Detailed Balance - 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:

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 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.