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

Abstract

We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) β− model. The standard quartic interaction is generalized through a coupling constant that decays as 1/r α (α ≥ 0)(with strength characterized by b > 0). In the α → ∞ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For α ≥ 1 the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators N (thus yielding ergodicity), whereas, for 0 ≤ α < 1, it asymptotically decreases as N −κ(α) (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for α large enough, whereas it is well approached by a q-Gaussian, with the index q(α) monotonically decreasing from about 1.5 to 1 (Gaussian) when α increases from zero to close to one. For α 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 ∝ b δ /t γ c with γ > 0 and δ > 0, in such a way that the q = 1 (q > 1) behavior dominates in the lim N →∞ lim t→∞ ordering (lim t→∞ lim N →∞ ordering).

Dynamics and statistics of the Fermi–Pasta–Ulamβ-model with different ranges of particle interactions

“…In the complete absence of harmonic terms the MLE appears to vanish in the thermodynamic limit [9]. Moreover, when harmonic terms are included in the potential, a similar behavior of the MLE is observed [10], which nevertheless tends to saturate to a non-zero value above a characteristic size N .…”

“…Note that there are three ways to introduce long-range interactions in our model: (a) only in the quadratic potential V 2 , (b) only in the quartic potential V 4 and (c) both in V 2 and V 4 . Case (b) was the one studied in [10] and gave the results mentioned above, where a "phase transition" occurs between q-statistics and BG thermostatistics near the value α 2 = 1 that separates the short term α 2 > 1 from the long term 0 ≤ α 2 < 1 interaction range.…”

Inter-occurrence times and universal laws in finance, earthquakes and genomes

“…We will specially review some available results in finance [130,131], earthquakes [132], and biology [133], among others [80,81,134]. We will not include in the present occasion the discussion of long-range-interacting Hamiltonian classical systems [82,135,136,137,138,139,140,141,142,143] which surely deserve a detailed study by themselves (in the present context we do not refer to systems whose elements short-range-interact, but only to those whose elements long-range-interact).…”

Fermi-Pasta-Ulam chains with harmonic and anharmonic long-range interactions

“…Coupled oscillators with power-law LRI were also studied [8][9][10][11][12][13][14][15][16]. The attention was mainly focused on long-range interactions in DNA [8,10], the existence of standing localized solutions like breathers [9], deriving the continuum counterpart of the discrete long-range models, which implies the use of fractional derivatives [11][12][13], weakly chaotic [14] and thermalization [15] properties caused by the long-range character of the interactions and the existence of solitons in a long-range extension of the quartic FPU chain [16].…”