The principle ofdetailed balance can be used inkinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that atequilibrium, each elementary process is in equilibrium with its reverse process.

The principle of detailed balance was explicitly introduced for collisions byLudwig Boltzmann. In 1872, he proved hisH-theorem using this principle.^[1] The arguments in favor of this property are founded uponmicroscopic reversibility.^[2]

Five years before Boltzmann,James Clerk Maxwell used the principle of detailed balance forgas kinetics with the reference to theprinciple of sufficient reason.^[3] 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 forchemical kinetics.^[4] In particular, he demonstrated that the irreversible cycles${\displaystyle {\text{A}}_1^{}⟶{\text{A}}_2^{}⟶\cdots ⟶{\text{A}}_{\mathit{\text{n}}}^{}⟶{\text{A}}_1^{}}$ 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,^[5] for which he was awarded the 1968Nobel Prize in Chemistry.

The principle of detailed balance has been used inMarkov chain Monte Carlo methods since their invention in 1953.^[6] In particular, in theMetropolis–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.^[7][8][9]

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

${\displaystyle \sum _i{\alpha }_i{\text{A}}_i⟶\sum _j{\beta }_j{\text{B}}_j}$ transforms into${\displaystyle \sum _j{\beta }_j{\text{B}}_j⟶\sum _i{\alpha }_i{\text{A}}_i}$

and conversely. (Here,${\displaystyle {\text{A}}_i,{\text{B}}_j}$ are symbols of components or states,${\displaystyle {\alpha }_i,{\beta }_j\ge 0}$ 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:

1. ${\displaystyle {\text{A}}_i}$ does not change under time reversal;
2. Equilibrium is invariant under time reversal;
3. The macroscopic elementary processes are microscopically distinguishable. That is, they represent disjoint sets of microscopic events.

Any of these assumptions may be violated.^[10] For example, Boltzmann's collision can be represented as${\displaystyle {\text{A}}_{\mathit{\text{v}}}^{}+{\text{A}}_{\mathit{\text{w}}}^{}⟶{\text{A}}_{{\mathit{\text{v}}}^{\prime }}^{}+{\text{A}}_{{\mathit{\text{w}}}^{\prime }}^{}}$, where${\displaystyle {\text{A}}_v}$ is a particle with velocityv. Under time reversal${\displaystyle {\text{A}}_v}$ transforms into${\displaystyle {\text{A}}_{-v}}$. Therefore, the collision is transformed into the reverse collision by thePT transformation, whereP is the space inversion andT is the time reversal. Detailed balance for Boltzmann's equation requiresPT-invariance of collisions' dynamics, not justT-invariance. Indeed, after the time reversal the collision${\displaystyle {\text{A}}_{\mathit{\text{v}}}^{}+{\text{A}}_{\mathit{\text{w}}}^{}⟶{\text{A}}_{{\mathit{\text{v}}}^{\prime }}^{}+{\text{A}}_{{\mathit{\text{w}}}^{\prime }}^{}}$, transforms into${\displaystyle {\text{A}}_{\mathit{-}{\mathit{\text{v}}}^{\prime }}^{}+{\text{A}}_{\mathit{-}{\mathit{\text{w}}}^{\prime }}^{}⟶{\text{A}}_{\mathit{-}\mathit{\text{v}}}^{}+{\text{A}}_{\mathit{-}\mathit{\text{w}}}^{}}$. For the detailed balance we need transformation into${\displaystyle {\text{A}}_{{\mathit{\text{v}}}^{\prime }}^{}+{\text{A}}_{{\mathit{\text{w}}}^{\prime }}^{}⟶{\text{A}}_{\mathit{\text{v}}}^{}+{\text{A}}_{\mathit{\text{w}}}^{}}$. For this purpose, we need to apply additionally the space reversalP. Therefore, for the detailed balance in Boltzmann's equation notT-invariance butPT-invariance is needed.

