Detailed balance
Encyclopedia
The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions): At equilibrium
, each elementary process should be equilibrated by its reverse process.
. In 1872, he proved his H-theorem
using this principle. The arguments in favor of this property are founded upon microscopic reversibility
. A. Einstein
in 1916 used this principle in a background for his quantum theory of emission and absorption of radiation.
In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics. In particular, he demonstrated that the irreversible cycles
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 awarded in 1968 by the Nobel Prize in Chemistry
.
Now, the principle of detailed balance is a standard part of the university courses in statistical mechanics, physical chemistry, chemical and physical kinetics.
transforms into 
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.
is said to have detailed balance if the transition probability, P, between each pair of states i and j in the state space obey
where P is the Markov transition matrix (transition probability), i.e., Pij = P(Xt = j | Xt − 1 = i); and πi and πj are the equilibrium probabilities of being in states i and j, respectively. When Pr(Xt−1 = i) = πi for all i, this is equivalent to the joint probability matrix, Pr(Xt−1 = i, Xt = j) being symmetric in i and j; or symmetric in t − 1 and t.
The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and P(s′, s) a transition kernel probability density from state s′ to state s:
A Markov process that has detailed balance is said to be a reversible Markov process or reversible Markov chain.
The detailed balance condition is stronger than that required merely for a stationary distribution
; that is, there are Markov processes with stationary distributions that do not have detailed balance. Detailed balance implies that, around any closed cycle of states, there is no net flow of probability. For example, it implies that, for all a, b and c,
This can be proved by substitution from the definition. In the case of a positive transition matrix, the "no net flow" condition implies detailed balance.
Transition matrices that are symmetric (Pij = Pji or P(s′, s) = P(s, s′)) always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution. For continuous systems with detailed balance, it may be possible to continuously transform the coordinates until the equilibrium distribution is uniform, with a transition kernel which then is symmetric. In the case of discrete states, it may be possible to achieve something similar by breaking the Markov states into a degeneracy of sub-states.
states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance 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 the entropy growth. For example, in the linear irreversible cycle
the entropy production is positive but the principle of detailed balance does not hold.
The principle of detailed balance is a sufficient but not necessary condition for the entropy growth in the Boltzmann kinetics. These relations between the principle of detailed balance and the Second law of thermodynamics were clarified in 1887 when Hendrik Lorentz
objected the Boltzmann 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 the entropy growth. In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was explicitely proved later. These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was discussed as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semi-detailed balance" or the "complex balance". In 1981, Carlo Cercignani and Maria Lampis proved that the Lorenz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules. Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.
, the elementary 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). The stoichiometric vector 
is the rth row of 
with coordinates 
.
According to the generalized mass action law, the reaction rate
for an elementary reaction is
where
is the activity
of
.
The reaction mechanism includes reactions with the reaction rate constants
. For each r the following notations are used: 
, 
, 
is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not, 
is the reaction rate for the reverse reaction if it is in the reaction mechanism and 0 if it is not. For a reversible reaction, 
is the equilibrium constant.
The principle of detailed balance for the generalized mass action law is: For given values
there exists a positive equilibrium 
with detailed balance, 
. This means that the system of linear detailed balance equations
is solvable (
). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium 
with detailed balance (see, for example, the textbook).
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
the Wegscheider's identity 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:
There are two nontrivial independent Wegscheider's identities for this system:
and 
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).
s cj and temperature
. For this purpose, let us the representation of the activity through the chemical potential:
where μi is the chemical potential
of the species under the conditions of interest, μoi is the chemical potential of that species in the chosen standard state
, R is the gas constant
and T is the thermodynamic temperature
.
The chemical potential can be represented as a function of c and T, where c is the vector of concentrations with components cj. For the ideal systems,
and 
: the activity is the concentration and the generalized mass action law is the usual law of mass action.
Let us consider a system in isothermal
(T=const) isochoric
(the volume V=const) condition. For these conditions, the Helmholtz free energy
F(T,V,N) measures the “useful” work obtainable from a system. It is a functions of the temperature T, the volume V and the amounts of chemical components Nj (usually measured in mole
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 ceq:
Elementary algebra gives
where
For the dissipation we obtain from these formulas:
The inequality holds because ln is a monotone function and, hence, the expressions
and 
have always the same sign.
