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.
Contents
- History
- Microscopical background
- Reversible Markov chains
- Detailed balance and entropy increase
- Wegscheiders conditions for the generalized mass action law
- Dissipation in systems with detailed balance
- Onsager reciprocal relations and detailed balance
- Semi detailed balance
- Dissipation in systems with semi detailed balance
- Detailed balance for systems with irreversible reactions
- References
History
The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle. The arguments in favor of this property are founded upon microscopic reversibility.
Five years before Boltzmann, James Clerk Maxwell used the principle of detailed balance for gas kinetics with the reference to the principle of sufficient reason. 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 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
The principle of detailed balance has been used in Markov chain Monte Carlo methods since their invention in 1953. In particular, in the Metropolis–Hastings algorithm and in its important particular case, Gibbs sampling, it is used as a simple and reliable condition to provide the desirable equilibrium state.
Now, the principle of detailed balance is a standard part of the university courses in statistical mechanics, physical chemistry, chemical and physical kinetics.
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
and conversely. (Here,
This reasoning is based on three assumptions:
-
A i - 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. For example, Boltzmann's collision can be represented as
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
A Markov process is called a reversible Markov process or reversible Markov chain precisely 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 Pij is the Markov transition probability) from state i to state j, 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:
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 (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 appropriately-sized degenerate sub-states.
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
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's condition holds for all Markov processes, irrespective of time-reversibility. Later, entropy increase was proved for all Markov processes by a direct method. These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was referred to as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semi-detailed balance" or the "complex balance". In 1981, Carlo Cercignani and Maria Lampis proved that the Lorentz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules. 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 kinetics, the elementary reactions are represented by the stoichiometric equations
where
The stoichiometric matrix is
According to the generalized mass action law, the reaction rate for an elementary reaction is
where
The reaction mechanism includes reactions with the reaction rate constants
The principle of detailed balance for the generalized mass action law is: For given values
is solvable (
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
- If
k r + > 0 thenk r − > 0 and, conversely, ifk r − > 0 thenk r + > 0 (reversibility); - For any solution
λ = ( λ r ) 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:
-
A 1 ↽ − − ⇀ A 2 -
A 2 ↽ − − ⇀ A 3 -
A 3 ↽ − − ⇀ A 1 -
A 1 + A 2 ↽ − − ⇀ 2 A 3
There are two nontrivial independent Wegscheider's identities for this system:
They correspond to the following linear relations between the stoichiometric vectors:
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 concentrations cj and temperature. For this purpose, use 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,
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 moles), 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
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 systems) the entropy increases as well as for isobaric systems with the constant enthalpy.
Onsager reciprocal relations and detailed balance
Let the principle of detailed balance be valid. Then, 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
The coefficient matrix
These symmetry relations,
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 each
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 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
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.
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.
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
is
where
An auxiliary function
This function
Direct calculation gives that according to the kinetic equations
This is the general dissipation formula for the generalized mass action law.
Convexity of
The semi-detailed balance condition can be transformed into identity
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 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
Gorban–Yablonsky theorem. A system of reactions with some irreversible reactions is a limit of systems with detailed balance when some constants tend to zero if and only if (i) the reversible part of this system satisfies the principle of detailed balance and (ii) 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.