Equilibrium may be notT- orPT-invariant even if the laws of motion are invariant. This non-invariance may be caused by thespontaneous symmetry breaking. There existnonreciprocal media (for example, somebi-isotropic materials) withoutT andPT invariance.^[10]

If different macroscopic processes are sampled from the same elementary microscopic events then macroscopic detailed balance^{[clarification needed]} may be violated even when microscopic detailed balance holds.^[10][11]

Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.

AMarkov process is called areversible Markov process orreversible Markov chain if there exists a positive stationary distribution π that satisfies thedetailed balance equations^[12]$${\displaystyle {\pi }_i{P}_{ij}={\pi }_j{P}_{ji},}$$whereP_ij is the Markov transition probability from statei to statej, 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 statesi andj, respectively.^[12] WhenPr(X_t−1 = i) = π_i for alli, this is equivalent to the joint probability matrix,Pr(X_t−1 = i, X_t = j) being symmetric ini andj; or symmetric int − 1 and t.

The definition carries over straightforwardly to continuous variables, where π becomes a probability density, andP(s′, s) a transition kernel probability density from states′ to state s:$${\displaystyle \pi (s^{\prime })P(s^{\prime },s)=\pi (s)P(s,s^{\prime }).}$$The detailed balance condition is stronger than that required merely for astationary distribution, because there are Markov processes with stationary distributions that do not have detailed balance.

Transition matrices that are symmetric(P_ij = P_ji orP(s′, s) = P(s, s′)) always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution.

Reversibility is equivalent toKolmogorov'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 alla,b andc,$${\displaystyle P(a,b)P(b,c)P(c,a)=P(a,c)P(c,b)P(b,a).}$$For example, if we have a Markov chain with three states such that only these transitions are possible:${\displaystyle A\to B,B\to C,C\to A,B\to A}$, then they violate Kolmogorov's criterion.

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^{[clarification needed]} 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.

For many systems of physical and chemical kinetics, detailed balance providessufficient conditions for the strict increase of entropy in isolated systems. For example, the famous BoltzmannH-theorem^[1] 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^[1][2] served as a prototype of many similar formulas for dissipation in mass action kinetics^[13] and generalized mass action kinetics^[14] with detailed balance.

Nevertheless, the principle of detailed balance is not necessary for entropy growth. For example, in the linear irreversible cycle${\displaystyle {\text{A}}_1^{}⟶{\text{A}}_2^{}⟶{\text{A}}_3^{}⟶{\text{A}}_1^{}}$, 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 thesecond law of thermodynamics were clarified in 1887 whenHendrik Lorentz objected to the Boltzmann H-theorem for polyatomic gases.^[15] 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.^[16] 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.^[17][18] 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.^[19] Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.

Inchemical kinetics, theelementary reactions are represented by thestoichiometric equations$${\displaystyle \sum _i{\alpha }_{ri}{\text{A}}_i⟶\sum _j{\beta }_{rj}{\text{A}}_j(r=1,\dots ,m),}$$where${\displaystyle {\text{A}}_i}$ are the components and${\displaystyle {\alpha }_{ri},{\beta }_{rj}\ge 0}$ 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 thereaction mechanism.

Thestoichiometric matrix is${\displaystyle \mathbf{\Gamma }=({\gamma }_{ri})}$,${\displaystyle {\gamma }_{ri}={\beta }_{ri}-{\alpha }_{ri}}$ (gain minus loss). This matrix need not be square. Thestoichiometric vector${\displaystyle {\gamma }_r}$ is therth row of${\displaystyle \mathbf{\Gamma }}$ with coordinates${\displaystyle {\gamma }_{ri}={\beta }_{ri}-{\alpha }_{ri}}$.

According to thegeneralizedmass action law, thereaction rate for an elementary reaction is$${\displaystyle w_r=k_r\prod _{i=1}^n{a}_i^{{\alpha }_{ri}},}$$where${\displaystyle a_i\ge 0}$ is theactivity (the "effective concentration") of${\displaystyle A_i}$.