Similar inequalities are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the Gibbs free energy
decreases, for the isochoric systems with the constant internal energy
(isolated system
s) the entropy
increases as well as for isobaric systems with the constant enthalpy
.
Onsager reciprocal relations
Let the principle of detailed balance be valid. Then, in the linear approximation near equilibrium the reaction rates for the generalized mass action law are
Therfore, in the linear approximation near equilibrium, the kinetic equations are (
):
This is exacltly 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
of the stoichiometric vectors
.
So, the Onsager relations follow from the principle of detailed balance in the linear approximation near equilibrium.
Let us use the notations
, 
for the input and the output vectors of the stoichiometric coefficients of the rth 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 rth elementary reaction;
if and only if 
is the vector of the output stoichiometric coefficients 
for the rth elementary reaction.
The principle of semi-detailed balance means that in equilibrium the semi-detailed balance condition holds: for every
The semi-detailded balance condition is sufficient for the stationarity: it implies that
.
For the Markov kinetics the semi-detailed balance condition is just the elementary balance equation
and holds for any steady state. For the nonlinear mass action law it is, in general, sufficient but not necessary condition for stationarity.
The 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 inroduced by Horn and Jackson in 1972.
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
for the Markov microkinetics according to the Michaelis-Menten-Stueckelberg theorem.
is
where
is the chemical potential and 
is the Helmholtz free energy
. 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 
is convenient for the representation of dissipation for the mass action law
This function
may be considered as the sum of the reaction rates for deformed input stoichiometric coefficients 
. For 
it is just the sum of the reaction rates. The function 
is convex because 
.
Direct calculation gives that according to the kinetic equations
This is the general dissipation formula for the generalized mass action law.
Convexity of
gives the sufficient and necessary conditions for the proper dissipation inequality:
The semi-detailed balance condition can be transformed into identity
. Therefore, for the systems with semi-detailed balance 
.
cannot be obtained as such a limit but the reaction mechanism 
can.
A system of reactions with some irreversible reactions is a limit of systems with detailed balance when some constants tend to zero if and only if (i) the reversible part of this system satisfies the principle of detailed balance and (ii) the convex hull
of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span
of the stoichiometric vectors of the reversible reactions. Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
, each elementary process should be equilibrated by its reverse process.
History
The principle of detailed balance was explicitly introduced for collisions by Ludwig BoltzmannLudwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
. In 1872, he proved his H-theorem
H-theorem
In Classical Statistical Mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process. H-theorem follows from considerations of Boltzmann's equation...
using this principle. The arguments in favor of this property are founded upon microscopic reversibility
Microscopic reversibility
The principle of Microscopic reversibility in physics and chemistry is twofold:* First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with respect to inversion in time ;* Second, it relates to...
. A. Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
in 1916 used this principle in a background for his quantum theory of emission and absorption of radiation.
In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics. In particular, he demonstrated that the irreversible cycles
are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance. In 1931, Lars OnsagerLars Onsager
Lars Onsager was a Norwegian-born American physical chemist and theoretical physicist, winner of the 1968 Nobel Prize in Chemistry.He held the Gibbs Professorship of Theoretical Chemistry at Yale University....
used these relations in his works awarded in 1968 by the Nobel Prize in Chemistry
Nobel Prize in Chemistry
The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...
.
Now, the principle of detailed balance is a standard part of the university courses in statistical mechanics, physical chemistry, chemical and physical kinetics.
Microscopical background
The microscopic "reversing of time" turns at the kinetic level into the "reversing of arrows": the elementary processes transform into their reverse processes. For example, the reaction transforms into and conversely. (Here,
are symbols of components or states, are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of microreversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.Reversible Markov chains
A Markov processMarkov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
is said to have detailed balance if the transition probability, P, between each pair of states i and j in the state space obey
where P is the Markov transition matrix (transition probability), i.e., Pij = P(Xt = j | Xt − 1 = i); and πi and πj are the equilibrium probabilities of being in states i and j, respectively. When Pr(Xt−1 = i) = πi for all i, this is equivalent to the joint probability matrix, Pr(Xt−1 = i, Xt = j) being symmetric in i and j; or symmetric in t − 1 and t.
