Detailed balance is the equality of the equilibrium probability flows between each pair of states of a Markov chain.
For a transition matrix 
and a statistical
distribution 
of state probabilities, its equations are
 |
Summing over 
shows that 
,
so detailed balance implies that 
is a stationary distribution.
The converse is not generally true. If 
for every state, detailed balance makes the Markov
chain reversible when initialized with statistical
distribution 
(Norris 1997). Reversibility means that the probabilities
of finite sequences of consecutive
states are unchanged by reversing their order.
For a continuous-time Markov process with a finite set of states and transition
rates 
,
the corresponding equations are
 |
These are the equilibrium conditions around which the minimum entropy production principle is expanded (Maes and Netočný 2007).
