The principle of detailed balance 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.

A1 -> A2 -> \cdots -> A_\mathit -> 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

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 $\backslash ce\; A\_v$ is a particle with velocity ''v''. Under time reversal $\backslash ce\; A\_v$ transforms into $\backslash ce\; A\_$. 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. Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.

_{''ij''} is the Markov transition probability from state ''i'' to state ''j'', i.e. , and π_{''i''} and π_{''j''} 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'':
:$\backslash pi(s\text{'})\; P(s\text{'},s)\; =\; \backslash pi(s)\; P(s,s\text{'})\backslash ,.$
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'',
:$P(a,b)\; P(b,c)\; P(c,a)\; =\; P(a,c)\; P(c,b)\; P(b,a)\backslash ,.$
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. Indeed, a necessary and sufficient condition for the reversibility condition is Kolmogorov's criterion, which demands that for the reversible chains the product of transition rates over any closed loop of states must be the same in both directions.
Transition matrices that are symmetric or 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 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 more details see Closest reversible Markov chain

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 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. Hugh Everettbr>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

Extended detailed balance for systems with irreversible reactions

Chemical Engineering Science 66, 5388–5399

holds: :$\backslash prod\_^m\; (k\_r^+)^=\backslash prod\_^m\; (k\_r^-)^\; \backslash ,\; .$ ''Remark.'' It is sufficient to use in the Wegscheider conditions a basis of solutions of the system $\backslash boldsymbol\; =0$. 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: #A1 <=> A2
# A2 <=> A3
# A3 <=> A1
# +A2 <=> 2A3
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:
:$\backslash gamma\_1+\backslash gamma\_2+\backslash gamma\_3=0$ and $\backslash gamma\_3+\backslash gamma\_4-\backslash gamma\_2=0$.
The computational aspect of the Wegscheider conditions was studied by D. Colquhoun with co-authors.
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).

_{j}'' and temperature. For this purpose, use the representation of the activity through the chemical potential:
:$a\_i\; =\; \backslash exp\backslash left\; (\backslash frac\backslash right\; )$
where ''μ_{i}'' is the _{j}''. For the ideal systems, $\backslash mu\_i=RT\backslash ln\; c\_i+\backslash mu^\_i$ and $a\_j=c\_j$: the activity is the concentration and the generalized mass action law is the usual _{j}'' (usually measured in _{j}''. For the ideal systems,
:$F=RT\; \backslash sum\_i\; N\_i\; \backslash left(\backslash ln\backslash left(\backslash frac\backslash right)-1+\backslash frac\backslash right)$.
The chemical potential is a partial derivative: $\backslash mu\_i=\backslash partial\; F(T,V,N)/\backslash partial\; N\_j$.
The chemical kinetic equations are
:$\backslash frac=V\; \backslash sum\_r\; \backslash gamma\_(w^+\_r-w^-\_r)\; .$
If the principle of detailed balance is valid then for any value of ''T'' there exists a positive point of detailed balance ''c''^{eq}:
:$w^+\_r(c^,T)=w^-\_r(c^,T)=w^\_r$
Elementary algebra gives
:$w^+\_r=w^\_r\; \backslash exp\; \backslash left(\backslash sum\_i\; \backslash frac\backslash right);\; \backslash ;\backslash ;\; w^-\_r=w^\_r\; \backslash exp\; \backslash left(\backslash sum\_i\; \backslash frac\backslash right);$
where $\backslash mu^\_i=\backslash mu\_i(c^,T)$
For the dissipation we obtain from these formulas:
:$\backslash frac=\backslash sum\_i\; \backslash frac\; \backslash frac=\backslash sum\_i\; \backslash mu\_i\; \backslash frac\; =\; -VRT\; \backslash sum\_r\; (\backslash ln\; w\_r^+-\backslash ln\; w\_r^-)\; (w\_r^+-w\_r^-)\; \backslash leq\; 0$
The inequality holds because ln is a monotone function and, hence, the expressions $\backslash ln\; w\_r^+-\backslash ln\; w\_r^-$ and $w\_r^+-w\_r^-$ have always the same sign.
Similar inequalities are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the

The Michaelis–Menten–Stueckelberg Theorem.

Entropy 13, no. 5, 966–1019.

A1 -> A2 -> A3 -> A1 cannot be obtained as such a limit but the reaction mechanism A1 -> A2 -> A3 <- A1 can.Chu, Ch. (1971), Gas absorption accompanied by a system of first-order reactions, Chem. Eng. Sci. 26(3), 305–312.