The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and P(s′, s) a transition kernel probability density from state s′ to state s:
A Markov process that has detailed balance is said to be a reversible Markov process or reversible Markov chain.
The detailed balance condition is stronger than that required merely for a stationary distribution
Stationary distribution
Stationary distribution may refer to:* The limiting distribution in a Markov chain* The marginal distribution of a stationary process or stationary time series* The set of joint probability distributions of a stationary process or stationary time series...
; that is, there are Markov processes with stationary distributions that do not have detailed balance. Detailed balance implies that, around any closed cycle of states, there is no net flow of probability. For example, it implies that, for all a, b and c,
This can be proved by substitution from the definition. In the case of a positive transition matrix, the "no net flow" condition implies detailed balance.
Transition matrices that are symmetric (Pij = Pji or P(s′, s) = P(s, s′)) always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution. For continuous systems with detailed balance, it may be possible to continuously transform the coordinates until the equilibrium distribution is uniform, with a transition kernel which then is symmetric. In the case of discrete states, it may be possible to achieve something similar by breaking the Markov states into a degeneracy of sub-states.
Detailed balance and the entropy growth
For many systems of physical and chemical kinetics, detailed balance provides sufficient conditions for the entropy growth in isolated systems. For example, the famous Boltzmann H-theoremH-theorem
In Classical Statistical Mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process. H-theorem follows from considerations of Boltzmann's equation...
states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance 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 the entropy growth. For example, in the linear irreversible cycle
the entropy production is positive but the principle of detailed balance does not hold.The principle of detailed balance is a sufficient but not necessary condition for the entropy growth in the Boltzmann kinetics. These relations between the principle of detailed balance and the Second law of thermodynamics were clarified in 1887 when Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
objected 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 the entropy growth. In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was explicitely proved later. These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was discussed as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semi-detailed balance" or the "complex balance". In 1981, Carlo Cercignani and Maria Lampis proved that the Lorenz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules. 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
In chemical kineticsChemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...
, the elementary reaction
Elementary reaction
An elementary reaction is a chemical reaction in which one or more of the chemical species react directly to form products in a single reaction step and with a single transition state....
s are represented by the stoichiometric equations
Chemical equation
A chemical equation is the symbolic representation of a chemical reaction where the reactant entities are given on the left hand side and the product entities on the right hand side. The coefficients next to the symbols and formulae of entities are the absolute values of the stoichiometric numbers...
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
Stoichiometry
Stoichiometry is a branch of chemistry that deals with the relative quantities of reactants and products in chemical reactions. In a balanced chemical reaction, the relations among quantities of reactants and products typically form a ratio of whole numbers...
is
, (gain minus loss). The stoichiometric vector is the rth row of with coordinates .According to the generalized mass action law, the reaction rate
Reaction rate
The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
for an elementary reaction is
where
is the activityActivity (chemistry)
In chemical thermodynamics, activity is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.By convention, activity...
of
.The reaction mechanism includes reactions with the reaction rate constants
. For each r the following notations are used: , , is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not, is the reaction rate for the reverse reaction if it is in the reaction mechanism and 0 if it is not. For a reversible reaction, is the equilibrium constant.The principle of detailed balance for the generalized mass action law is: For given values
there exists a positive equilibrium with detailed balance, . This means that the system of linear detailed balance equationsis solvable (
). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium with detailed balance (see, for example, the textbook).Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
- If
then 
(reversibility); - For any solution
of the system
the Wegscheider's identity 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:
There are two nontrivial independent Wegscheider's identities for this system:
and 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 concentrationConcentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
s cj and temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. For this purpose, let us the representation of the activity through the chemical potential:
where μi is the chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
of the species under the conditions of interest, μ
Standard state
In chemistry, the standard state of a material is a reference point used to calculate its properties under different conditions. In principle, the choice of standard state is arbitrary, although the International Union of Pure and Applied Chemistry recommends a conventional set of standard states...
