Doppelgänger entropies

KirwanJr., A. D.; Seitz, W. A.

doi:10.1007/s10910-016-0658-z

Cited by 5 publications

(4 citation statements)

References 7 publications

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“…We have previously observed [7] that there is a high degree of degeneracy among the Boltzmann states with respect to S [λ] , so that while S [λ] is a scalar, the Boltzmann states are only partially ordered by entropy. In other words, states that have the same value of S [λ] cannot be uniquely compared or ordered by entropy.…”

Section: The Mixing Partial Ordermentioning

confidence: 99%

Boltzmann Complexity: An Emergent Property of the Majorization Partial Order

Seitz

Kirwan

2016

Entropy

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Abstract: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.

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Section: The Mixing Partial Ordermentioning

confidence: 99%

Boltzmann Complexity: An Emergent Property of the Majorization Partial Order

Seitz

Kirwan

2016

Entropy

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“…Here, the partitions

and

have the same permutation number, 210. Moreover, for large N most

are degenerate [ 13 ]. This turns out to be a crucial aspect of our analysis that was not recognized by Gibbs [ 18 ]), Ruch [ 19 ], or Lieb and Yngvason [ 23 ].…”

Section: Mixed-up-ness and The Mathematics Of Integer Partitionsmentioning

confidence: 99%

“…Figure 3 shows an extremely thin slice of Figure 2 between

. The noteworthy feature of this figure is the vertical stripes indicating the presence of degenerate

or doppelgängers [ 13 ]. This suggests that the thickness of Figure 2 is due to a large number of degenerate permutation numbers.…”

Section: The Law Of Mixed-up-ness For Integer Partitionsmentioning

confidence: 99%

“…First, as preceptively suggested by [ 12 ], entropy may not be a natural gauge of ignorance. Second, the Boltzmann entropy, the most commonly used version of entropy in non-thermodynamic applications of the SL, has numerous degenerate or isentropic states [ 13 ]. In traditional thermodynamics, there are state variables such as temperature and pressure, etc,.…”

Section: Introductionmentioning

confidence: 99%

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Mixed-Up-Ness or Entropy?

Seitz

Kirwan

2022

Entropy

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Mixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect “order” to states of total “disorder”. Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems.

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The Rise and Fall of Thermodynamic Complexity and the Arrow of Time

KirwanJr.

Seitz

2017

Advances in Nonlinear Geosciences

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Doppelgänger entropies

Cited by 5 publications

References 7 publications

Boltzmann Complexity: An Emergent Property of the Majorization Partial Order

Boltzmann Complexity: An Emergent Property of the Majorization Partial Order

Mixed-Up-Ness or Entropy?

The Rise and Fall of Thermodynamic Complexity and the Arrow of Time

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