History

The principle of detailed balance was explicitly introduced for collisions byLudwig Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...

. 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
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and lig ...

used the principle of detailed balance for gas kinetics with the reference to the principle of sufficient reason
The principle of sufficient reason states that everything must have a reason or a cause. The principle was articulated and made prominent by Gottfried Wilhelm Leibniz, with many antecedents, and was further used and developed by Arthur Schopenhau ...

. 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 (pg. 64).
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

in 1916 used the principle of detailed balance in a background for his quantum theory of emission and absorption of radiation.
In 1901, Rudolf Wegscheider introduced the principle of detailed balance for chemical kinetics. In particular, he demonstrated that the irreversible cycles 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
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then "M ...

.
The principle of detailed balance has been used in Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a ...

methods since their invention in 1953. In particular, in the Metropolis–Hastings algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequ ...

and in its important particular case, Gibbs sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is diff ...

, 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
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical ...

, 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 :$\backslash sum\_i\; \backslash alpha\_i\; \backslash ce\; A\_i\; \backslash ce\; \backslash sum\_j\; \backslash beta\_j\; \backslash ce\; B\_j$ transforms into $\backslash sum\_j\; \backslash beta\_j\; \backslash ce\; B\_j\; \backslash ce\; \backslash sum\_i\; \backslash alpha\_i\; \backslash ce\; A\_i$ and conversely. (Here, $\backslash ce\; A\_i,\; \backslash ce\; B\_j$ are symbols of components or states, $\backslash alpha\_i,\; \backslash beta\_j\; \backslash geq\; 0$ 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: # $\backslash ce\; A\_i$ 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. (2014Detailed 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 $\backslash ce\; A\_v$ is a particle with velocity ''v''. Under time reversal $\backslash ce\; A\_v$ transforms into $\backslash ce\; A\_$. 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. Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.

Reversible Markov chains

AMarkov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

is called a ''reversible Markov process'' or '' reversible Markov chain'' if it satisfies the detailed balance equations. These equations require that the transition probability matrix, ''P'', for the Markov process possess a stationary distribution (i.e. equilibrium probability distribution) π such that
:$\backslash pi\_\; P\_\; =\; \backslash pi\_\; P\_\backslash ,,$
where ''P''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 kinetics and generalized mass action kinetics with detailed balance. Nevertheless, the principle of detailed balance is not necessary for entropy growth. For example, in the linear irreversible cycleThesis, Princeton University, (1956, 1973), Appendix I, pp 121 ff. In his thesis, Everett used the term "detailed balance" unconventionally, instead of

balance equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probabi ...

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
Carlo Cercignani (17 June 1939 in Teulada – 7 January 2010 in Milan) was an Italian mathematician known for his work on the kinetic theory of gases. His contributions to the study of Boltzmann's equation include the proof of the H-theorem f ...

and Maria Lampis proved that the Lorentz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules. 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

Inchemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...

, the elementary reaction
An elementary reaction is a chemical reaction in which one or more chemical species react directly to form products in a single reaction step and with a single transition state. In practice, a reaction is assumed to be elementary if no reaction ...

s are represented by the stoichiometric equations
:$\backslash sum\_i\; \backslash alpha\_\; \backslash ce\; A\_i\; \backslash ce\; \backslash sum\_j\; \backslash beta\_\; \backslash ce\; A\_j\; \backslash ;\backslash ;\; (r=1,\; \backslash ldots,\; m)\; \backslash ,\; ,$
where $\backslash ce\; A\_i$ are the components and $\backslash alpha\_,\; \backslash beta\_\backslash geq\; 0$ 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 $\backslash boldsymbol=(\backslash gamma\_)$, $\backslash gamma\_=\backslash beta\_-\backslash alpha\_$ (gain minus loss). This matrix need not be square. The ''stoichiometric vector'' $\backslash gamma\_r$ is the ''r''th row of $\backslash boldsymbol$ with coordinates $\backslash gamma\_=\backslash beta\_-\backslash alpha\_$.
According to the ''generalized mass action law'', the reaction rate
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...

for an elementary reaction is
:$w\_r=k\_r\; \backslash prod\_^n\; a\_i^\; \backslash ,\; ,$
where $a\_i\backslash geq\; 0$ is the activity (the "effective concentration") of $A\_i$.
The reaction mechanism includes reactions with the reaction rate constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction.
For a reaction between reactants A and B to form product C
the reaction rate is often found to have the ...

s $k\_r>0$. 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 equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...

