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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. Lewis put forward this general principle in 1925:

Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process.[1]
According to Ter Haar [2] the essence of the detailed balance is:
...at equilibrium the number of processes destroying situation $A$ and creating situation $B$ will be equal to the number of processes producing $A$ and destroying $B$

The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle [3]. The arguments in favour of this property are founded upon microscopic reversibility [4]. In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.[5] In particular, he demonstrated that the irreversible cycles $A_1 \to A_2 \to ... \to A_n \to A_1$ are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.

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

$\sum_i \alpha_i A_i \to \sum_j \beta_j B_j$

transforms into

$\sum_j \beta_j B_j \to \sum_i \alpha_i A_i$

and conversely. (Here, $A_i, B_j$ are symbols of components or states, $\alpha_i, \beta_j \geq 0$ are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of micro-reversibility 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

$\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,$

where $P$ is the Markov transition matrix (transition probability), i.e., Pij = P(Xt = j | Xt − 1 = i); and πi and πj are the equilibrium probabilities of being in states i and j, respectively.[6] When Pr(Xt−1 = i) = πi for all i, this is equivalent to the joint probability matrix, Pr(Xt−1 = iXt = 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:

$\pi(s') P(s',s) = \pi(s) P(s,s')\,.$

A Markov process that has detailed balance is said to be a reversible Markov process or reversible Markov chain [6].

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,

$P(a,b) P(b,c) P(c,a) = P(a,c) P(c,b) P(b,a)\,.$

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 (Pij = Pji or P(s′, s) = P(ss′)) 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 sub-states.

Detailed balance and the entropy growth

For many systems that treat physical and chemical kinetics, detailed balance provides sufficient conditions for the entropy growth in isolated systems. For example, the famous Boltzmann H-theorem[3] 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[3][4] served as a prototype of many similar formulas for dissipation in mass action kinetics [7] and generalised mass action kinetics [8] with detailed balance.

Nevertheless, the principle of detailed balance is not necessary for entropy growth. For example, in the linear irreversible cycle $A_1 \to A_2 \to A_3 \to A_1$ 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 H-theorem for polyatomic gases [9]. 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. [10]. In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was explicitly proved later [11] [12]. 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 "semi-detailed 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 [13]. Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the generalisation of detailed balance.

Wegscheider's conditions for the generalized mass action law

In chemical kinetics, the elementary reactions are represented by the stoichiometric equations

$\sum_i \alpha_{ri} A_i \to \sum_j \beta_{rj} A_j \;\; (r=1, \ldots, m) \, ,$

where $A_i$ are the components and $\alpha_{ri}, \beta_{rj}\geq 0$ 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 $\boldsymbol{\Gamma}=(\gamma_{ri})$, $\gamma_{ri}=\beta_{ri}-\alpha_{ri}$ (gain minus loss). The stoichiometric vector $\gamma_r$ is the rth row of $\boldsymbol{\Gamma}$ with coordinates $\gamma_{ri}=\beta_{ri}-\alpha_{ri}$.

According to the generalised mass action law, the reaction rate for an elementary reaction is

$w_r=k_r \prod_{i=1}^n a_i^{\alpha_{ri}} \, ,$

where $a_i\geq 0$ 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: $k_r^+=k_r$, $w_r^+=w_r$, $k_r^-$ is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not, $w_r^-$ 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, $K_r=k_r^+/k_r^-$ 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 $a_i^{\rm eq}>0$ with detailed balance, $w_r^+=w_r^-$. This means that the system of linear detailed balance equations

$\sum_i \gamma_{ri} x_i = \ln k_r^+-\ln k_r^-=\ln K_r$

is solvable ($x_i=\ln a_i^{\rm eq}$). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium $a_i^{\rm eq}>0$ with detailed balance [14].

Two conditions are sufficient and necessary in order to solve the system of detailed balance equations:

1. If $k_r^+>0$ then $k_r^->0$ (reversibility);
2. For any solution $\boldsymbol{\lambda}=(\lambda_r)$ of the system
$\boldsymbol{\lambda \Gamma} =0 \;\; \left(\mbox{i.e.}\;\; \sum_r \lambda_r \gamma_{ri}=0\;\; \mbox{for all} \;\; i\right)$

the Wegscheider's identity [15] holds:

$\prod_{r=1}^m (k_r^+)^{\lambda_r}=\prod_{r=1}^m (k_r^-)^{\lambda_r} \, .$

Remark. It is sufficient to use in the Wegscheider conditions a basis of solutions of the system $\boldsymbol{\lambda \Gamma} =0$.

