 Detailed balance

The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions): At equilibrium, each elementary process should be equilibrated by its reverse process.
History
The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his Htheorem using this principle.^{[1]} The arguments in favor of this property are founded upon microscopic reversibility.^{[2]} A. Einstein in 1916 used this principle in a background for his quantum theory of emission and absorption of radiation.^{[3]}
In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.^{[4]} In particular, he demonstrated that the irreversible cycles are impossible and found explicitely the relations between kinetic constants that follow from the principle of detailed balance. In 1931, Lars Onsager used these relations in his works^{[5]} awarded in 1968 by the Nobel Prize in Chemistry.
Now, the principle of detailed balance is a standard part of the university courses in statistical mechanics, physical chemistry, chemical and physical kinetics.^{[6]}^{[7]}^{[8]}
Microscopical 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, A_{i},B_{j} 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.
Reversible Markov chains
A Markov process is said to have detailed balance if the transition probability, P, between each pair of states i and j in the state space obey
where P is the Markov transition matrix (transition probability), i.e., P_{ij} = P(X_{t} = j  X_{t − 1} = i); and π_{i} and π_{j} are the equilibrium probabilities of being in states i and j, respectively.^{[9]} When Pr(X_{t−1} = i) = π_{i} for all i, this is equivalent to the joint probability matrix, Pr(X_{t−1} = i, X_{t} = j) being symmetric in i and j; or symmetric in t − 1 and t.
The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and P(s′, s) a transition kernel probability density from state s′ to state s:
A Markov process that has detailed balance is said to be a reversible Markov process or reversible Markov chain.^{[9]}
The detailed balance condition is stronger than that required merely for a stationary distribution; that is, there are Markov processes with stationary distributions that do not have detailed balance. Detailed balance implies that, around any closed cycle of states, there is no net flow of probability. For example, it implies that, for all a, b and c,
This can be proved by substitution from the definition. In the case of a positive transition matrix, the "no net flow" condition implies detailed balance.
Transition matrices that are symmetric (P_{ij} = P_{ji} or P(s′, s) = P(s, s′)) always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution. 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 a degeneracy of substates.
Detailed balance and the entropy growth
For many systems of physical and chemical kinetics, detailed balance provides sufficient conditions for the entropy growth in isolated systems. For example, the famous Boltzmann Htheorem^{[1]} states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance^{[1]}^{[10]} served as a prototype of many similar formulas for dissipation in mass action kinetics^{[11]} and generalized mass action kinetics^{[12]} with detailed balance.
Nevertheless, the principle of detailed balance is not necessary for the entropy growth. For example, in the linear irreversible cycle the entropy production is positive but the principle of detailed balance does not hold.
The principle of detailed balance is a sufficient but not necessary condition for the entropy growth in the Boltzmann kinetics. These relations between the principle of detailed balance and the Second law of thermodynamics were clarified in 1887 when Hendrik Lorentz objected the Boltzmann Htheorem for polyatomic gases.^{[13]} 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 the entropy growth.^{[14]} In particular, this condition is valid for all Markov processes without any relation to timereversibility. The entropy growth in all Markov processes was explicitely proved later^{[15]}^{[16]}. These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was discussed as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semidetailed balance" or the "complex balance". In 1981, Carlo Cercignani and Maria Lampis proved that the Lorenz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules.^{[17]} Nevertheless, the extended semidetailed 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 A_{i} 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 , γ_{ri} = β_{ri} − α_{ri} (gain minus loss). The stoichiometric vector γ_{r} is the rth row of with coordinates γ_{ri} = β_{ri} − α_{ri}.
According to the generalized mass action law, the reaction rate for an elementary reaction is
where is the activity of A_{i}.
The reaction mechanism includes reactions with the reaction rate constants k_{r} > 0. For each r the following notations are used: , , is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not, is the reaction rate for the reverse reaction if it is in the reaction mechanism and 0 if it is not. For a reversible reaction, is the equilibrium constant.
The principle of detailed balance for the generalized mass action law is: For given values k_{r} there exists a positive equilibrium with detailed balance, . This means that the system of linear detailed balance equations
is solvable (). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium with detailed balance (see, for example, the textbook^{[7]}).
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
 If then (reversibility);
 For any solution of the system
the Wegscheider's identity^{[18]} holds:
Remark. It is sufficient to use in the Wegscheider conditions a basis of solutions of the system .
In particular, for any cycle in the monomolecular (linear) reactions the product of the reaction rate constants in the clockwise direction is equal to the product of the reaction rate constants in the counterclockwise direction. The same condition is valid for the reversible Markov processes (it is equivalent to the "no net flow" condition).
A simple nonlinear example gives us a linear cycle supplemented by one nonlinear step^{[18]}:
There are two nontrivial independent Wegscheider's identities for this system:
 and
They correspond to the following linear relations between the stoichiometric vectors:
 γ_{1} + γ_{2} + γ_{3} = 0 and γ_{3} + γ_{4} − γ_{2} = 0.
