Rodríguez Antonio, Nobre Fernando D, Tsallis Constantino
GISC, Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros s/n, 28040 Madrid, Spain.
Department of Physics, University of Warwick, Coventry CV4 7AL, UK.
Entropy (Basel). 2019 Jan 4;21(1):31. doi: 10.3390/e21010031.
We numerically study the first-principle dynamics and thermostatistics of a -dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α ≥ 0 ), where the limit α = 0 ( α → ∞ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 ≤ α / d ≤ 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ ∼ N - κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by -Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by -exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.
我们对一维经典惯性海森堡铁磁模型（(d = 1,2,3)）的第一性原理动力学和热统计学进行了数值研究，其相互作用随距离(r_{ij})以(1 / r_{ij}^{\alpha})（(\alpha \geq 0)）衰减，其中(\alpha = 0)（(\alpha \to \infty)）的极限对应于无限范围（最近邻）相互作用，且(\alpha / d > 1)（(0 \leq \alpha / d \leq 1)）的比值表征短程（长程） regime。通过第一性原理分子动力学，我们研究：（i）最大李雅普诺夫指数(\lambda)随系统大小(N)的标度形式为(\lambda \sim N^{-\kappa})，其中(\kappa(\alpha / d))仅取决于(\alpha / d)的比值；（ii）长程 regime (0 \leq \alpha / d \leq 1)中典型情况下的时间平均单粒子角动量概率分布（结果发现由 -高斯分布很好地拟合），以及（iii）长程 regime (0 \leq \alpha / d \leq 1)中典型情况下的时间平均单粒子能量概率分布（结果发现由 -指数分布很好地拟合）。通过李雅普诺夫指数，我们观察到即使在(\alpha / d > 1) regime中，非玻尔兹曼行为也存在一种有趣的、可能与尺寸有关的持续性。我们观察到的概率分布关于(\alpha / d)比值的普适性使得该模型类似于(\alpha - XY)和(\alpha - Fermi - Pasta - Ulam)哈密顿模型以及渐近尺度不变增长网络。