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 counter-clockwise direction. The same condition is valid for the reversible Markov processes (it is equivalent to the "no net flow" condition).

A simple non-linear example gives us a linear cycle supplemented by one non-linear step [15]:

1. $A_1 \rightleftharpoons A_2$
2. $A_2 \rightleftharpoons A_3$
3. $A_3 \rightleftharpoons A_1$
4. $A_1+A_2 \rightleftharpoons 2A_3$

There are two nontrivial independent Wegscheider's identities for this system:

$k_1^+k_2^+k_3^+=k_1^-k_2^-k_3^-$ and $k_3^+k_4^+/k_2^+=k_3^-k_4^-/k_2^-$

They correspond to the following linear relations between the stoichiometric vectors:

$\gamma_1+\gamma_2+\gamma_3=0$ and $\gamma_3+\gamma_4-\gamma_2=0$.

The computational aspect of the Wegscheider conditions was studied by D. Colquhoun et al. [16].

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 generalised law of mass action).

Dissipation in systems with detailed balance

To describe dynamics of the systems that obey the generalised mass action law, one has to represent the activities as functions of the concentrations $c_j$ and the temperature. For this purpose, let us represent the activity in terms of the chemical potential:

$a_i = \exp\left (\frac{\mu_i - \mu^{\Theta}_i}{RT}\right )$

where $\mu_i$ is the chemical potential of the species under the conditions of interest, $\mu^{\Theta}_i$ 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 ideal systems, $\mu_i=RT\ln c_i+\mu^{\Theta}_i$ and $a_j=c_j$: the activity is the concentration.

For an isothermal, isochoric system the Helmholtz energy function $A(N, V, T)$ measures the "useful" work obtainable from a system. For an ideal system one has

$A=RT \sum_i N_i \left(\ln\left(\frac{N_i}{V}\right)-1+\frac{\mu^{\Theta}_i(T)}{RT}\right)$

The chemical potential is given by the partial derivative

$\mu_i=\frac{\partial A(N,V,T)}{\partial N_j}.$

The chemical kinetic equations are

$\frac{d N_i}{d t}=V \sum_r \gamma_{ri}(w^+_r-w^-_r) .$

If the principle of detailed balance is valid, then for any value of $T$ there exists a positive point of detailed balance $c^{\mathrm {eq}}$:

$w^+_r(c^{\mathrm {eq}},T)=w^-_r(c^{\rm eq},T)=w^{\rm eq}_r$

$w^+_r=w^{\rm eq}_r \exp \left(\sum_i \frac{\alpha_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right); \;\; w^-_r=w^{\rm eq}_r \exp \left(\sum_i \frac{\beta_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right);$

where $\mu^{\rm eq}_i=\mu_i(c^{\rm eq},T)$

For the dissipation one obtains

$\frac{d F}{d t}=\sum_i \frac{\partial F(T,V,N)}{\partial N_i} \frac{d N_i}{d t}=\sum_i \mu_i \frac{d N_i}{d t} = -VRT \sum_r (\ln w_r^+-\ln w_r^-) (w_r^+-w_r^-) \leq 0$

The inequality holds because the logarithm function is monotonic, hence, the expressions $\ln w_r^+-\ln w_r^-$ and $w_r^+-w_r^-$ always have the same sign.

Similar inequalities [14] are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the Gibbs energy function decreases, for isochoric systems with constant internal energy 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

$w^+_r=w^{\rm eq}_r \left(1+\sum_i \frac{\alpha_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right); \;\; w^-_r=w^{\rm eq}_r \left(1+ \sum_i \frac{\beta_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right);$

Therefore, in the linear approximation near equilibrium, the kinetic equations are ($\gamma_{rj}=\beta_{ri}-\alpha_{ri}$):

$\frac{d N_i}{d t}=-V \sum_j \left[\sum_r w^{\rm eq}_r \gamma_{ri}\gamma_{rj}\right] \frac{\mu_j-\mu^{\rm eq}_j}{RT}.$