The computational aspect of the Wegscheider conditions was studied by D. Colquhoun with coauthors^{[19]}.
The Wegscheider conditions demonstrate that whereas the principle of detailed balance states a local property of equilibrium, it implies the relations between the kinetic constants that are valid for all states far from equilibrium. This is possible because a kinetic law is known and relations between the rates of the elementary processes at equilibrium can be transformed into relations between kinetic constants which are used globally. For the Wegscheider conditions this kinetic law is the law of mass action (or the generalized law of mass action).
Dissipation in systems with detailed balance
To describe dynamics of the systems that obey the generalized mass action law, one has to represent the activities as functions of the concentrations c_{j} and temperature. For this purpose, let us the representation of the activity through the chemical potential:
where μ_{i} is the chemical potential of the species under the conditions of interest, μ^{o}_{i} is the chemical potential of that species in the chosen standard state, R is the gas constant and T is the thermodynamic temperature. The chemical potential can be represented as a function of c and T, where c is the vector of concentrations with components c_{j}. For the ideal systems, and a_{j} = c_{j}: the activity is the concentration and the generalized mass action law is the usual law of mass action.
Let us consider a system in isothermal (T=const) isochoric (the volume V=const) condition. For these conditions, the Helmholtz free energy F(T,V,N) measures the “useful” work obtainable from a system. It is a functions of the temperature T, the volume V and the amounts of chemical components N_{j} (usually measured in moles), N is the vector with components N_{j}. For the ideal systems,
The chemical potential is a partial derivative: .
The chemical kinetic equations are
If the principle of detailed balance is valid then for any value of T there exists a positive point of detailed balance c^{eq}:
Elementary algebra gives
where
For the dissipation we obtain from these formulas:
The inequality holds because ln is a monotone function and, hence, the expressions and have always the same sign.
Similar inequalities^{[7]} are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the Gibbs free energy decreases, for the isochoric systems with the constant internal energy (isolated systems) the entropy increases as well as for isobaric systems with the constant enthalpy.
Onsager reciprocal relations and detailed balance
Let the principle of detailed balance be valid. Then, in the linear approximation near equilibrium the reaction rates for the generalized mass action law are
Therfore, in the linear approximation near equilibrium, the kinetic equations are (γ_{rj} = β_{ri} − α_{ri}):
This is exacltly the Onsager form: following the original work of Onsager^{[5]}, we should introduce the thermodynamic forces X_{j} and the matrix of coefficients L_{ij} in the form
The coefficient matrix L_{ij} is symmetric:
These symmetry relations, L_{ij} = L_{ji}, are exactly the Onsager reciprocal relations. The coefficient matrix L is nonpositive. It is negative on the linear span of the stoichiometric vectors γ_{r}.
So, the Onsager relations follow from the principle of detailed balance in the linear approximation near equilibrium.
Semidetailed balance
To formulate the principle of semidetailed balance, it is convenient to count the direct and inverse elementary reactions separately. In this case, the kinetic equations have the form:
Let us use the notations α_{r} = α_{ri}, β_{r} = β_{ri} for the input and the output vectors of the stoichiometric coefficients of the rth elementary reaction. Let Y be the set of all these vectors α_{r},β_{r}.
For each , let us define two sets of numbers:
if and only if ν is the vector of the input stoichiometric coefficients α_{r} for the rth elementary reaction; if and only if ν is the vector of the output stoichiometric coefficients β_{r} for the rth elementary reaction.
The principle of semidetailed balance means that in equilibrium the semidetailed balance condition holds: for every
The semidetailded balance condition is sufficient for the stationarity: it implies that
 .
For the Markov kinetics the semidetailed balance condition is just the elementary balance equation and holds for any steady state. For the nonlinear mass action law it is, in general, sufficient but not necessary condition for stationarity.
The semidetailed balance condition is weaker than the detailed balance one: if the principle of detailed balance holds then the condition of semidetailed balance also holds.
For systems that obey the generalized mass action law the semidetailed balance condition is sufficient for the dissipation inequality (for the Helmholtz free energy under isothermal isochoric conditions and for the dissipation inequalities under other classical conditions for the corresponding thermodynamic potentials).
Boltzmann introduced the semidetailed balance condition for collisions in 1887^{[14]} and proved that it guaranties the positivity of the entropy production. For chemical kinetics, this condition (as the complex balance condition) was inroduced by Horn and Jackson in 1972^{[20]}.
The microscopic backgrounds for the semidetailed 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^{[21]}. Under these microscopic assumptions, the semidetailed balance condition is just the balance equation for the Markov microkinetics according to the MichaelisMentenStueckelberg theorem^{[22]}.
