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.
Read more about this topic: Detailed Balance
Famous quotes containing the words detailed, balance, entropy and/or growth:
“... every event has had its cause, and nothing, not the least wind that blows, is accident or causeless. To understand what happens now one must find the cause, which may be very long ago in its beginning, but is surely there, and therefore a knowledge of history as detailed as possible is essential if we are to comprehend the past and be prepared for the future.”
—Pearl S. Buck (1892–1973)
“Not in vain is Ireland pouring itself all over the earth. Divine Providence has a mission for her children to fulfill; though a mission unrecognized by political economists. There is ever a moral balance preserved in the universe, like the vibrations of the pendulum. The Irish, with their glowing hearts and reverent credulity, are needed in this cold age of intellect and skepticism.”
—Lydia M. Child (1802–1880)
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)
“The windy springs and the blazing summers, one after another, had enriched and mellowed that flat tableland; all the human effort that had gone into it was coming back in long, sweeping lines of fertility. The changes seemed beautiful and harmonious to me; it was like watching the growth of a great man or of a great idea. I recognized every tree and sandbank and rugged draw. I found that I remembered the conformation of the land as one remembers the modelling of human faces.”
—Willa Cather (1873–1947)