In thermodynamics, the principle of detailed balance says that every process of energy transfer in one direction must also allow energy transfer in the opposite direction, and in equilibrium, the flux in both directions must be equal. For example, the principle 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.Boltzmann, L. (1964), Lectures on gas theory, Berkeley, CA, USA: U. of California Press. The arguments in favor of this property are founded upon microscopic reversibility.Tolman, R. C. (1938). The Principles of Statistical Mechanics. Oxford University Press, London, UK.
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.Maxwell, J. C. (1867), On the dynamical theory of gases, Philos. Trans. R. Soc. London, 157, pp. 49–88. 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 (p. 64).
In 1901, Rudolf Wegscheider introduced the principle of detailed balance for chemical kinetics.Wegscheider, R. (1901) Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme , Monatshefte für Chemie / Chemical Monthly 32(8), 849–906. 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,Onsager, L. (1931), Reciprocal relations in irreversible processes. I (), Phys. Rev. 37, 405–426; II, 38, 2265–2279. 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.van Kampen, N. G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).Yablonskii, G. S., Bykov, V. I., Gorban, A. N., Elokhin, V. I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, the Netherlands: Elsevier.
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.Gorban, A.N. (2014),Detailed balance in micro- and macrokinetics and micro-distinguishability of macro-processes, Results in Physics 4, 142–147 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.Joshi, B. (2013), Deterministic detailed balance in chemical reaction networks is sufficient but not necessary for stochastic detailed balance, arXiv:1312.4196 [math.PR].
Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.
Detailed balance
Reversibility A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equationswhere Pij is the Markov transition probability from state i to state j, i.e. , and πi and πj are the equilibrium probabilities of being in states i and j, respectively. When for all i, this is equivalent to the joint probability matrix, being symmetric in i and j; or symmetric in and t.
The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and a transition kernel probability density from state s′ to state s:The detailed balance condition is stronger than that required merely for a stationary distribution, because there are Markov processes with stationary distributions that do not have detailed balance.
Transition matrices that are symmetric or always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution.
Kolmogorov's criterion 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 example, it implies that, for all a, b and c,For example, if we have a Markov chain with three states such that only these transitions are possible: , then they violate Kolmogorov's criterion.
Closest reversible Markov chain
For continuous systems with detailed balance, it may be possible to continuously transform the coordinates until the equilibrium distribution is uniform, with a transition kernel which then is symmetric. In the case of discrete states, it may be possible to achieve something similar by breaking the Markov states into appropriately-sized degenerate sub-states.
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. The closest Matrix can be computed by solving a quadratic-convex optimization problem.
Detailed balance and entropy increase
For many systems of physical and chemical kinetics, detailed balance provides sufficient conditions for the strict increase of entropy in isolated systems. For example, the famous Boltzmann H-theorem states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of entropy production. The Boltzmann formula (1872) for entropy production in rarefied gas kinetics with detailed balance served as a prototype of many similar formulas for dissipation in mass action kineticsVolpert, A.I., Khudyaev, S.I. (1985), Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht, The Netherlands: Nijoff. (Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.) and generalized mass action kineticsSchuster, S., Schuster R. (1989). A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation. J. Math. Chem, 3 (1), 25–42. with detailed balance.
Nevertheless, the principle of detailed balance is not necessary for entropy growth. For example, in the linear irreversible cycle A1 -> A2 -> A3 -> A1, entropy production is positive but the principle of detailed balance does not hold.
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.Lorentz H.-A. (1887) Über das Gleichgewicht der lebendigen Kraft unter Gasmolekülen. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien. 95 (2), 115–152. 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.Boltzmann L. (1887) Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien. 95 (2), 153–164. Boltzmann's condition holds for all Markov processes, irrespective of time-reversibility. Later, entropy increase was proved for all Markov processes by a direct method.Shannon, C.E. (1948) A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656. Hugh Everett Theory of the Universal Wavefunction, Thesis, Princeton University, (1956, 1973), Appendix I, pp 121 ff. In his thesis, Everett used the term "detailed balance" unconventionally, instead of balance equation These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was referred to as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semi-detailed balance" or the "complex balance". 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.Cercignani, C. and Lampis, M. (1981). On the H-theorem for polyatomic gases, Journal of Statistical Physics, V. 26 (4), 795–801. Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.
Wegscheider's conditions for the generalized mass action law
In chemical kinetics, the elementary reactions are represented by the stoichiometric equations
where are the components and are the stoichiometric coefficients. Here, the reverse reactions with positive constants are included in the list separately. We need this separation of direct and reverse reactions to apply later the general formalism to the systems with some irreversible reactions. The system of stoichiometric equations of elementary reactions is the reaction mechanism.
The stoichiometric matrix is , (gain minus loss). This matrix need not be square. The stoichiometric vector is the rth row of with coordinates .
According to the generalized mass action law, the reaction rate for an elementary reaction is
where is the activity (the "effective concentration") of .