Boltzmann Complexity: An Emergent Property of the Majorization Partial Order
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2016-09-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/18/10/347 |
| _version_ | 1798038711029465088 |
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| author | William Seitz A. D. Kirwan |
| author_facet | William Seitz A. D. Kirwan |
| author_sort | William Seitz |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T21:44:00Z |
| format | Article |
| id | doaj.art-82c923615b7b4633aee47a6716d746b7 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T21:44:00Z |
| publishDate | 2016-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-82c923615b7b4633aee47a6716d746b72022-12-22T04:01:28ZengMDPI AGEntropy1099-43002016-09-01181034710.3390/e18100347e18100347Boltzmann Complexity: An Emergent Property of the Majorization Partial OrderWilliam Seitz0A. D. Kirwan1Department of Marine Sciences, Texas A&M University at Galveston, P.O. Box 1675, Galveston, TX 77553, USASchool of Marine Science and Policy, University of Delaware, Newark, DE 19716, USABoltzmann 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.http://www.mdpi.com/1099-4300/18/10/347Boltzmann entropyincomparabilitycomplexity |
| spellingShingle | William Seitz A. D. Kirwan Boltzmann Complexity: An Emergent Property of the Majorization Partial Order Entropy Boltzmann entropy incomparability complexity |
| title | Boltzmann Complexity: An Emergent Property of the Majorization Partial Order |
| title_full | Boltzmann Complexity: An Emergent Property of the Majorization Partial Order |
| title_fullStr | Boltzmann Complexity: An Emergent Property of the Majorization Partial Order |
| title_full_unstemmed | Boltzmann Complexity: An Emergent Property of the Majorization Partial Order |
| title_short | Boltzmann Complexity: An Emergent Property of the Majorization Partial Order |
| title_sort | boltzmann complexity an emergent property of the majorization partial order |
| topic | Boltzmann entropy incomparability complexity |
| url | http://www.mdpi.com/1099-4300/18/10/347 |
| work_keys_str_mv | AT williamseitz boltzmanncomplexityanemergentpropertyofthemajorizationpartialorder AT adkirwan boltzmanncomplexityanemergentpropertyofthemajorizationpartialorder |