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.