**Detailed Balance and The Entropy Growth**

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 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 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 the entropy growth. For example, in the linear irreversible cycle, the 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 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. 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. In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was explicitly proved later. 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. Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.

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