The reaction mechanism includes reactions with thereaction rate constants${\displaystyle k_r>0}$. For eachr the following notations are used:${\displaystyle k_r^{+}=k_r}$;${\displaystyle w_r^{+}=w_r}$;${\displaystyle k_r^{-}}$ is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not;${\displaystyle w_r^{-}}$ 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,${\displaystyle K_r=k_r^{+}/k_r^{-}}$ is theequilibrium constant.

The principle of detailed balance for the generalized mass action law is: For given values${\displaystyle k_r}$ there exists a positive equilibrium${\displaystyle a_i^{\mathrm{e}\mathrm{q}}>0}$ that satisfies detailed balance, that is,${\displaystyle w_r^{+}=w_r^{-}}$. This means that the system oflinear detailed balance equations$${\displaystyle \sum _i{\gamma }_{ri}{x}_i=\ln k_r^{+}-\ln k_r^{-}=\ln K_r}$$is solvable (${\displaystyle x_i=\ln a_i^{\mathrm{e}\mathrm{q}}}$). The following classical result gives the necessary and sufficient conditions for the existence of a positive equilibrium${\displaystyle a_i^{\mathrm{e}\mathrm{q}}>0}$ with detailed balance (see, for example, the textbook^[8]).

Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:

1. If${\displaystyle k_r^{+}>0}$ then${\displaystyle k_r^{-}>0}$ and, conversely, if${\displaystyle k_r^{-}>0}$ then${\displaystyle k_r^{+}>0}$ (reversibility);
2. For any solution${\displaystyle \mathit{\lambda }=({\lambda }_r)}$ of the system$${\displaystyle \mathit{\lambda }\mathbf{\Gamma }=0\left(\text{i.e.}\sum _r{\lambda }_r{\gamma }_{ri}=0\text{for all}i\right)}$$

the Wegscheider's identity^[20] holds:$${\displaystyle \prod _{r=1}^m(k_r^{+}{)}^{{\lambda }_r}=\prod _{r=1}^m(k_r^{-}{)}^{{\lambda }_r}.}$$

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

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 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:^[20]

1. 
2. 
3. 
4. 

There are two nontrivial independent Wegscheider's identities for this system:$${\displaystyle k_1^{+}{k}_2^{+}{k}_3^{+}=k_1^{-}{k}_2^{-}{k}_3^{-}}$$ and$${\displaystyle k_3^{+}{k}_4^{+}/k_2^{+}=k_3^{-}{k}_4^{-}/k_2^{-}}$$They correspond to the following linear relations between the stoichiometric vectors:$${\displaystyle {\gamma }_1+{\gamma }_2+{\gamma }_3=0}$$ and$${\displaystyle {\gamma }_3+{\gamma }_4-{\gamma }_2=0.}$$

The computational aspect of the Wegscheider conditions was studied by D. Colquhoun with co-authors.^[21]

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).

To describe dynamics of the systems that obey the generalized mass action law, one has to represent the activities as functions of theconcentrationsc_j andtemperature. For this purpose, use the representation of the activity through the chemical potential:$${\displaystyle a_i=\exp \left(\frac{{\mu }_i-{\mu }_i^{\ominus }}{RT}\right)}$$whereμ_i is thechemical potential of the species under the conditions of interest,⁠${\displaystyle {\mu }_i^{\ominus }}$⁠ is the chemical potential of that species in the chosenstandard state,R is thegas constant andT is thethermodynamic temperature. The chemical potential can be represented as a function ofc andT, wherec is the vector of concentrations with componentsc_j. For the ideal systems,${\displaystyle {\mu }_i=RT\ln c_i+{\mu }_i^{\ominus }}$ and${\displaystyle a_j=c_j}$: the activity is the concentration and the generalized mass action law is the usuallaw of mass action.

