Detailed balance


In thermodynamics, the principle of detailed balance says that every process of energy transfer in one direction must also allow energy transfer in the opposite direction, and in equilibrium, the flux in both directions must be equal. For example, the principle can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.
The principle is a consequence of the second law of thermodynamics.

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 and found that "Now it is impossible to assign a reason" why detailed balance should be rejected.
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, for which he was awarded the 1968 Nobel Prize in Chemistry.
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.

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
and conversely.. The equilibrium ensemble should be invariant with respect to this transformation because of microreversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.
This reasoning is based on three assumptions:
  1. does not change under time reversal;
  2. Equilibrium is invariant under time reversal;
  3. The macroscopic elementary processes are microscopically distinguishable. That is, they represent disjoint sets of microscopic events.
Any of these assumptions may be violated. For example, Boltzmann's collision can be represented as where is a particle with velocity v. Under time reversal transforms into. 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 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.

Detailed balance

Reversibility

A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equationswhere Pij is the Markov transition probability from state i to state j, i.e., and πi and πj 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, because there are Markov processes with stationary distributions that do not have detailed balance.
Transition matrices that are symmetric or always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution.

Kolmogorov's criterion

Reversibility is equivalent to Kolmogorov's criterion: the product of transition rates over any closed loop of states is the same in both directions.
For example, it implies that, for all a, b and c,For example, if we have a Markov chain with three states such that only these transitions are possible:, then they violate Kolmogorov's criterion.

Closest reversible Markov chain

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.

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 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'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 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 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, . This matrix need not be square. 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 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.
Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
  1. If then and, conversely, if then ;
  2. 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 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.
A simple nonlinear example gives us a linear cycle supplemented by one nonlinear step:
  1. A1 <=> A2
  2. A2 <=> A3
  3. A3 <=> A1
  4. +A2 <=> 2A3
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.