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      Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

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          Abstract

          We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) \(\beta-\) model. The standard quartic interaction is generalized through a coupling constant that decays as \(1/r^\alpha\) (\(\alpha \ge 0\))(with strength characterized by \(b>0\)). In the \(\alpha \to\infty\) limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For \(\alpha \geq 1\) the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators \(N\) (thus yielding ergodicity), whereas, for \(0 \le \alpha <1\), it asymptotically decreases as \(N^{- \kappa(\alpha)}\) (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for \(\alpha\) large enough, whereas it is well approached by a \(q\)-Gaussian, with the index \(q(\alpha)\) monotonically decreasing from about 1.5 to 1 (Gaussian) when \(\alpha\) increases from zero to close to one. For \(\alpha\) small enough, the whole picture is consistent with a crossover at time \(t_c\) from \(q\)-statistics to Boltzmann-Gibbs (BG) thermostatistics. More precisely, we construct a "phase diagram" for the system in which this crossover occurs through a frontier of the form \(1/N \propto b^\delta /t_c^\gamma\) with \(\gamma >0\) and \(\delta >0\), in such a way that the \(q=1\) (\(q>1\)) behavior dominates in the \(\lim_{N \to\infty} \lim_{t \to\infty}\) ordering (\(\lim_{t \to\infty} \lim_{N \to\infty}\) ordering).

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          Breakdown of Exponential Sensitivity to Initial Conditions: Role of the Range of Interactions

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            An introduction to nonadditive entropies and a thermostatistical approach of inanimate and living matter

            The possible distinction between inanimate and living matter has been of interest to humanity since thousands of years. Clearly, such a rich question can not be answered in a single manner, and a plethora of approaches naturally do exist. However, during the last two decades, a new standpoint, of thermostatistical nature, has emerged. It is related to the proposal of nonadditive entropies in 1988, in order to generalise the celebrated Boltzmann-Gibbs additive functional, basis of standard statistical mechanics. Such entropies have found deep fundamental interest and uncountable applications in natural, artificial and social systems. In some sense, this perspective represents an epistemological paradigm shift. These entropies crucially concern complex systems, in particular those whose microscopic dynamics violate ergodicity. Among those, living matter and other living-like systems play a central role. We briefly review here this approach, and present some of its predictions, verifications and applications.
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              Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

              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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                Author and article information

                Journal
                14 May 2014
                Article
                10.1209/0295-5075/108/40006
                1405.3528
                8c2920a0-f3ac-4457-9361-e0bbb4de07d2

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                8 pages, 5 fugures
                nlin.CD cond-mat.stat-mech

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