Dissipation in systems with semidetailed balance
Let us represent the generalized mass action law in the equivalent form: the rate of the elementary process
is
where is the chemical potential and F(T,V,N) is the Helmholtz free energy. The exponential term is called the Boltzmann factor and the multiplier is the kinetic factor.^{[22]} Let us count the direct and reverse reaction in the kinetic equation separately:
An auxiliary function θ(λ) of one variable is convenient for the representation of dissipation for the mass action law
This function θ(λ) may be considered as the sum of the reaction rates for deformed input stoichiometric coefficients . For λ = 1 it is just the sum of the reaction rates. The function θ(λ) is convex because .
Direct calculation gives that according to the kinetic equations
This is the general dissipation formula for the generalized mass action law.^{[22]}
Convexity of θ(λ) gives the sufficient and necessary conditions for the proper dissipation inequality:
The semidetailed balance condition can be transformed into identity . Therefore, for the systems with semidetailed balance .^{[20]}
Detailed balance for systems with irreversible reactions
Detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process and required reversibility of all elementary processes. For many real physicochemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. If one represents irreversible reactions as limits of reversible steps, then it become obvious that not all reaction mechanisms with irreversible reactions can be obtained as limits of systems or reversible reactions with detailed balance. For example, the irreversible cycle cannot be obtained as such a limit but the reaction mechanism can^{[23]}.
A system of reactions with some irreversible reactions is a limit of systems with detailed balance when some constants tend to zero if and only if (i) the reversible part of this system satisfies the principle of detailed balance and (ii) the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reactions.^{[18]} Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.
See also
 asymmetry
 Atomic spectral line (deduction of the Einstein coefficients)
 Balance equation
 DMP
 Gibbs sampling
 Master equation
 Random walks on graphs
References
 ^ ^{a} ^{b} ^{c} Boltzmann, L. (1964), Lectures on gas theory, Berkeley, CA, USA: U. of California Press.
 ^ Tolman, R. C. (1938). The Principles of Statistical Mechanics. Oxford University Press, London, UK.
 ^ Einstein, A. (1916). StrahlungsEmission und Absorption nach der Quantentheorie [=Emission and absorption of radiation in quantum theory], Verhandlungen der Deutschen Physikalischen Gesellschaft 18 (13/14). Braunschweig: Vieweg, 318323.
 ^ Wegscheider, R. (1911) Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme, Monatshefte für Chemie / Chemical Monthly 32(8), 849906.
 ^ ^{a} ^{b} Onsager, L. (1931), Reciprocal relations in irreversible processes. I, Phys. Rev. 37, 405426; II 38, 22652279
 ^ van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).
 ^ ^{a} ^{b} ^{c} Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, The Netherlands: Elsevier.
 ^ Lifshitz, E. M.; and Pitaevskii, L. P. (1981). Physical kinetics. London: Pergamon. ISBN 0080264808 ISBN 0750626356. Vol. 10 of the Course of Theoretical Physics(3rd Ed).
 ^ ^{a} ^{b} O'Hagan, Anthony; Forster, Jonathan (2004) "Section 10.3" Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference New York: Oxford University Press p. 263 ISBN 0 340 807520
 ^ Tolman, R. C. (1938). The Principles of Statistical Mechanics. Oxford University Press, London, UK.
 ^ Volpert, 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.)
 ^ Schuster, S., Schuster R. (1989). A generalization of Wegscheider's condition. Implications for properties of steady states and for quasisteadystate approximation. J. Math. Chem, 3 (1), 2542.
 ^ Lorentz H.A. (1887) Üeber das Gleichgewicht der lebendigen Kraft unter Gasmolekülen. S.A.W. [Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne.] 95, 115152.
 ^ ^{a} ^{b} Boltzmann L. (1887) Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen. S.A.W. [Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne.] 95, 153164.
 ^ Shannon, C.E. (1948) A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656. [1] [2]
 ^ 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
 ^ Cercignani, C. and Lampis, M. (1981). On the Htheorem for polyatomic gases, Journal of Statistical Physics, V. 26 (4), 795801.
 ^ ^{a} ^{b} ^{c} Gorban, A.N., Yablonsky, G.S. (2011) Extended detailed balance for systems with irreversible reactions, Chemical Engineering Science 66, 5388–5399.
 ^ Colquhoun, D., Dowsland, K.A., Beato, M., and Plested, A.J.R. (2004) How to Impose Microscopic Reversibility in Complex Reaction Mechanisms, Biophysical 86, June 2004, 3510–3518
 ^ ^{a} ^{b} Horn, F., Jackson, R. (1972) General mass action kinetics. Arch. Ration. Mech. Anal. 47, 87116.
 ^ Stueckelberg, E.C.G. (1952) Theoreme H et unitarite de S. Helv. Phys. Acta 25, 577580
 ^ ^{a} ^{b} ^{c} Gorban, A.N., Shahzad, M. (2011) The MichaelisMentenStueckelberg Theorem. Entropy 13, no. 5, 9661019.
 ^ Chu, Ch. (1971), Gas absorption accompanied by a system of firstorder reactions, Chem. Eng. Sci. 26(3), 305312.
External links
 David Aldous, James A. Fill, Reversible Markov Chains (draft).
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