, R is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
and T is the thermodynamic temperature
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
.
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.Let us consider a system in isothermal
Isothermal process
An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir , and the change occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir...
(T=const) isochoric
Isochoric process
An isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant...
(the volume V=const) condition. For these conditions, the Helmholtz free energy
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
F(T,V,N) measures the “useful” work obtainable from a system. It is a functions of the temperature T, the volume V and the amounts of chemical components Nj (usually measured in mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
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 ceq:
Elementary algebra gives
where
For the dissipation we obtain from these formulas:
The inequality holds because ln is a monotone function and, hence, the expressions
and have always the same sign.Similar inequalities are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the Gibbs free energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
decreases, for the isochoric systems with the constant internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
(isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...
s) the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
increases as well as for isobaric systems with the constant enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
.
Onsager reciprocal relationsOnsager reciprocal relationsIn thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists....
and detailed balance
Let the principle of detailed balance be valid. Then, in the linear approximation near equilibrium the reaction rates for the generalized mass action law areTherfore, in the linear approximation near equilibrium, the kinetic equations are (
):This is exacltly the Onsager form: following the original work of Onsager, we should introduce the thermodynamic forces
and the matrix of coefficients in the formThe 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 spanLinear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
of the stoichiometric vectors
.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:Let us use the notations
, for the input and the output vectors of the stoichiometric coefficients of the rth 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 rth elementary reaction; if and only if is the vector of the output stoichiometric coefficients for the rth elementary reaction.The principle of semi-detailed balance means that in equilibrium the semi-detailed balance condition holds: for every
The semi-detailded balance condition is sufficient for the stationarity: it implies that
.For the Markov kinetics the semi-detailed balance condition is just the elementary balance equation
Balance equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states.-Global balance:...
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 inroduced by Horn and Jackson in 1972.
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
Balance equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states.-Global balance:...
for the Markov microkinetics according to the Michaelis-Menten-Stueckelberg theorem.
Dissipation in systems with semi-detailed balance
Let us represent the generalized mass action law in the equivalent form: the rate of the elementary processis
where
is the chemical potential and is the Helmholtz free energyHelmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
. 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 is convenient for the representation of dissipation for the mass action lawThis function
may be considered as the sum of the reaction rates for deformed input stoichiometric coefficients . For it is just the sum of the reaction rates. The function is convex because .Direct calculation gives that according to the kinetic equations
This is the general dissipation formula for the generalized mass action law.
Convexity of
gives the sufficient and necessary conditions for the proper dissipation inequality:The semi-detailed balance condition can be transformed into identity
. Therefore, for the systems with semi-detailed balance .Detailed balance for systems with irreversible reactions
Detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process and required reversibility of all elementary processes. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. If one represents irreversible reactions as limits of reversible steps, then it become obvious that not all reaction mechanisms with irreversible reactions can be obtained as limits of systems or reversible reactions with detailed balance. For example, the irreversible cycle cannot be obtained as such a limit but the reaction mechanism can.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
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
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
- asymmetryAsymmetryAsymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....
- Atomic spectral lineAtomic spectral lineIn physics, atomic spectral lines are of two types:* An emission line is formed when an electron makes a transition from a particular discrete energy level of an atom, to a lower energy state, emitting a photon of a particular energy and wavelength...
(deduction of the Einstein coefficients) - Balance equationBalance equationIn probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states.-Global balance:...
- DMPDynamics of Markovian ParticlesDynamics of Markovian particles is the basis of a theory for kinetics of particles in open heterogeneous systems. It can be looked upon as an application of the notion of stochastic process conceived as a physical entity; e.g...
- Gibbs samplingGibbs samplingIn statistics and in statistical physics, Gibbs sampling or a Gibbs sampler is an algorithm to generate a sequence of samples from the joint probability distribution of two or more random variables...
- Master equationMaster equationIn physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...
- Random walks on graphs
External links
- David Aldous, James A. Fill, Reversible Markov Chains (draft).