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==Semi-detailed balance== | ==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: | 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: | ||
:<math>\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r </math> | :<math>\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r </math> | ||
Let us use the notations <math>\alpha_r=\alpha_{ri}</math>, <math>\beta_r=\beta_{ri}</math> for the input and the output vectors of the stoichiometric coefficients of the ''r''th elementary reaction. Let <math>Y</math> be the set of all these vectors <math>\alpha_r, \beta_r</math>. | |||
For each <math>\nu \in Y</math>, let us define two sets of numbers: | For each <math>\nu \in Y</math>, let us define two sets of numbers: | ||
:<math>R_{\nu}^+=\{r|\alpha_r=\nu \}; \;\;\; R_{\nu}^-=\{r|\beta_r=\nu \}</math> | :<math>R_{\nu}^+=\{r|\alpha_r=\nu \}; \;\;\; R_{\nu}^-=\{r|\beta_r=\nu \}</math> | ||
<math>r \in R_{\nu}^+</math> if and only if <math>\nu</math> is the vector of the input stoichiometric coefficients <math>\alpha_r</math> for the ''r''th elementary reaction;<math>r \in R_{\nu}^-</math> if and only if <math>\nu</math> is the vector of the output stoichiometric coefficients <math>\beta_r</math> for the ''r''th elementary reaction. | <math>r \in R_{\nu}^+</math> if and only if <math>\nu</math> is the vector of the input stoichiometric coefficients <math>\alpha_r</math> for the ''r''th elementary reaction;<math>r \in R_{\nu}^-</math> if and only if <math>\nu</math> is the vector of the output stoichiometric coefficients <math>\beta_r</math> for the ''r''th elementary reaction. | ||
The principle of ''semi-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>\sum_{r\in R_{\nu}^+}w_r=\sum_{r\in R_{\nu}^+}w_r</math> | :<math>\sum_{r\in R_{\nu}^+}w_r=\sum_{r\in R_{\nu}^+}w_r</math> | ||
The semi-detailded balance condition is sufficient for the stationarity: it implies that | The semi-detailded balance condition is sufficient for the stationarity: it implies that | ||
:<math>\frac{d N}{dt}=V \sum_r \gamma_r w_r=0 | :<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. | |||
For systems that obey the generalized mass action law the semi-detailed balance condition is sufficient for the dissipation inequality <math>d F/ dt \geq 0</math> (for the Helmholtz free energy under isothermal isochoric conditions and for the dissipation inequalities under other classical conditions for the corresponding thermodynamic potentials). | |||
For systems that obey the | |||
Boltzmann introduced the semi-detailed balance condition for collisions in 1887 <ref name=Boltzmann1887 /> and proved that it guaranties the positivity of the entropy production. For chemical kinetics, this condition (as the ''complex balance'' condition) was | Boltzmann introduced the semi-detailed balance condition for collisions in 1887<ref name=Boltzmann1887 /> 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.<ref name="HornJackson1972">''Horn, F., Jackson, R.'' (1972) General mass action kinetics. Arch. Ration. Mech. Anal. 47, 87-116.</ref> | ||
<ref name="HornJackson1972"> | |||
The microscopic backgrounds for the semi-detailed balance were found in the Markov | 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> | ||
<ref> | |||
Under these microscopic assumptions, the semi-detailed balance condition | |||
== Dissipation in systems with semi-detailed balance == | == Dissipation in systems with semi-detailed balance == |