# Detailed Balance - Dissipation in Systems With Semi-detailed Balance

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 is the chemical potential and is the Helmholtz free energy. The exponential term is called the Boltzmann factor and the multiplier is the kinetic factor. Let us count the direct and reverse reaction in the kinetic equation separately:

An auxiliary function of one variable is convenient for the representation of dissipation for the mass action law

This function may be considered as the sum of the reaction rates for deformed input stoichiometric coefficients . For it is just the sum of the reaction rates. The function is convex because .

Direct calculation gives that according to the kinetic equations

This is the general dissipation formula for the generalized mass action law.

Convexity of gives the sufficient and necessary conditions for the proper dissipation inequality:

The semi-detailed balance condition can be transformed into identity . Therefore, for the systems with semi-detailed balance .

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