Boltzmann Complexity: An Emergent Property of the Majorization Partial Order
AbstractBoltzmann macrostates, which are in 1:1 correspondence with the partitions of integers, are investigated. Integer partitions, unlike entropy, uniquely characterize Boltzmann states, but their use has been limited. Integer partitions are well known to be partially ordered by majorization. It is less well known that this partial order is fundamentally equivalent to the “mixedness” of the set of microstates that comprise each macrostate. Thus, integer partitions represent the fundamental property of the mixing character of Boltzmann states. The standard definition of incomparability in partial orders is applied to the individual Boltzmann macrostates to determine the number of other macrostates with which it is incomparable. We apply this definition to each partition (or macrostate) and calculate the number C with which that partition is incomparable. We show that the value of C complements the value of the Boltzmann entropy, S, obtained in the usual way. Results for C and S are obtained for Boltzmann states comprised of up to
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Seitz, W.; Kirwan, A.D. Boltzmann Complexity: An Emergent Property of the Majorization Partial Order. Entropy 2016, 18, 347.
Seitz W, Kirwan AD. Boltzmann Complexity: An Emergent Property of the Majorization Partial Order. Entropy. 2016; 18(10):347.Chicago/Turabian Style
Seitz, William; Kirwan, A. D. 2016. "Boltzmann Complexity: An Emergent Property of the Majorization Partial Order." Entropy 18, no. 10: 347.
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