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Boltzmann Complexity: An Emergent Property of the Majorization Partial Order

by William Seitz 1,† and A. D. Kirwan 2,*,†
Department of Marine Sciences, Texas A&M University at Galveston, P.O. Box 1675, Galveston, TX 77553, USA
School of Marine Science and Policy, University of Delaware, Newark, DE 19716, USA
Author to whom correspondence should be addressed.
Academic Editor: J.A. Tenreiro Machado
Entropy 2016, 18(10), 347;
Received: 31 July 2016 / Revised: 18 September 2016 / Accepted: 19 September 2016 / Published: 23 September 2016
(This article belongs to the Section Complexity)
Boltzmann 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 N = 50 microstates where there are 204,226 Boltzmann macrostates. We note that, unlike mixedness, neither C nor S uniquely characterizes macrostates. Plots of C vs. S are shown. The results are surprising and support the authors’ earlier suggestion that C be regarded as the complexity of the Boltzmann states. From this we propose that complexity may generally arise from incomparability in other systems as well. View Full-Text
Keywords: Boltzmann entropy; incomparability; complexity Boltzmann entropy; incomparability; complexity
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Seitz, W.; Kirwan, A.D. Boltzmann Complexity: An Emergent Property of the Majorization Partial Order. Entropy 2016, 18, 347.

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