Editing Detailed balance

According to Ter Haar <ref>[http://dx.doi.org/10.1103/RevModPhys.27.289 D. Ter Haar  "Foundations of Statistical Mechanics", Reviews of Modern Physics '''27''' pp. 289-338 (1955)] Appendix VII</ref> the essence of the detailed balance is: "...at equilibrium the number of processes destroying situation <math>A</math> and creating situation <math>B</math> will be equal to the number of processes producing <math>A</math> and destroying <math>B</math>.

A Markov process that has detailed balance is said to be a '''reversible Markov process''' or '''reversible Markov chain'''.<ref name=OHagan />

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-theorem<ref name = "Boltzmann1872"/> 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<ref name = "Boltzmann1872"/><ref name=Tolman1938/> served as a prototype of many similar formulas for dissipation in mass action kinetics<ref>Volpert, A.I., Khudyaev, S.I. (1985), Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht, The Netherlands: Nijoff. (Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.)</ref> and generalized mass action kinetics<ref>Schuster, S., Schuster R. (1989). [http://www.springerlink.com/content/g3m2177v4344065q/ A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation.] J. Math. Chem, 3 (1), 25-42.</ref> with detailed balance.

Nevertheless, the principle of detailed balance is not necessary for the entropy growth. For example, in the linear irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> 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.<ref>Lorentz H.-A. (1887) Üeber das Gleichgewicht der lebendigen Kraft unter Gasmolekülen. S.A.W. [Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne.] 95, 115-152.</ref> 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.<ref name=Boltzmann1887>Boltzmann L. (1887) Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen. S.A.W. [Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne.] 95, 153-164.</ref> 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.<ref>Shannon, C.E. (1948) A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp.&nbsp;379–423, 623–656. [http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-379.pdf] [http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-4-623.pdf]</ref> 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.<ref>Cercignani, C. and Lampis, M. (1981). On the H-theorem for polyatomic gases, Journal of Statistical Physics, V. 26 (4), 795-801.</ref> Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.

According to the ''generalized mass action law'', the reaction rate for an elementary reaction is

is solvable (<math>x_i=\ln a_i^{\rm eq}</math>). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium <math>a_i^{\rm eq}>0</math> with  detailed balance (see, for example, the textbook<ref name=Yab1991/>).

Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:

the Wegscheider's identity<ref name=GorbanYablonsky2011>Gorban, A.N., Yablonsky, G.S. (2011) [http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.5280v3.pdf Extended detailed balance for systems with irreversible reactions], Chemical Engineering Science 66, 5388–5399.</ref> holds:

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<ref name=GorbanYablonsky2011/>:  

The computational aspect of the Wegscheider conditions was studied by D. Colquhoun with co-authors.<ref>Colquhoun, D., Dowsland, K.A., Beato, M., and Plested, A.J.R. (2004) [http://www.ucl.ac.uk/Pharmacology/dc-bits/colquhoun-biophysj-04.pdf How to Impose Microscopic Reversibility in Complex Reaction Mechanisms], Biophysical 86, June 2004, 3510–3518</ref>

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).

:<math>a_i = \exp\left (\frac{\mu_i - \mu^{\ominus}_i}{RT}\right )</math>

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 ''N_j'' (usually measured in moles), ''N'' is the vector with components ''N_j''. For the ideal systems, <math>F=RT \sum_i N_i \left(\ln\left(\frac{N_i}{V}\right)-1+\frac{\mu^{\ominus}_i(T)}{RT}\right) </math>

The chemical potential is a partial derivative: <math> \mu_i=\partial F(T,V,N)/\partial N_j</math>.

If the principle of detailed balance is valid then for any value of ''T'' there exists a positive point of detailed balance ''c''^eq:

:<math>w^+_r(c^{\rm eq},T)=w^-_r(c^{\rm eq},T)=w^{\rm eq}_r</math>

For the dissipation we obtain from these formulas:

Similar inequalities<ref name=Yab1991/> 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 ==

The principle of '''semi-detailed balance''' means that in equilibrium the semi-detailed balance condition holds: for every <math>\nu \in Y</math>

:<math>\frac{d N}{dt}=V \sum_r \gamma_r w_r=0</math>.

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.

 

 

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.<ref>''Stueckelberg, E.C.G.'' (1952) Theoreme ''H'' et unitarite de ''S''. Helv. Phys. Acta  25, 577-580</ref> 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.<ref name="GorbanShahzad2011">''Gorban, A.N., Shahzad, M.'' (2011) [http://arxiv.org/pdf/1008.3296v3 The Michaelis-Menten-Stueckelberg Theorem.] Entropy 13, no. 5, 966-1019.</ref>

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 <math>A_1 \to A_2 \to A_3 \to A_1</math> cannot be obtained as such a limit but the reaction mechanism <math>A_1 \to A_2 \to A_3 \leftarrow A_1</math> can.<ref>Chu, Ch. (1971), Gas absorption accompanied by a system of first-order reactions, Chem. Eng. Sci. 26(3), 305-312.</ref>

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