.
The principle of detailed balance for the generalized mass action law is: For given values $k\_r$ there exists a positive equilibrium $a\_i^>0$ that satisfies detailed balance, that is, $w\_r^+=w\_r^-$. This means that the system of ''linear'' detailed balance equations
: $\backslash sum\_i\; \backslash gamma\_\; x\_i\; =\; \backslash ln\; k\_r^+-\backslash ln\; k\_r^-=\backslash ln\; K\_r$
is solvable ($x\_i=\backslash ln\; a\_i^$). The following classical result gives the necessary and sufficient conditions for the existence of a positive equilibrium $a\_i^>0$ with detailed balance (see, for example, the textbook).
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
# If $k\_r^+>0$ then $k\_r^->0$ and, conversely, if $k\_r^->0$ then $k\_r^+>0$ (reversibility);
# For any solution $\backslash boldsymbol=(\backslash lambda\_r)$ of the system
:$\backslash boldsymbol\; =0\; \backslash ;\backslash ;\; \backslash left(\backslash mbox\backslash ;\backslash ;\; \backslash sum\_r\; \backslash lambda\_r\; \backslash gamma\_=0\backslash ;\backslash ;\; \backslash mbox\; \backslash ;\backslash ;\; i\backslash right)$
the Wegscheider's identity Gorban, A.N, Yablonsky, G.S. (2011Extended detailed balance for systems with irreversible reactions

Chemical Engineering Science 66, 5388–5399

holds: :$\backslash prod\_^m\; (k\_r^+)^=\backslash prod\_^m\; (k\_r^-)^\; \backslash ,\; .$ ''Remark.'' It is sufficient to use in the Wegscheider conditions a basis of solutions of the system $\backslash boldsymbol\; =0$. 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: #

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 theconcentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', '' number concentration'', ...

s ''cchemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

of the species under the conditions of interest, is the chemical potential of that species in the chosen standard state
In chemistry, the standard state of a material (pure substance, mixture or solution) is a reference point used to calculate its properties under different conditions. A superscript circle ° (degree symbol) or a Plimsoll (⦵) character is use ...

, ''R'' is the gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...

and ''T'' is the thermodynamic temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wo ...

.
The chemical potential can be represented as a function of ''c'' and ''T'', where ''c'' is the vector of concentrations with components ''claw of mass action
In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dy ...

.
Consider a system in isothermal (''T''=const) isochoric (the volume ''V''=const) condition. For these conditions, the Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ener ...

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 ''Nmole
Mole (or Molé) may refer to:
Animals
* Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America
* Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...

s), ''N'' is the vector with components ''NGibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...

decreases, for the isochoric systems with the constant internal energy (isolated system
In physical science, an isolated system is either of the following:
# a physical system so far removed from other systems that it does not interact with them.
# a thermodynamic system enclosed by rigid immovable walls through which neither ma ...

s) the entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

increases as well as for isobaric systems with the constant enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...

.
Onsager reciprocal relations and detailed balance

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: :$w^+\_r=w^\_r\; \backslash left(1+\backslash sum\_i\; \backslash frac\backslash right);\; \backslash ;\backslash ;\; w^-\_r=w^\_r\; \backslash left(1+\; \backslash sum\_i\; \backslash frac\backslash right);$ Therefore, again in the linear response regime near equilibrium, the kinetic equations are ($\backslash gamma\_=\backslash beta\_-\backslash alpha\_$): :$\backslash frac=-V\; \backslash sum\_j\; \backslash left;\; href="/html/ALL/s/sum\_r\_w^\_r\_\backslash gamma\_\backslash gamma\_\backslash right.html"\; ;"title="sum\_r\; w^\_r\; \backslash gamma\_\backslash gamma\_\backslash right">sum\_r\; w^\_r\; \backslash gamma\_\backslash gamma\_\backslash right$ This is exactly the Onsager form: following the original work of Onsager, we should introduce the thermodynamic forces $X\_j$ and the matrix of coefficients $L\_$ in the form :$X\_j\; =\; \backslash frac;\; \backslash ;\backslash ;\; \backslash frac=\backslash sum\_j\; L\_X\_j$ The coefficient matrix $L\_$ is symmetric: :$L\_=-\backslash frac\backslash sum\_r\; w^\_r\; \backslash gamma\_\backslash gamma\_$ These symmetry relations, $L\_=L\_$, are exactly the Onsager reciprocal relations. The coefficient matrix $L$ is non-positive. It is negative on thelinear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...