This is exactly the Onsager form: following the original work of Lars Onsager [17][18] we should introduce the thermodynamic forces $X_j$ and the matrix of coefficients $L_{ij}$ in the form

$X_j = \frac{\mu_j-\mu^{\rm eq}_j}{T}; \;\; \frac{d N_i}{d t}=\sum_j L_{ij}X_j$

The coefficient matrix $L_{ij}$ is symmetric:

$L_{ij}=-\frac{V}{R}\sum_r w^{\rm eq}_r \gamma_{ri}\gamma_{rj}$

These symmetry relations, $L_{ij}=L_{ji}$, are exactly the Onsager reciprocal relations. The coefficient matrix $L$ is non-positive. It is negative on the linear span of the stoichiometric vectors $\gamma_{r}$. So, the Onsager relations follow from the principle of detailed balance in the linear approximation near equilibrium.

Semi-detailed balance

To formulate the principle of semi-detailed balance, it is convenient to count the direct and inverse elementary reactions separately. In this case, the kinetic equations have the form:

$\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r$

Let us use the notation $\alpha_r=\alpha_{ri}$, $\beta_r=\beta_{ri}$ 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 $\alpha_r, \beta_r$. For each $\nu \in Y$, let us define two sets of numbers:

$R_{\nu}^+=\{r|\alpha_r=\nu \}; \;\;\; R_{\nu}^-=\{r|\beta_r=\nu \}$

$r \in R_{\nu}^+$ if and only if $\nu$ is the vector of the input stoichiometric coefficients $\alpha_r$ for the rth elementary reaction;$r \in R_{\nu}^-$ if and only if $\nu$ is the vector of the output stoichiometric coefficients $\beta_r$ for the rth elementary reaction.

The principle of semi-detailed balance implies that when in equilibrium, for every $\nu \in Y$

$\sum_{r\in R_{\nu}^+}w_r=\sum_{r\in R_{\nu}^+}w_r$

The semi-detailded balance condition is sufficient for the stationarity: it implies that

$\frac{d N}{dt}=V \sum_r \gamma_r w_r=0.$

For the Markov kinetics the semi-detailed balance condition is simply the elementary balance equation and holds for any steady state. For the non-linear mass action law it is, in general, a sufficient but not necessary condition for stationarity. The semi-detailed balance condition is weaker than that of detailed balance: if the principle of detailed balance holds then the condition of semi-detailed balance also holds. For systems that obey the generalised mass action law the semi-detailed balance condition is sufficient for the dissipation inequality $d F/ dt \geq 0$ (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 semi-detailed balance condition for collisions in 1887 [10] and proved that it guaranties the positivity of the entropy production. For chemical kinetics, this condition (as the complex balance condition) was introduced by Horn and Jackson in 1972 [19].

The microscopic backgrounds for the semi-detailed balance were found in the Markov micro-kinetics of the intermediate compounds that are present in small amounts and whose concentrations are in quasiequilibrium with the main components [20]. Under these microscopic assumptions, the semi-detailed balance condition becomes the balance equation for the Markov microkinetics according to the Michaelis-Menten-Stueckelberg theorem[21].

Dissipation in systems with semi-detailed balance

Let us represent the generalized mass action law in the equivalent form: the rate of the elementary process

$\sum_i \alpha_{ri} A_i \to \sum_i \beta_{ri} A_i$

is

$w_r=\varphi_r \exp\left(\sum_i\frac{\alpha_{ri} \mu_i}{RT}\right)$

where $\mu_i=\partial F(T,V,N)/ \partial N_i$ 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 $\varphi_r \geq 0$ is the kinetic factor.[21] Let us count the direct and reverse reaction in the kinetic equation separately:

$\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r$

An auxiliary function $\theta(\lambda)$ of one variable $\lambda\in [0,1]$ is convenient for the representation of dissipation for the mass action law

$\theta(\lambda)=\sum_{r}\varphi_{r}\exp\left(\sum_i\frac{(\lambda \alpha_{ri}+(1-\lambda)\beta_{ri}))\mu_i}{RT}\right)$

This function $\theta(\lambda)$ may be considered as the sum of the reaction rates for deformed input stoichiometric coefficients $\tilde{\alpha}_{\rho}(\lambda)=\lambda \alpha_{\rho}+(1-\lambda)\beta_{\rho}$. For $\lambda=1$ it is just the sum of the reaction rates. The function $\theta(\lambda)$ is convex because $\theta''(\lambda) \geq 0$.