Consider a system inisothermal (T=const)isochoric (the volumeV=const) condition. For these conditions, theHelmholtz free energy⁠${\displaystyle F(T,V,N)}$⁠ measures the “useful” work obtainable from a system. It is a functions of the temperatureT, the volumeV and the amounts of chemical componentsN_j (usually measured inmoles),N is the vector with componentsN_j. For the ideal systems,$${\displaystyle F=RT\sum _i{N}_i\left(\ln \left(\frac{N_i}{V}\right)-1+\frac{{\mu }_i^{\ominus }(T)}{RT}\right).}$$

The chemical potential is a partial derivative:${\displaystyle {\mu }_i=\mathrm{\partial }F(T,V,N)/\mathrm{\partial }{N}_i}$.

The chemical kinetic equations are$${\displaystyle \frac{d{N}_i}{dt}=V\sum _r{\gamma }_{ri}(w_r^{+}-w_r^{-}).}$$

If the principle of detailed balance is valid then for any value ofT there exists a positive point of detailed balancec^eq:$${\displaystyle w_r^{+}(c^{\mathrm{e}\mathrm{q}},T)=w_r^{-}(c^{\mathrm{e}\mathrm{q}},T)=w_r^{\mathrm{e}\mathrm{q}}}$$Elementary algebra gives$${\displaystyle w_r^{+}=w_r^{\mathrm{e}\mathrm{q}}\exp \left(\sum _i\frac{{\alpha }_{ri}({\mu }_i-{\mu }_i^{\mathrm{e}\mathrm{q}})}{RT}\right);w_r^{-}=w_r^{\mathrm{e}\mathrm{q}}\exp \left(\sum _i\frac{{\beta }_{ri}({\mu }_i-{\mu }_i^{\mathrm{e}\mathrm{q}})}{RT}\right);}$$where${\displaystyle {\mu }_i^{\mathrm{e}\mathrm{q}}={\mu }_i(c^{\mathrm{e}\mathrm{q}},T)}$

For the dissipation we obtain from these formulas:$${\displaystyle \frac{dF}{dt}=\sum _i\frac{\mathrm{\partial }F(T,V,N)}{\mathrm{\partial }{N}_i}\frac{d{N}_i}{dt}=\sum _i{\mu }_i\frac{d{N}_i}{dt}=-VRT\sum _r(\ln w_r^{+}-\ln w_r^{-})(w_r^{+}-w_r^{-})\le 0}$$The inequality holds because ln is a monotone function and, hence, the expressions${\displaystyle \ln w_r^{+}-\ln w_r^{-}}$ and${\displaystyle w_r^{+}-w_r^{-}}$ have always the same sign.

Similar inequalities^[8] are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions theGibbs free energy decreases, for the isochoric systems with the constantinternal energy (isolated systems) theentropy increases as well as for isobaric systems with the constantenthalpy.

Let the principle of detailed balance be valid. Then, for small deviations from equilibrium, the kinetic response of the system can be approximated as linearly related to its deviation from chemical equilibrium, giving the reaction rates for the generalized mass action law as:$${\displaystyle w_r^{+}=w_r^{\mathrm{e}\mathrm{q}}\left(1+\sum _i\frac{{\alpha }_{ri}({\mu }_i-{\mu }_i^{\mathrm{e}\mathrm{q}})}{RT}\right);w_r^{-}=w_r^{\mathrm{e}\mathrm{q}}\left(1+\sum _i\frac{{\beta }_{ri}({\mu }_i-{\mu }_i^{\mathrm{e}\mathrm{q}})}{RT}\right);}$$

Therefore, again in the linear response regime near equilibrium, the kinetic equations are (${\displaystyle {\gamma }_{ri}={\beta }_{ri}-{\alpha }_{ri}}$):$${\displaystyle \frac{d{N}_i}{dt}=-V\sum _j\left[\sum _r{w}_r^{\mathrm{e}\mathrm{q}}{\gamma }_{ri}{\gamma }_{rj}\right]\frac{{\mu }_j-{\mu }_j^{\mathrm{e}\mathrm{q}}}{RT}.}$$

