized Boltzmann weights give this logistic function. Random selection of one unit at a time satisfies detailed balance. For finite weights and finite positive $\beta$, all conditional probabilities lie strictly between zero and one, so the finite Markov chain can reach every state and can remain in place. Its unique stationary distribution is $p_{\beta }$. A short run need not yet sample that distribution accurately.

As $\beta \to \infty$, a nonzero field selects its sign, approaching asynchronous Hopfield energy descent. At zero field the stochastic rule remains a fair coin, whereas the deterministic convention above retains the old value. At finite inverse temperature, uphill moves permit exploration beyond one local minimum. Recall and learning are different operations: sampling changes the units at fixed weights; learning changes the weights to fit the training data.

A stable equilibrium M-M-s queue has a Poisson process of departures with the same rate as its input. Its present population is also independent of past departures, a component of quasireversibility. The birth-death process satisfies detailed balance; reversed upward jumps have a constant arrival rate, so they are generated by an independent Poisson process.

Carrier-sense multiple access lets a station attempt transmission only when interfering neighbors are inactive. An ideal continuous-time model gives inactive station $r$ an exponential attempt rate $e^{{\theta }_r}$ and unit-rate exponential transmission completions. Detailed balance then gives activation weights $e^{\theta \cdot n}$ on the independent sets of the interference graph.

A dimensionless atomic collision measure used to parameterize electron-impact transition coefficients. In the Maxwellian approximation used here, $q_{ji}=8.6\times 10^{-12}{\Omega }_{ji}/(g_j\sqrt{T_e})$ in ${\mathrm{m}}^3{\mathrm{s}}^{-1}$, and detailed balance fixes the excitation coefficient. Its value and temperature dependence encode atomic scattering physics beyond the elementary Boltzmann factor.

At a positive concentration vector $c$, the total incoming deterministic reaction flux equals the total outgoing flux at every reaction complex. Here $c^y={\prod }_i{c}_i^{y_i}$ is the law of mass action monomial. This is stronger than balancing net production of each chemical species, but does not require pairwise detailed balance between reverse chemical reactions. A directed cycle can have nonzero circulation while satisfying complex balance.

A single-coordinate Gibbs sampler update preserves the remaining coordinates and independently redraws the selected coordinate from its full conditional distribution. The joint measure of old and new states factors into a marginal and two identical conditional factors, making it symmetric. A fixed-probability mixture of such kernels therefore satisfies detailed balance.

Detailed balance does not imply that a Markov chain is an irreducible Markov chain: the identity transition matrix is reversible for every probability distribution. Even for an irreducible Markov chain with aperiodicity, the two-state matrix with diagonal $1-\epsilon$ and off-diagonal $\epsilon$, $0<\epsilon <1/2$, has uniform stationary distribution and nontrivial eigenvalue $1-2\epsilon$. From either fixed state its total variation distance after $n$ steps is $\frac{1}{2}(1-2\epsilon {)}^n$, so convergence can be arbitrarily slow. Invariance, reachability and mixing time are distinct questions.

A time-homogeneous Discrete-time Markov chain is a sequence of random variables $Y_0,Y_1,\dots$ indexed by nonnegative integers, with the displayed conditional probabilities. Here ${\mathcal{F}}_n$ is its natural filtration and $P$ is a row-stochastic transition matrix. The Markov property removes dependence on earlier observations once $Y_n$ is specified; the time-homogeneous Markov property makes $P$ independent of $n$.

Consider states $S$ (sunny) and $R$ (rainy), with the constructed transition matrixRows give today's state and columns tomorrow's, in order $S,R$. These numbers define an illustrative model, with one step interpreted as one day; they are not estimates of real weather. Let $q_n=\mathbb{P}(Y_n=S)$. The law of total probability gives the recurrence relationSubtracting $2/3$ makes the deviation multiply by $0.4$ at each step. ThusStarting sunny gives $q_0=1$, $q_1=0.8$, $q_2=0.72$ and $q_3=0.688$; starting rainy gives $q_0=0$ and $q_2=0.56$. The two predictions approach the same stationary distribution while retaining a shrinking trace of the initial probability distribution.

An expected hitting time answers a different question. Starting sunny, let $T_R=\inf \{n\ge 0:Y_n=R\}$. Every day before the first rain remains sunny, soThis geometric distribution has finite expected value. A first-step calculation gives $h_S=1+0.8h_S+0.2h_R$, with $h_R=0$, hence $h_S=5$ days. The same result follows from the geometric series ${\mathbb{E}}_S{T}_R={\sum }_{n\ge 0}0.8^n=5$. This is a mean waiting time, not a prediction that rain first occurs exactly five days later.

A probability row vector $\pi$ is a stationary distribution when $\pi P=\pi$. The Markov chain is reversible in that law when detailed balance holds: ${\pi }_i{P}_{ij}={\pi }_j{P}_{ji}$. Multiplying those equations along a finite path gives equal forward and reversed stationary path probabilities.

