The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reverse process.
History
The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle. The arguments in favor of this property are founded upon microscopic reversibility.
Five years before Boltzmann, James Clerk Maxwell used the principle of detailed balance for gas kinetics with the reference to the principle of sufficient reason. He compared the idea of detailed balance with other types of balancing (like cyclic balance) and found that "Now it is impossible to assign a reason" why detailed balance should be rejected (pg. 64).
Albert Einstein in 1916 used the principle of detailed balance in a background for his quantum theory of emission and absorption of radiation.
In 1901, Rudolf Wegscheider introduced the principle of detailed balance for chemical kinetics. In particular, he demonstrated that the irreversible cycles A1 -> A2 -> \cdots -> A_\mathit{ n} -> A1 are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance. In 1931, Lars Onsager used these relations in his works, for which he was awarded the 1968 Nobel Prize in Chemistry.
The principle of detailed balance has been used in Markov chain Monte Carlo methods since their invention in 1953. In particular, in the Metropolis–Hastings algorithm and in its important particular case, Gibbs sampling, it is used as a simple and reliable condition to provide the desirable equilibrium state.
Now, the principle of detailed balance is a standard part of the university courses in statistical mechanics, physical chemistry, chemical and physical kinetics.
Microscopic background
The microscopic "reversing of time" turns at the kinetic level into the "reversing of arrows": the elementary processes transform into their reverse processes. For example, the reaction
transforms into
and conversely. (Here, are symbols of components or states, are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of microreversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.
This reasoning is based on three assumptions: does not change under time reversal; Equilibrium is invariant under time reversal; The macroscopic elementary processes are microscopically distinguishable. That is, they represent disjoint sets of microscopic events.
Any of these assumptions may be violated. For example, Boltzmann's collision can be represented as where is a particle with velocity v. Under time reversal transforms into . Therefore, the collision is transformed into the reverse collision by the PT transformation, where P is the space inversion and T is the time reversal. Detailed balance for Boltzmann's equation requires PT-invariance of collisions' dynamics, not just T-invariance. Indeed, after the time reversal the collision transforms into For the detailed balance we need transformation into For this purpose, we need to apply additionally the space reversal P. Therefore, for the detailed balance in Boltzmann's equation not T-invariance but PT-invariance is needed.
Equilibrium may be not T- or PT-invariant even if the laws of motion are invariant. This non-invariance may be caused by the spontaneous symmetry breaking. There exist nonreciprocal media (for example, some bi-isotropic materials) without T and PT invariance.
If different macroscopic processes are sampled from the same elementary microscopic events then macroscopic detailed balance may be violated even when microscopic detailed balance holds.
Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.
Reversible Markov chains
A Markov process is called a reversible Markov process or reversible Markov chain if it satisfies the detailed balance equations. These equations require that the transition probability matrix, P, for the Markov process possess a stationary distribution (i.e. equilibrium probability distribution) π such that