Thermostatistics in the neighbourhood of the π-mode solution for the Fermi–Pasta–Ulam β system: from weak to strong chaos

Leo, M.; Leo, R. A.; Tempesta, Piergiulio

doi:10.1088/1742-5468/2010/04/p04021

Cited by 18 publications

(27 citation statements)

References 35 publications

(49 reference statements)

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“…Such is the case for the Fermi-Pasta-Ulam β model for finite N , as recently shown [95,96]: Gaussian distributions of momenta are observed for time averages far from its π-orbit (consistently with the CLT), whereas (strict or approximate) q-Gaussians emerge very close to it. See Figure 13.…”

Section: Remarks On Paradigmatic Long-range-interacting Many-body Clasupporting

confidence: 55%

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Tsallis

2011

Entropy

171

184

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The nonadditive entropy S q has been introduced in 1988 focusing on a generalization of Boltzmann-Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that S q has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (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, we exhibit 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.

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Section: Remarks On Paradigmatic Long-range-interacting Many-body Clasupporting

confidence: 55%

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Tsallis

2011

Entropy

171

184

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“…(18) therein]. We also note that our results do not exclude the possibility of Tsallis distributions in nonergodic systems such as those composed of classical longrange interacting particles or in some regions of weak chaos [27][28][29][30] since the ergodicity of the total system is assumed in the present work.…”

Section: Discussionmentioning

confidence: 73%

Tsallis power laws and finite baths with negative heat capacity

Bagci

Oikonomou

2013

Phys. Rev. E

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It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for classical Hamiltonian systems. By using well-known fundamental statistical mechanics expressions, we rigorously show that Tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity, while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium. We comment on the implications of the finite bath for the recent attempts towards a q-generalized central limit theorem.

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“…Following the approach of [34], we verify that the pdfs of chaotic orbits starting very close to the unstable π-mode are well approximated by a q-Gaussian over time intervals of the order of t ≈ 10 6 . These trajectories, however, represent a QSS: Their distributions for longer times deviate from a q-Gaussian and show a tendency to converge to a Gaussian as q → 1.…”

Section: Introductionmentioning

confidence: 72%

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

Antonopoulos

Bountis²,

Basios

2011

Physica A: Statistical Mechanics and its Applications

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We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-$\beta$) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within "small size" phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to $t\approx10^6$) by a $q$-Gaussian ($1

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Thermostatistics in the neighbourhood of the π-mode solution for the Fermi–Pasta–Ulam β system: from weak to strong chaos

Cited by 18 publications

References 35 publications

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Tsallis power laws and finite baths with negative heat capacity

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

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