Macroscopic thermodynamics of equilibrium characterized by power law canonical distributions

Abe, Sumiyoshi; Rajagopal, A. K.

doi:10.1209/epl/i2001-00373-4

Cited by 28 publications

(29 citation statements)

References 15 publications

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“…The convolution of these distributions was shown to converge to the limiting Lévy distribution [30], according to the Lévy-Gnedenko generalized central-limit theorem [31]. Furthermore, the interrelationship between microscopic statistical and macroscopic thermal equilibria for such power-law systems was shown to be a natural extension of the corresponding Boltzmann-Gibbs theory [32].…”

Section: Applications To Special Functions Used In Statisticsmentioning

confidence: 99%

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

Aksenov

Savageau

Becher

et al. 2003

Computer Physics Communications

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We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.

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Section: Applications To Special Functions Used In Statisticsmentioning

confidence: 99%

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

Aksenov

Savageau

Becher

et al. 2003

Computer Physics Communications

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“…This is in fact the original and most frequently invoked motivation of the formulation and, consequently, constitutes its genuine source of validation. Our results clarify whether Tsallis's statistics is relevant to an aspect that-in spite of its essential role in thermodynamics-has been scarcely treated in the profuse literature on Tsallis's theory [16], namely, thermal equilibrium between interacting systems. Assume to have a system which, due to its nonextensive nature, exhibits an energy distribution of the form of Eq.…”

mentioning

confidence: 61%

“…The value of the parameters in the thermometer distribution predicted by Tsallis's formalism, however, results to be wrong. Abe and Rajagopal [16] have shown that a formal extension of equilibrium conditions for two systems in thermal contact yields, in Tsallis's theory, β q = β q ′ , where q and q ′ are the nonextensivity indices corresponding to the two systems [17]. Our results show that this relation is generally not satisfied.…”

mentioning

confidence: 69%

Thermal measurements of stationary nonequilibrium systems: a test for generalized thermostatistics

Zanette

Montemurro

2003

Physics Letters A

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We show that a gas thermometer in contact with a stationary classical system out of thermal (Boltzmann) equilibrium evolves, under very general conditions, towards a state characterized by a Lévy velocity distribution. Our approach is based on a kinetic-like equation that applies to a wide class of models for the systemthermometer interaction. The results clarify the role of non-exponential energy distributions as possible generalizations of the Boltzmann distribution for systems where the usual formulation of thermostatistics may not apply. In particular, they show that the power-law distributions derived from Tsallis's nonextensive formalism are irrelevant to the stationary state of the thermometer, thus failing to give a consistent description of the system-thermometer equilibrium. We point out the need of a generalized thermostatistical formulation able to give a unified frame to Lévy and Maxwell distributions.

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“…We use the relation (12) in finding the most probable partition of energy in conformity with the principle of equal a priori probability. We find that the two attain the same "temperature" β β Consider as before the case when S 1 is a small system immersed in the very large heat bath (reservoir) S 2 .…”

Section: -Amentioning

confidence: 99%

“…(2') This is the new factor in place of the Boltzmann factor [12]. As before, this derivation is based entirely on macroscopic considerations and probability concepts.…”

Section: -Amentioning

confidence: 99%

Statistical mechanical foundations of power-law distributions

Rajagopal¹,

Abe²

2004

Physica D: Nonlinear Phenomena

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The foundations of the Boltzmann-Gibbs (BG) distributions for describing equilibrium statistical mechanics of systems are examined. Broadly, they fall into: (i) probabilistic approaches based on the principle of equal a priori probability (counting technique and method of steepest descents), law of large numbers, or the state density considerations and (ii) a variational scheme -maximum entropy principle (due to Gibbs and Jaynes) subject to certain constraints. A minimum set of requirements on each of these methods are briefly pointed out: in the first approach, the function space and the counting algorithm while in the second, "additivity" property of the entropy with respect to the composition of statistically independent systems. In the past few decades, a large number of systems, which are not necessarily in thermodynamic equilibrium (such as glasses, for example), have been found to display power-law distributions, which are not describable by the above-mentioned methods. In this paper, parallels to all the inquiries underlying the BG program described above are given in a brief form. In particular, in the probabilistic derivations, one employs a different function space and one gives up "additivity" in the variational scheme with a different form for the entropy. The requirement of stability makes the entropy choice to be that proposed by Tsallis. From this a generalized thermodynamics description of the system in a quasi-equilibrium state is derived. A brief account of a unified consistent formalism associated with systems obeying power-law distributions precursor to the exponential form associated with thermodynamic equilibrium of systems is presented here.

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Macroscopic thermodynamics of equilibrium characterized by power law canonical distributions

Cited by 28 publications

References 15 publications

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

Thermal measurements of stationary nonequilibrium systems: a test for generalized thermostatistics

Statistical mechanical foundations of power-law distributions

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