For many physical systems initial distributions of a stochastic variable y tend to equilibrium distributions: w(y, t)\rightarrow w_0(y) as t\rightarrow\infty . In equilibrium detailed balance constrains the transition rates
W(y\rightarrow y')w(y ) = W(y'\rightarrow y)w_0 (y),where W(y'\rightarrow y) is the probability, per unit time, that the system changes from a state |y\rangle , characterized by the value y for the stochastic variable Y , to a state |y'\rangle .
Note that for a system in equilibrium the transition rate W(y'\rightarrow y) and the reverse W(y\rightarrow y') may be very different.