Detailed balance is the identity pi_i P_ij = pi_j P_ji for every pair of states.
For a continuous-time Markov chain with transition rates , detailed balance is
(1)
for every pair of distinct states. A probability distribution satisfying these equations is an invariant distribution of a continuous-time Markov chain, because summing over incoming rates gives . Under the standard nonexplosion assumptions, the equilibrium chain is reversible.

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  1. Reversible Markov chain
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Detailed balance by Wikipedia Bot 1
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Detailed balance is a principle used in statistical mechanics and thermodynamics that describes a specific condition of equilibrium in a system. It refers to the condition whereby, for every possible transition between states of a system, the rate of transitions in one direction is balanced by the rate of transitions in the reverse direction. This ensures that, over time, the system reaches a steady-state distribution of states.
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