Detailed balance and time reversal

Let $w_{ij}$ be a continuous-time transition rate from state i to state j, with units of inverse seconds. For probabilities p, define the oriented edge current $J_{ij}=p_iw_{ij}-p_jw_{ji}$. The master equation adds incoming and subtracts outgoing currents at each node. A stationary distribution pi makes every node's total accumulation vanish. Detailed balance instead requires $\pi_iw_{ij}=\pi_jw_{ji}$ on every connected pair. Equal total inflow and outflow need not mean equal flow on each edge.

In a two-state network with both rates positive, stationarity forces the only edge current to zero, so the stationary chain obeys detailed balance. At least one cycle is needed to sustain an independent stationary circulation in such a finite-state model. For a three-state ring, clockwise currents can enter and leave each node at the same rate, preserving all populations. The absence of population change then says nothing about whether forward and reverse transitions are individually balanced. Counting transitions as well as occupations reveals the hidden motion.

Multiplying detailed-balance relations around a closed cycle cancels the stationary probabilities. The product of forward rates must equal the product of reverse rates. Conversely, on an irreducible finite graph with bidirectional edges, satisfying this condition on all cycles allows a consistent reversible stationary distribution to be constructed. The logarithm of the product ratio is a dimensionless cycle affinity. A nonzero value rules out detailed balance even when the stationary probabilities are known exactly and appear symmetric.

For a thermally consistent transition model, local detailed balance relates log rate ratios to energy and reservoir entropy exchanges. A cycle affinity can then represent a chemical or mechanical driving force divided by thermal energy. Purely assigning a Markov matrix does not automatically specify that thermodynamic interpretation. Without energies, reservoirs, and coupling assumptions, the cycle test diagnoses stochastic irreversibility, while an actual heat or fuel consumption rate requires more physical information. This separation prevents a convenient mathematical chain from being mistaken for a fully calibrated thermodynamic machine.

For states representing positions or other time-even variables and time-independent reversible dynamics, detailed balance equates stationary probabilities of a path and its reversed sequence. If velocities are included, physical time reversal also changes their signs. Magnetic fields or moving boundary protocols may require corresponding reversals. Simply playing a movie backward without transforming the appropriate external controls is not the same experiment. The mathematical definition of a reverse path must match the physical parity and the chosen dynamics.

Reversibility is also distinct from convergence to stationarity. A discrete chain that alternates deterministically between two states obeys detailed balance with equal stationary weights, yet an ensemble started in one state never converges to those weights. Aperiodicity or appropriate continuous-time assumptions are needed for relaxation. These qualifications matter when a laboratory instrument uses discrete transitions while a physical model uses continuous rates. Both can teach balance, but their transition parameters, clock units, and convergence behavior must not be silently identified.