Classical gas in nonextensive optimal Lagrange multipliers formalism

Abe, Sumiyoshi; Martı́nez, S.; Pennini, F.; Plastino, A.

doi:10.1016/s0375-9601(00)00780-5

Cited by 45 publications

(28 citation statements)

References 22 publications

(56 reference statements)

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“…This nonextensive index coincides with that of [23], the index deduced on the basis of additive energy.…”

supporting

confidence: 80%

Thermoequilibrium statistics for a finite system with energy nonextensivity

Zheng

2011

Chin. Sci. Bull.

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A first-principles derivation is presented of canonical distributions for a finite thermostat taking into account nonextensive energy.Parameterizing this energy by , we derive an explicit form for the distribution functions by regulating , and then explore the nontrivial relationship between these functions and energy nonextensivity, as well other system parameters such as system size. A variational entropy function is also derived from these distribution functions.. finite system, energy nonextensivity, first-principles method, canonical distribution function Citation:Zheng L, Li W. Thermoequilibrium statistics for a finite system with energy nonextensivity. Chinese Sci Bull, 2011Bull, , 56: 3666-3670, doi: 10.1007 Studies into the family of small or finite-size systems reveal that these systems show complex statistical characteristics very different from large systems, and expose failings in our understanding of thermostatistics for finite systems and their nonequilibrium dynamics [1][2][3][4]. One of these characteristics is nonextensivity (refer to [2] and references therein for examples), which means that macroscopic quantities of such systems may not be proportional to system size. A thermodynamic quantity of a nonextensive system is called nonadditive if this quantity is not the sum of that of its subsystems. Nonextensivity might arise from surface effects or interactions between subsystems. Thus, extensive energy and entropy may become inappropriate in treating finite systems. There has already been much discussion on the properties of finite systems that has raised many questions and controversies [5][6][7]. To describe the behaviors of these systems, one point worth noting is that the Boltzmann-Gibbs statistical mechanics, which is based on the concept of extensive entropy, is not applicable [1,8,9]. Consequently, it is questionable to assume an exponential distribution for finite systems. Therefore, one of the first things to work out for a statistical description of finite systems is to find the appropriate probability distribution. Some published results obtained by first principles (see [8] for example) have shown that a small system, in equilibrium with a finite reservoir, may follow a q-exponential distribution of nonextensive statistical mechanics (NSM), as proposed by Tsallis [9]. This NSM has been used widely in many areas, and considered effective in solving many physical problems [10][11][12][13]. Results from mathematical proofs are able to demonstrate a connection between the system's finiteness and nonextensivity of the theory. However, most proofs have relied on the additivity of energy. Clearly, for large systems, this assumption is acceptable and provides a helpful approximation in obtaining the statistics in the thermodynamic limit. The assumption is, however, questionable when establishing statistics for finite systems. Related problems arising from this additive energy assumption can be found in [14][15][16][17].Work in [18] represents a first attempt to build statistics for fin...

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“…This nonextensive index coincides with that of [23], the index deduced on the basis of additive energy.…”

supporting

confidence: 80%

Thermoequilibrium statistics for a finite system with energy nonextensivity

Zheng

2011

Chin. Sci. Bull.

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“…Numerical complications often ensue, which has encouraged the development of different, alternative lines of NET research. See, for instance (a by no means exhaustive list), [16,17,18,19,20,21,22] and references therein.…”

Section: Reasons For Using Weighted Mean Valuesmentioning

confidence: 99%

Equivalence of the four versions of Tsallis’s statistics

Ferri¹,

Martı́nez²,

Plastino³

2005

J. Stat. Mech.

106

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In spite of its undeniable success, there are still open questions regarding Tsallis' non-extensive statistical formalism, whose founding stone was laid in 1988 in JSTAT. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes' MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) 1988 Tsallis-original (TO), (ii) Curado-Tsallis (CT), (iii) Tsallis-MendesPlastino (TMP), and (iv) the same as (iii), but using centered operators as constraints (OLM).The 1988 was promptly abandoned and replaced, mostly with versions ii) and iii). We will here (a) show that the 1988 is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyze some consequences that emerge from these four choices.PACS: 05.30.Jp

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“…Moreover, as mentioned earlier, in recent studies, the entropic index is related to the number of particles of the system [29,30]. Therefore it is expected that q, the entropic index, could be related to the density, which will be the subject of another study in near future and the variation of s near T N I , nematic-isotropic phase transition temperature, could be explained hopefully by GMST.…”

Section: Discussionmentioning

confidence: 74%

“…In Table 1, the variations of the density of PAA with respect to temperature are given. In addition, in the recent studies, the entropic index is related to the number of particles of the system [29,30]. Then a better aggreement with the experimental data could be obtained, which will be the subject of a forthcoming study.…”

Section: Generalization Of the Maier-saupe Theorymentioning

confidence: 92%

Generalization of the Maier–Saupe theory of the nematics within Tsallis thermostatistics

Kayacan

Büyükkılıç

Demirhan

2001

Physica A: Statistical Mechanics and its Applications

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In this study, one of the mean-field theories in nematics, the Maier-Saupe theory (MST), is generalized within Tsallis Thermostatistics (TT). The variation of the order parameter versus temperature has been investigated and compared with experimental data for PAA (p-azoxyanisole). So far we believe that this is the first attempt of the application of TT in liquid crystals. It is well known that MST fails to explain the experimental data for some of the nematics, one of which is PAA. However generalized MST (GMST) is able to account for the experimental data for a long range of temperatures. Also in this study, the effect of nonextensivity is shown for various values of the entropic index.

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Classical gas in nonextensive optimal Lagrange multipliers formalism

Abstract: Based on the prescription termed the optimal Lagrange multipliers formalism for extremizing the Tsallis entropy indexed by q, it is shown that key aspects of the treatment of the ideal gas problem are identical in both the nonextensive q = 1 and extensive q = 1 cases.

Cited by 45 publications

References 22 publications

Thermoequilibrium statistics for a finite system with energy nonextensivity

Thermoequilibrium statistics for a finite system with energy nonextensivity

Equivalence of the four versions of Tsallis’s statistics

Generalization of the Maier–Saupe theory of the nematics within Tsallis thermostatistics

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