On the Thermodynamics of Classical Micro-Canonical Systems

Baeten, Maarten; Naudts, Jan

doi:10.3390/e13061186

Cited by 13 publications

(17 citation statements)

References 42 publications

(88 reference statements)

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“…This happens for Ising-type spin or laser systems in the population inverted system, as well is in Hamiltonian systems exhibiting singular points in their DoS that separate regions of ν(E, Z) > 0 with regions with negative-valued ν(E, Z). An instructive ergodic 1D example [17] is the classical pendulum (mass m, length L, gravitational acceleration g) with Hamiltonian…”

Section: B Positive and Negative Absolute Temperatures: An Examplementioning

confidence: 99%

“…The DoS ω can be given analytically in terms of complete elliptic integrals of the first kind (see in Ref. [17] and Fig. 1 therein).…”

Section: B Positive and Negative Absolute Temperatures: An Examplementioning

confidence: 99%

“…The above discussion implicitly assumed that all derivatives exist and are well behaved. This is typically the case for classical Hamiltonian systems with the exception of critical points [13], as also encountered in the pendulum example above [17]. For quantum systems, the problem is generally more subtle since quantum-mechanical energy spectra can be partially or completely discrete, and are typically very sensitive to small perturbations that can break symmetry-related degeneracies.…”

Section: Open Questionsmentioning

confidence: 99%

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Meaning of temperature in different thermostatistical ensembles

Hänggi

Hilbert

Dunkel

2016

Phil. Trans. R. Soc. A.

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Depending on the exact experimental conditions, the thermodynamic properties of physical systems can be related to one or more thermostatistical ensembles. Here, we survey the notion of thermodynamic temperature in different statistical ensembles, focusing in particular on subtleties that arise when ensembles become non-equivalent. The 'mother' of all ensembles, the microcanonical ensemble, uses entropy and internal energy (the most fundamental, dynamically conserved quantity) to derive temperature as a secondary thermodynamic variable. Over the past century, some confusion has been caused by the fact that several competing microcanonical entropy definitions are used in the literature, most commonly the volume and surface entropies introduced by Gibbs. It can be proved, however, that only the volume entropy satisfies exactly the traditional form of the laws of thermodynamics for a broad class of physical systems, including all standard classical Hamiltonian systems, regardless of their size. This mathematically rigorous fact implies that negative 'absolute' temperatures and Carnot efficiencies more than 1 are not achievable within a standard thermodynamical framework. As an important offspring of microcanonical thermostatistics, we shall briefly consider the canonical ensemble and comment on the validity of the Boltzmann weight factor. We conclude by addressing open mathematical problems that arise for systems with discrete energy spectra.

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Section: B Positive and Negative Absolute Temperatures: An Examplementioning

confidence: 99%

“…The DoS ω can be given analytically in terms of complete elliptic integrals of the first kind (see in Ref. [17] and Fig. 1 therein).…”

Section: B Positive and Negative Absolute Temperatures: An Examplementioning

confidence: 99%

Section: Open Questionsmentioning

confidence: 99%

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Meaning of temperature in different thermostatistical ensembles

Hänggi

Hilbert

Dunkel

2016

Phil. Trans. R. Soc. A.

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“…Another paper by Castelló et al [4] presents a study and a comparison of the use of different information-theoretic measures for polygonal mesh simplification by applying generalized measures from Information Theory such as Havrda-Charvát-Tsallis entropy and mutual information. Baeten and Naudts [5] present two arguments why the thermodynamic entropy of non-extensive systems involves Renyi's entropy function rather than that of Tsallis. The first argument is that the temperature of the configurational subsystem of a mono-atomic gas is equal to that of the kinetic subsystem, and the second one is that the instability of the pendulum, which occurs for energies close to the rotation threshold, is correctly reproduced.…”

mentioning

confidence: 99%

Special Issue: Tsallis Entropy

Anastasiadis

2012

Entropy

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One of the crucial properties of the Boltzmann-Gibbs entropy in the context of classical thermodynamics is extensivity, namely proportionality with the number of elements of the system. The Boltzmann-Gibbs entropy satisfies this prescription if the subsystems are statistically (quasi-) independent, or typically if the correlations within the system are essentially local. In such cases the energy of the system is typically extensive and the entropy is additive. In general, however, the situation is not of this type and correlations may be far from negligible at all scales. Tsallis in 1988 introduced an entropic expression characterized by an index q which leads to a non-extensive statistics. Tsallis entropy, Sq, is the basis of the so called non-extensive statistical mechanics, which generalizes the Boltzmann-Gibbs theory. Tsallis statistics have found applications in a wide range of phenomena in diverse disciplines such as physics, chemistry, biology, medicine, economics, geophysics, etc. The focus of this special issue of Entropy was to solicit contributions that apply Tsallis entropy in various scientific fields.This special issue consists of nine regular papers, covering various aspects and applications of Tsallis non-additive entropy, and an invited review paper written by Tsallis [1]. In this review, the following aspects of Tsallis entropy are discussed: (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results are presented.In their paper, Zhang and Wu [2] propose a global multi-level thresholding method for image segmentation by applying the Tsallis entropy, as a general information theory entropy formalism, and using an artificial bee colony algorithm. They demonstrate that Tsallis entropy is superior to traditional maximum entropy thresholding, based on Shannon entropy, and that the artificial bee colony is more rapid than either genetic algorithm or particle swarm optimization. Vila, Bardera, Feixas and Sbert [3] investigate the application of three different Tsallis-based generalizations of mutual information to OPEN ACCESS

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“…Despite its wide spread applications, the Shannon entropy has a limitation to be confined to extensive systems [ 4 , 6 ]. These systems obey the Boltzmann–Gibbs (B-G) function [ 10 ] and are usually assumed to be in equilibrium from the thermodynamics point of view. Thermodynamics assumes that these systems will depend on their initial state whenever they undergo transformation.…”

Section: Introductionmentioning

confidence: 99%

Tsallis Entropy for Assessing Spatial Uncertainty Associated with Mean Annual Runoff of Quaternary Catchments of the Middle Vaal Basin in South Africa

Ilunga

2020

Entropy

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This study assesses mainly the uncertainty of the mean annual runoff (MAR) for quaternary catchments (QCs) considered as metastable nonextensive systems (from Tsalllis entropy) in the Middle Vaal catchment. The study is applied to the surface water resources (WR) of the South Africa 1990 (WR90), 2005 (WR2005) and 2012 (WR2012) data sets. The q-information index (from the Tsalllis entropy) is used here as a deviation indicator for the spatial evolution of uncertainty for the different QCs, using the Shannon entropy as a baseline. It enables the determination of a (virtual) convergence point, zone of positive and negative uncertainty deviation, zone of null deviation and chaotic zone for each data set. Such a determination is not possible on the basis of the Shannon entropy alone as a measure for the MAR uncertainty of QCs, i.e., when they are viewed as extensive systems. Finally, the spatial distributions for the zones of the q-uncertainty deviation (gain or loss in information) of the MAR are derived and lead to iso q-uncertainty deviation maps.

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On the Thermodynamics of Classical Micro-Canonical Systems

Cited by 13 publications

References 42 publications

Meaning of temperature in different thermostatistical ensembles

Meaning of temperature in different thermostatistical ensembles

Special Issue: Tsallis Entropy

Tsallis Entropy for Assessing Spatial Uncertainty Associated with Mean Annual Runoff of Quaternary Catchments of the Middle Vaal Basin in South Africa

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