I. Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status

Tsallis, Constantino

doi:10.1007/3-540-40919-x_1

Cited by 159 publications

(183 citation statements)

References 192 publications

(257 reference statements)

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“…This result was found employing the unnormalized expectation values of Curado and Tsallis (see Ref. [5]). Analogously, if one uses the normalized expectation values of Ref.…”

Section: Introductionmentioning

confidence: 57%

“…Tsallis' thermostatistics [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] is by now known to offer a nonextensive generalization of traditional BoltzmannGibbs statistical mechanics. A key ingredient in this formalism is the introduction of a particular definition of expectation value termed the normalized q-expectation value [4].…”

Section: Introductionmentioning

confidence: 99%

“…A quite interesting fact is to be emphasized concerning the entropy constant k T , which is usually identified simply with Boltzmann's k B in the literature: the only certified fact one can be sure of is "k T → k B for q → 1" [5], which entails that there is room to choose k T = k(q) in a suitable way. It is seen [1] that if one chooses (with Z q the generalized partition function)…”

Section: Introductionmentioning

confidence: 99%

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Classical gas in nonextensive optimal Lagrange multipliers formalism

et al. 2001

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“…This result was found employing the unnormalized expectation values of Curado and Tsallis (see Ref. [5]). Analogously, if one uses the normalized expectation values of Ref.…”

Section: Introductionmentioning

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Classical gas in nonextensive optimal Lagrange multipliers formalism

et al. 2001

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“…Nonextensive behavior has been found in many physical systems, and may be produced by a variety of physical ingredients, like long-range interactions and long-time memory, among others. Clear limitations have been identified on standard formalisms, e.g., the powerful Boltzmann-Gibbs statistical mechanics was shown to fail in providing a satisfactory description of a wide variety of theoretical models and experimental realizations [11,12,13], in such a way that a completely new physics is emerging for dealing with such systems, in the recent years.…”

Section: Introductionmentioning

confidence: 99%

Competing long-range bonds and site dilution in the one-dimensional bond-percolation problem

et al. 2003

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The long-range bond-percolation problem, on a linear chain (d = 1), in the presence of diluted sites (with an occupancy probability p s for an active site) is studied by means of a Monte Carlo simulation. The occupancy probability for a bond between two active sites i and j, separated by a distance rij is given by pij = p/r α ij , where p represents the usual occupancy probability between nearest-neighbor sites. This model allows one to analyse the competition between long-range bonds (which enhance percolation) and diluted sites (which weaken percolation). By varying the parameter α (α ≥ 0), one may find a crossover between a nonextensive regime and an extensive regime; in particular, the cases α = 0 and α → ∞ represent, respectively, two well-known limits, namely, the mean-field (infinite-range bonds) and first-neighbor-bond limits. The percolation order parameter, P ∞ , was investigated numerically for different values of α and p s . Two characteristic values of α were found, which depend on the site-occupancy probability ps, namely, α1(ps) and α2(ps) (α2(ps) > α1(ps) ≥ 0). The parameter P ∞ equals unit, ∀p > 0, for 0 ≤ α ≤ α 1 (p s ) and vanishes, ∀p < 1, for α > α 2 (p s ). In the interval α1(ps) < α < α2(ps), the parameter P∞ displays a familiar behavior, i.e., 0 for p ≤ pc(α) and finite otherwise. It is shown that both α 1 (p s ) and α 2 (p s ) decrease with the inclusion of diluted sites. For a fixed p s , it is shown that a convenient variable, p * ≡ p * (p, α, N ), may be defined in such a way that plots of P ∞ versus p * collapse for different sizes and values of α in the nonextensive regime. I IntroductionPercolation [1, 2] represents one of the most interesting problems in statistical physics. Many physical situations depend essentially on the geometric properties of random clusters, e.g., random resistor networks, forest fires, and the flow of fluids in porous media. The study of such clusters, and, in particular, the existence of an infinite connected cluster which spans the system in question, is the subject of percolation theory. The percolation order parameter, P ∞ , is defined as the fraction of sites of the system that belong to the infinite cluster. Obviously, P ∞ attains its maximum value (P ∞ = 1) when all the sites of the system appear inside the infinite cluster, whereas P ∞ = 0 below a certain threshold, when it is not possible to produce an infinite cluster. For the bond-percolation problem, where p represents the usual occupancy probability between nearest-neighbor sites, the one-dimensional problem is trivial: one has that P ∞ = 1 for p = 1 and P ∞ = 0 ∀p < 1. Also, if one introduces a dilution of sites in the system (with an occupancy probability p s for an active site), one gets that P ∞ = 0 (even for p = 1) ∀p s < 1. However, this trivial situation changes completely if one introduces long-range effects, i.e., bond-occupancy probabilities associated with two active sites further than nearest-neighboring ones. An interesting competition may occur in such a case between the dilution...

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“…For instance, metastable phases of some long-range interacting systems, where the Lyapunov exponent vanishes in the large N limit, exhibit breakdown of ergodicity, anomalous diffusion, and non-Maxwell velocity distributions [1]. An extension of standard statistical mechanics is required for the theoretical explanation of such phenomena [2].…”

Section: Introductionmentioning

confidence: 99%

Theoretical estimates for the largest Lyapunov exponent of many-particle systems

Vallejos

Anteneodo

2002

Phys. Rev. E

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The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N -particle Hamiltonian system, with a smooth Hamiltonian of the type p 2 +V(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution. In this way a stochastic process V(t) is defined, and the evolution equation for tangent vectors can now be seen as a stochastic differential equation. An equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cummulant expansion. We show that if cummulants higher than the second one are discarded, the Lyapunov exponent can be obtained by diagonalizing a smalldimension matrix, which, in some cases, can be as small as 3×3. In all cases the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process. We discuss the connection between our approach and an alternative theory, the so-called geometric method.

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I. Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status

Cited by 159 publications

References 192 publications

Classical gas in nonextensive optimal Lagrange multipliers formalism

Classical gas in nonextensive optimal Lagrange multipliers formalism

Competing long-range bonds and site dilution in the one-dimensional bond-percolation problem

Theoretical estimates for the largest Lyapunov exponent of many-particle systems

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