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==Detailed balance and the entropy growth== | ==Detailed balance and the entropy growth== | ||
For many systems | 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. | ||
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with detailed balance. | |||
Nevertheless, the principle of detailed balance is not necessary for | 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 | 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. 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. | ||
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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>L. | |||
In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was | |||
<ref>[http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-379.pdf | |||
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> | |||
Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the | |||
==Wegscheider's conditions for the generalized mass action law== | ==Wegscheider's conditions for the generalized mass action law== |