This is exactly the Onsager form: following the original work of Onsager,^[5] we should introduce the thermodynamic forces${\displaystyle X_j}$ and the matrix of coefficients${\displaystyle L_{ij}}$ in the form$${\displaystyle X_j=\frac{{\mu }_j-{\mu }_j^{\mathrm{e}\mathrm{q}}}{T};\frac{d{N}_i}{dt}=\sum _j{L}_{ij}{X}_j}$$

The coefficient matrix${\displaystyle L_{ij}}$ is symmetric:$${\displaystyle L_{ij}=-\frac{V}{R}\sum _r{w}_r^{\mathrm{e}\mathrm{q}}{\gamma }_{ri}{\gamma }_{rj}}$$

These symmetry relations,${\displaystyle L_{ij}=L_{ji}}$, are exactly theOnsager reciprocal relations. The coefficient matrix${\displaystyle L}$ is non-positive. It is negative on thelinear span of the stoichiometric vectors${\displaystyle {\gamma }_r}$.

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. In this case, the kinetic equations have the form:$${\displaystyle \frac{d{N}_i}{dt}=V\sum _r{\gamma }_{ri}{w}_r=V\sum _r({\beta }_{ri}-{\alpha }_{ri})w_r}$$Let us use the notations${\displaystyle {\alpha }_r={\alpha }_{ri}}$,${\displaystyle {\beta }_r={\beta }_{ri}}$ for the input and the output vectors of the stoichiometric coefficients of therth elementary reaction. Let${\displaystyle Y}$ be the set of all these vectors${\displaystyle {\alpha }_r,{\beta }_r}$.

For each${\displaystyle \nu \in Y}$, let us define two sets of numbers:$${\displaystyle R_{\nu }^{+}=\{r|{\alpha }_r=\nu \};R_{\nu }^{-}=\{r|{\beta }_r=\nu \}}$$

${\displaystyle r\in R_{\nu }^{+}}$ if and only if${\displaystyle \nu }$ is the vector of the input stoichiometric coefficients${\displaystyle {\alpha }_r}$ for therth elementary reaction;${\displaystyle r\in R_{\nu }^{-}}$ if and only if${\displaystyle \nu }$ is the vector of the output stoichiometric coefficients${\displaystyle {\beta }_r}$ for therth elementary reaction.

The principle ofsemi-detailed balance means that in equilibrium the semi-detailed balance condition holds: for every${\displaystyle \nu \in Y}$$${\displaystyle \sum _{r\in R_{\nu }^{-}}{w}_r=\sum _{r\in R_{\nu }^{+}}{w}_r}$$

The semi-detailed balance condition is sufficient for the stationarity: it implies that$${\displaystyle \frac{dN}{dt}=V\sum _r{\gamma }_r{w}_r=0.}$$

For the Markov kinetics the semi-detailed balance condition is just the elementarybalance 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 semi-detailed balance condition is weaker than the detailed balance one: if the principle of detailed balance holds then the condition of semi-detailed balance also holds.

For systems that obey the generalized mass action law the semi-detailed balance condition is sufficient for the dissipation inequality${\displaystyle dF/dt\ge 0}$ (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^[16] and proved that it guaranties the positivity of the entropy production. For chemical kinetics, this condition (as thecomplex balance condition) was introduced by Horn and Jackson in 1972.^[22]

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.^[23] Under these microscopic assumptions, the semi-detailed balance condition is just thebalance equation for the Markov microkinetics according to theMichaelis–Menten–Stueckelberg theorem.^[24]