of the stoichiometric vectors $\backslash gamma\_$.
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: :$\backslash frac=V\backslash sum\_r\; \backslash gamma\_\; w\_r=V\backslash sum\_r\; (\backslash beta\_-\backslash alpha\_)w\_r$ Let us use the notations $\backslash alpha\_r=\backslash alpha\_$, $\backslash beta\_r=\backslash beta\_$ for the input and the output vectors of the stoichiometric coefficients of the ''r''th elementary reaction. Let $Y$ be the set of all these vectors $\backslash alpha\_r,\; \backslash beta\_r$. For each $\backslash nu\; \backslash in\; Y$, let us define two sets of numbers: :$R\_^+=\backslash ;\; \backslash ;\backslash ;\backslash ;\; R\_^-=\backslash $ $r\; \backslash in\; R\_^+$ if and only if $\backslash nu$ is the vector of the input stoichiometric coefficients $\backslash alpha\_r$ for the ''r''th elementary reaction;$r\; \backslash in\; R\_^-$ if and only if $\backslash nu$ is the vector of the output stoichiometric coefficients $\backslash beta\_r$ for the ''r''th elementary reaction. The principle of semi-detailed balance means that in equilibrium the semi-detailed balance condition holds: for every $\backslash nu\; \backslash in\; Y$ :$\backslash sum\_w\_r=\backslash sum\_w\_r$ The semi-detailed balance condition is sufficient for the stationarity: it implies that :$\backslash frac=V\; \backslash sum\_r\; \backslash gamma\_r\; w\_r=0$. For the Markov kinetics the semi-detailed balance condition is just the elementarybalance equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probabi ...

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 $d\; F/\; dt\; \backslash geq\; 0$ (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 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.''Horn, F., Jackson, R.'' (1972) General mass action kinetics. Arch. Ration. Mech. Anal. 47, 87–116.
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. Under these microscopic assumptions, the semi-detailed balance condition is just the balance equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probabi ...

for the Markov microkinetics according to the Michaelis– Menten– Stueckelberg theorem.''Gorban, A.N., Shahzad, M.'' (2011The Michaelis–Menten–Stueckelberg Theorem.

Entropy 13, no. 5, 966–1019.

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 :$\backslash sum\_i\; \backslash alpha\_\; \backslash ce\; A\_i\; \backslash ce\; \backslash sum\_i\; \backslash beta\_\; \backslash ce\; A\_i$ is :$w\_r=\backslash varphi\_r\; \backslash exp\backslash left(\backslash sum\_i\backslash frac\backslash right)$ where $\backslash mu\_i=\backslash partial\; F(T,V,N)/\; \backslash partial\; N\_i$ is the chemical potential and $F(T,V,N)$ is theHelmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ener ...

. The exponential term is called the ''Boltzmann factor'' and the multiplier $\backslash varphi\_r\; \backslash geq\; 0$ is the kinetic factor.
Let us count the direct and reverse reaction in the kinetic equation separately:
:$\backslash frac=V\backslash sum\_r\; \backslash gamma\_\; w\_r$
An auxiliary function $\backslash theta(\backslash lambda)$ of one variable $\backslash lambda\backslash in;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$Cone theorem and local equivalence of detailed and complex balance

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'' $\backslash mathbf\_(N)=\backslash ,$ where cone stands for theconical hull Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 10 ...

and the piecewise-constant functions $(w\_r^+(N)-w\_r^-(N))$ do not depend on (positive) values of equilibrium reaction rates $w\_r^$ and are defined by thermodynamic quantities under assumption of detailed balance.
The cone theorem states that for the given reaction mechanism and given positive equilibrium, the velocity (''dN/dt'') at a state ''N'' for a system with complex balance belongs to the cone $\backslash mathbf\_(N)$. 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''. 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. This means ''local equivalence of detailed and complex balance.''
Detailed balance for systems with irreversible reactions

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 cycleGorban
Gorban is a commune in Iași County, Western Moldavia, Romania
Romania ( ; ro, România ) is a country located at the crossroads of Central, Eastern, and Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hun ...

– 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) the convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...

of the stoichiometric vectors of the reversible reactions.'' Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.
See also

*T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal,
: T: t \mapsto -t.
Since the second law of thermodynamics states that entropy increases as time flows toward the futu ...

* Microscopic reversibility
* Master equation
* Balance equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probabi ...

* Gibbs sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is diff ...

* Metropolis–Hastings algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequ ...

* Atomic spectral line
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...

(deduction of the Einstein coefficients)
* Random walks on graphs
References

{{Reflist Non-equilibrium thermodynamics Statistical mechanics Markov models Chemical kinetics