A minimal bacterial flagellar motor model combines angular velocity $\omega >0$ with bound and detached probabilities $P_A+P_B=1$. With detachment rate $k_0$ in the first half-turn and attachment rate $k_0$ in the second, the periodic stationary bound probability is $P_A=\alpha e^{-\beta \theta }$ on $0\le \theta \le \pi$, where $\alpha =(1+e^{-\pi \beta }{)}^{-1}$, and $P_A(\theta +\pi )=1-P_A(\theta )$. The one-way switching windows describe externally driven kinetics rather than detailed balance with a passive heat bath.

The Ehrenfest urn model moves one uniformly chosen ball between two boxes at each step. For $k\ge 1$ balls, the count in one box is a Markov chain on $\{0,\dots ,k\}$. Its stationary distribution is $2^{-k}\left(\genfrac{}{}{0ex}{}{k}{i}\right)$, and detailed balance makes it a reversible Markov chain. The chain has period two, but its mean recurrence time to the all-in-one-box state is still $2^k$.

For an ideal emitter able to exchange one fermion with a zero-chemical potential bath, Pauli blocking gives downward rate $A(1-f)$ rather than the bosonic $A(1+f)$. If $u_{\nu }=h\nu {\mathcal{D}}_{\nu }f$ and $C=A/(h\nu {\mathcal{D}}_{\nu })>0$, the rate is $A-C{u}_{\nu }$. Detailed balance with a Fermi-Dirac distribution implies $g_1{B}_{12}=g_{2}C$ and $A/C=h\nu {\mathcal{D}}_{\nu }$. For $d$ massless helicity states, ${\mathcal{D}}_{\nu }=d4\pi {\nu }^2/c^3$. Thus an additive induced-emission coefficient would be negative, expressing blocking rather than a negative total rate. The emitter must change its fermion parity; two ordinary electronic energy levels cannot exchange a single neutrino this way.

On a finite state space $F$ with finite objective function values, the fixed-$T$ target is a Boltzmann distribution:The partition function $Z_T$ normalizes the weights but cancels from the Metropolis–Hastings acceptance probability. A short detailed balance proof explains the target. With symmetric $q$ and $x\ne y$,Summing over $x$ gives ${\pi }_T{P}_T={\pi }_T$, so this is a stationary distribution. On a finite state space, a connected graph of positive symmetric proposals makes the accepted Markov chain an irreducible Markov chain. Convergence from any starting state additionally requires an aperiodic Markov chain; one positive self-transition suffices under irreducibility. Replacing $q$ by $(I+q)/2$ ensures this. For asymmetric $q$, require connectivity through bidirectional proposals: one-way proposals are rejected.^[ref]

In the five-state barrier example for simulated annealing, $T_{\ast }=1/\ln 2$ gives exact Boltzmann weights $2^{-E(x)}$. Multiplying by $16$ givesFor the uphill edge, detailed balance is visible numerically:Thus the global minimum has the largest individual stationary probability, $16/33$, but is far from certain at this positive $T_{\ast }$. At $T_{\ast }/2$ the exact stationary distribution is $(16,64,1,16,256)/353$.

For any finite $F$, let $M$ be the set of global minima and $E_{\min }$ the smallest objective function value. Cancelling $e^{-E_{\min }/T}$ from numerator and denominator shows that each nonminimal state's weight tends to zero as $T\downarrow 0$, while every state in $M$ has equal weight. Therefore ${\pi }_T$ tends to the uniform distribution on $M$. This is a statement about a family of equilibrium probability distributions; it does not prove that a cooling Markov chain follows that family.

In a positive recurrent birth-death-immigration jump chain, let $I_n$ and $A_n$ count immigration events and upward steps before time $n$. The centered immigration indicators form bounded martingale differences; the strong law for martingales with bounded increments and the countable-state Markov chain ergodic theorem give $I_n/n\to {\sum }_k{\pi }_k\theta /(k+\theta )$ almost surely. The analogous argument for upward steps gives $A_n/n\to {\sum }_k{\pi }_k{u}_k$. Summing detailed balance gives ${\sum }_k{\pi }_k{u}_k={\sum }_k{\pi }_k{d}_k=1/2$, since every step is upward or downward. Therefore $I_n/A_n$ converges to the displayed fraction. For this chain the explicit stationary distribution reduces the fraction to $1-p/(1-p)$.

An involutive Metropolis proposal uses a deterministic bijection $S$ satisfying $S^2=I$. For a volume-preserving $S$ and target probability density function $\rho$, accept $S(z)$ with probability $\min (1,\rho (S(z))/\rho (z))$. The identity $\rho (z)\alpha (z)=\min (\rho (z),\rho (S(z)))$ and a change of variables under $S$ prove detailed balance. A non-unit Jacobian determinant must be included when volume is not preserved.

A last-exit decomposition partitions a transient path event according to the final visit to a finite set. Reversing the finite path before that visit and using detailed balance converts last-exit probabilities into hitting probabilities weighted by an equilibrium measure of a finite set.

The Metropolis–Hastings algorithm proposes $y$ from $q(x,\cdot )$ at state $x$ and accepts it with probabilityThis acceptance rule enforces detailed balance with the target density $\pi$.