Let us represent the generalized mass action law in the equivalent form: the rate of the elementary process$${\displaystyle \sum _i{\alpha }_{ri}{\text{A}}_i⟶\sum _i{\beta }_{ri}{\text{A}}_i}$$is$${\displaystyle w_r={\phi }_r\exp \left(\sum _i\frac{{\alpha }_{ri}{\mu }_i}{RT}\right)}$$where${\displaystyle {\mu }_i=\mathrm{\partial }F(T,V,N)/\mathrm{\partial }{N}_i}$ is the chemical potential and${\displaystyle F(T,V,N)}$ is theHelmholtz free energy. The exponential term is called theBoltzmann factor and the multiplier${\displaystyle {\phi }_r\ge 0}$ is the kinetic factor.^[24]Let us count the direct and reverse reaction in the kinetic equation separately:$${\displaystyle \frac{d{N}_i}{dt}=V\sum _r{\gamma }_{ri}{w}_r}$$An auxiliary function${\displaystyle \theta (\lambda )}$ of one variable${\displaystyle \lambda \in [0,1]}$ is convenient for the representation of dissipation for the mass action law$${\displaystyle \theta (\lambda )=\sum _r{\phi }_r\exp \left(\sum _i\frac{(\lambda {\alpha }_{ri}+(1-\lambda ){\beta }_{ri}){\mu }_i}{RT}\right)}$$This function${\displaystyle \theta (\lambda )}$ may be considered as the sum of the reaction rates fordeformed input stoichiometric coefficients${\displaystyle {\tilde{\alpha }}_{\rho }(\lambda )=\lambda {\alpha }_{\rho }+(1-\lambda ){\beta }_{\rho }}$. For${\displaystyle \lambda =1}$ it is just the sum of the reaction rates. The function${\displaystyle \theta (\lambda )}$ is convex because${\displaystyle {\theta }^{″}(\lambda )\ge 0}$.

Direct calculation gives that according to the kinetic equations$${\displaystyle \frac{dF}{dt}=-VRT{\frac{d\theta (\lambda )}{d\lambda }|}_{\lambda =1}}$$This isthe general dissipation formula for the generalized mass action law.^[24]

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

The semi-detailed balance condition can be transformed into identity${\displaystyle \theta (0)\equiv \theta (1)}$. Therefore, for the systems with semi-detailed balance${\displaystyle dF/dt\le 0}$.^[22]

For any reaction mechanism and a given positive equilibrium acone of possible velocities for the systems with detailed balance is defined for any non-equilibrium stateN$${\displaystyle {\mathbf{Q}}_{\mathrm{D}\mathrm{B}}(N)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\{{\gamma }_r\mathrm{s}\mathrm{g}\mathrm{n}(w_r^{+}(N)-w_r^{-}(N))|r=1,\dots ,m\},}$$where cone stands for theconical hull and the piecewise-constant functions${\displaystyle \mathrm{s}\mathrm{g}\mathrm{n}(w_r^{+}(N)-w_r^{-}(N))}$ do not depend on (positive) values of equilibrium reaction rates${\displaystyle w_r^{\mathrm{e}\mathrm{q}}}$ and are defined by thermodynamic quantities under assumption of detailed balance.

Thecone theorem states that for the given reaction mechanism and given positive equilibrium, the velocity (dN/dt) at a stateN for a system with complex balance belongs to the cone${\displaystyle {\mathbf{Q}}_{\mathrm{D}\mathrm{B}}(N)}$. That is, there exists a system with detailed balance, the same reaction mechanism, the same positive equilibrium, that gives the same velocity at stateN.^[25] According to cone theorem, for a given stateN, 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. This meanslocal equivalence of detailed and complex balance.

Detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process and requires 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 becomes 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${\displaystyle {\text{A}}_1^{}⟶{\text{A}}_2^{}⟶{\text{A}}_3^{}⟶{\text{A}}_1^{}}$ cannot be obtained as such a limit but the reaction mechanism can.^[26]

Gorban–Yablonsky theorem.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) theconvex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with thelinear span of the stoichiometric vectors of the reversible reactions.^[20] Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.

• T-symmetry
• Microscopic reversibility
• Master equation
• Balance equation
• Gibbs sampling
• Metropolis–Hastings algorithm
• Atomic spectral line (deduction of the Einstein coefficients)
• Random walks on graphs

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• Markov models
• Chemical kinetics

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