Direct calculation gives that according to the kinetic equations

$\frac{d F}{d t}=-VRT \left.\frac{d \theta(\lambda)}{d \lambda}\right|_{\lambda=1}$

This is the general dissipation formula for the generalized mass action law.[21]

Convexity of $\theta(\lambda)$ gives the sufficient and necessary conditions for the proper dissipation inequality:

$\frac{d F}{d t}<0 \mbox{ if and only if } \theta(\lambda)< \theta(1) \mbox{ for some }\lambda <1; \;\;\; \frac{d F}{d t}\leq0 \mbox{ if and only if } \theta(\lambda)\leq \theta(1) \mbox{ for some }\lambda <1$

The semi-detailed balance condition can be transformed into identity $\theta(0)\equiv \theta(1)$. Therefore, for the systems with semi-detailed balance ${d F}/{d t}\leq 0$.[19]

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 physico-chemical 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 $A_1 \to A_2 \to A_3 \to A_1$ cannot be obtained as such a limit but the reaction mechanism $A_1 \to A_2 \to A_3 \leftarrow A_1$ can [22].

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.[15] Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.

References

1. Ludwig Boltzmann "Lectures on Gas Theory" (original title "Vorlesungen uber Gastheorie") (1896) ISBN 0486684555
2. Richard C. Tolman "The Principles of Statistical Mechanics" Oxford University Press (1938) ISBN 0486638960
3. Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie 22 pp. 849-906 (1901)
4. Anthony O'Hagan and Jonathan J. Forster "Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference", Wiley (2004) ISBN 0340807520
5. A.I. Volpert and S. I. Khudyaev "Analysis in classes of discontinuous functions and equations of mathematical physics", Springer (1985) ISBN 978-90-247-3109-1
6. Stefan Schuster and Ronny Schuster "A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation", Journal of Mathematical Chemistry 3 pp. 25-42 (1989)
7. Hendrik Antoon Lorentz "Üeber das Gleichgewicht der lebendigen Kraft unter Gasmolekülen" S.A.W. (Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne) 95 pp. 115-152 (1887)
8. L. Boltzmann "Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen" S.A.W. (Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne) 95 pp. 153-164 (1887)
9. Claude Elwood Shannon "A Mathematical Theory of Communication", Bell System Technical Journal 27 July pp. 379-423 (1948)
10. Claude Elwood Shannon "A Mathematical Theory of Communication", Bell System Technical Journal 27 October pp. 623-656 (1948)
11. Carlo Cercignani and Maria Lampis "On the H-theorem for polyatomic gases", Journal of Statistical Physics 26 pp. 795-801 (1981)
12. G. S. Yablonskii, V. I. Bykov, V. I. Elokhin, and A.N. Gorban "Kinetic Models of Catalytic Reactions", Elsevier (1991) ISBN 978-0-444-88802-0
13. A. N. Gorban and G. S. Yablonsky "Extended detailed balance for systems with irreversible reactions", Chemical Engineering Science 66 pp. 5388-5399 (2011)
14. David Colquhoun, Kathryn A. Dowsland, Marco Beato and Andrew J.R. Plested "How to Impose Microscopic Reversibility in Complex Reaction Mechanisms", Biophysical Journal 86 pp. 3510-3518 (2004)
15. Lars Onsager "Reciprocal Relations in Irreversible Processes. I.", Physical Review 37 pp. 405-426 (1931)
16. Lars Onsager "Reciprocal Relations in Irreversible Processes. II.", Physical Review 38 pp. 2265-2279 (1931)
17. F. Horn and R. Jackson "General mass action kinetics", Archive for Rational Mechanics and Analysis 47 pp. 81-116 (1972)
18. Ernst Carl Gerlach Stueckelberg "Theoreme H et unitarite de S", Helvetica Physica Acta 25 pp. 577-580 (1952)
19. Alexander N. Gorban and Muhammad Shahzad "The Michaelis-Menten-Stueckelberg Theorem", Entropy 13 pp. 966-1019 (2011)
20. Chieh Chu "Gas absorption accompanied by a system of first-order reactions", Chemical Engineering Science 26 pp. 305–312 (1971)