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.
The principle of detailed balance was explicitly introduced for collisions by
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 ( ; ; 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 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
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
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,
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 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.]
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
and conversely. (Here,
are symbols of components or states,
are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of microreversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.
This reasoning is based on three assumptions:
does not change under time reversal;
# Equilibrium is invariant under time reversal;
# The macroscopic elementary processes are microscopically distinguishable. That is, they represent disjoint sets of microscopic events.
Any of these assumptions may be violated.
[Gorban, A.N. (2014]
Detailed balance in micro- and macrokinetics and micro-distinguishability of macro-processes
Results in Physics 4, 142–147
For example, Boltzmann's collision can be represented as where
is a particle with velocity ''v''. Under time reversal
. 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 material
s) 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
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
where ''P''''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'':
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. 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
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 cycle A1 -> A2 -> A3 -> A1, entropy production is positive but the principle of detailed balance does not hold.
Thus, the principle of detailed balance is a sufficient but not necessary condition for entropy increase in Boltzmann kinetics. These relations between the principle of detailed balance and the second law of thermodynamics were clarified in 1887 when Hendrik Lorentz objected to the Boltzmann H-theorem for polyatomic gases. Lorentz 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] 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,
Thesis, Princeton University, (1956, 1973), Appendix I, pp 121 ff. In his thesis, Everett used the term "detailed balance" unconventionally, instead of
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 ... 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
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
where are the components and are the stoichiometric coefficients. Here, the reverse reactions with positive constants are included in the list separately. We need this separation of direct and reverse reactions to apply later the general formalism to the systems with some irreversible reactions. The system of stoichiometric equations of elementary reactions is the ''reaction mechanism''.
The '' stoichiometric matrix'' is , (gain minus loss). This matrix need not be square. The ''stoichiometric vector'' is the ''r''th row of with coordinates .
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
where is the activity (the "effective concentration") of .
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 . 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 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 there exists a positive equilibrium that satisfies detailed balance, that is, . 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 a positive equilibrium with detailed balance (see, for example, the textbook [).
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
# If then and, conversely, if then (reversibility);
# For any solution of the system
the Wegscheider's identity] [ Gorban, A.N, Yablonsky, G.S. (2011] 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:
Extended detailed balance for systems with irreversible reactions
Chemical Engineering Science 66, 5388–5399
# A1 <=> A2
# A2 <=> A3
# A3 <=> A1
# +A2 <=> 2A3
There are two nontrivial independent Wegscheider's identities for this system:
They correspond to the following linear relations between the stoichiometric vectors:
: and .
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).
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
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 ''cj'' and temperature. For this purpose, use the representation of the activity through the chemical potential:
where ''μi'' is the chemical 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 ''cj''. For the ideal systems, and : the activity is the concentration and the generalized mass action law is the usual law 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 ''Nj'' (usually measured in mole
Mole (or Molé) may refer to:
* 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 ''Nj''. 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
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 [ are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the ] Gibbs 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:
Therefore, again in the linear response regime near equilibrium, the kinetic equations are ():
This is exactly the Onsager form: following the original work of Onsager,
we should introduce the thermodynamic forces and the matrix of coefficients in the form
The coefficient matrix is symmetric:
These symmetry relations, , are exactly the Onsager reciprocal relations. The coefficient matrix is non-positive. It is negative on 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 .
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:
Let us use the notations , for the input and the output vectors of the stoichiometric coefficients of the ''r''th elementary reaction. Let be the set of all these vectors .
For each , let us define two sets of numbers:
if and only if is the vector of the input stoichiometric coefficients for the ''r''th elementary reaction; if and only if is the vector of the output stoichiometric coefficients for the ''r''th elementary reaction.
The principle of semi-detailed balance means that in equilibrium the semi-detailed balance condition holds: for every
The semi-detailed balance condition is sufficient for the stationarity: it implies that
For the Markov kinetics the semi-detailed balance condition is just the elementary
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 (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.'' (2011]
The 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
where is the chemical potential and is 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 .... The exponential term is called the ''Boltzmann factor'' and the multiplier is the kinetic factor.
Let us count the direct and reverse reaction in the kinetic equation separately:
An auxiliary function of one variable