In 1901, Rudolf Wegscheider introduced the principle of detailed balance for chemical kinetics.
are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.
Albert Einstein in 1916 used the principle of detailed balance in a background for his quantum theory of emission and absorption of radiation.
[6] The principle of detailed balance has been used in Markov chain Monte Carlo methods since their invention in 1953.
[7] 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.
Detailed balance for Boltzmann's equation requires PT-invariance of collisions' dynamics, not just T-invariance.
[11][12] Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.
Reversibility is equivalent to Kolmogorov's criterion: the product of transition rates over any closed loop of states is the same in both directions.
For a Markov transition matrix and a stationary distribution, the detailed balance equations may not be valid.
However, it can be shown that a unique Markov transition matrix exists which is closest according to the stationary distribution and a given norm.
Thus, the principle of detailed balance is a sufficient but not necessary condition for entropy increase in Boltzmann kinetics.
These relations between the principle of detailed balance and the second law of thermodynamics were clarified in 1887 when Hendrik Lorentz objected to the Boltzmann H-theorem for polyatomic gases.
[16] Lorentz stated that the principle of detailed balance is not applicable to collisions of polyatomic molecules.
Boltzmann immediately invented a new, more general condition sufficient for entropy growth.
In 1981, Carlo Cercignani and Maria Lampis proved that the Lorentz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules.
The principle of detailed balance for the generalized mass action law is: For given values
The following classical result gives the necessary and sufficient conditions for the existence of a positive equilibrium
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations: the Wegscheider's identity[21] holds:
To describe dynamics of the systems that obey the generalized mass action law, one has to represent the activities as functions of the concentrations cj and temperature.
is the chemical potential of that species in the chosen standard state, R is the gas constant and T is the thermodynamic temperature.
So, the Onsager relations follow from the principle of detailed balance in the linear approximation near equilibrium.
To formulate the principle of semi-detailed balance, it is convenient to count the direct and inverse elementary reactions separately.
For systems that obey the generalized mass action law the semi-detailed balance condition is sufficient for the dissipation inequality
Boltzmann introduced the semi-detailed balance condition for collisions in 1887[17] and proved that it guaranties the positivity of the entropy production.
[23] The microscopic backgrounds for the semi-detailed balance were found in the Markov microkinetics of the intermediate compounds that are present in small amounts and whose concentrations are in quasiequilibrium with the main components.
[25] Let us represent the generalized mass action law in the equivalent form: the rate of the elementary process
[23] For any reaction mechanism and a given positive equilibrium a cone of possible velocities for the systems with detailed balance is defined for any non-equilibrium state N
That is, there exists a system with detailed balance, the same reaction mechanism, the same positive equilibrium, that gives the same velocity at state N.[26] According to cone theorem, for a given state N, the set of velocities of the semidetailed balance systems coincides with the set of velocities of the detailed balance systems if their reaction mechanisms and equilibria coincide.
For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc.
[21] Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.