d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

Abstract

We numerically study the first-principle dynamics and thermostatistics of a d-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 q-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 q-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.

Beyond Boltzmann–Gibbs–Shannon in Physics and Elsewhere

“…We see here that both the thermal and the geometrical model exhibit the interesting scaling. The same happens for the -Heisenberg inertial ferromagnet [ 100 ], the -Fermi–Pasta–Ulam model [ 101 , 102 , 103 , 104 ], and other asymptotically scale-free networks [ 17 , 18 ], thus exhibiting the ubiquity of this grounding scaling law.…”

“…Let us end this section by mentioning other selected entropic applications beyond BG in physics: long-range interacting many-body classical Hamiltonian systems (XY model [ 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 ], Heisenberg model [ 98 , 99 , 100 ], Fermi–Pasta–Ulam (FPU) model [ 101 , 102 , 103 , 104 ]) (see [ 105 , 106 ] for earlier related approaches of the original FPU model and also [ 107 ], where the existence of non-Maxwellian compact-support momenta distributions are detected for special initial conditions); quantum-entangled low-dimensional Hamiltonian systems [ 108 , 109 , 110 ]; plasma physics [ 111 , 112 , 113 , 114 , 115 ]; turbulence [ 87 , 116 ]; astrophysics, cosmology, and black holes [ 89 , 117 , 118 , 119 , 120 , 121 , 122 ]; nonlinear dynamical systems [ 123 , 124 , 125 , 126 , 127 , 128 ]; nonlinear quantum mechanics [ 129 , 130 , 131 , 132 ]; anomalous diffusion, type II superconductors, and repulsive short-range interacting systems with overdamping […”

Nonlinear Fokker–Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

“…where the constant D depends only on the properties of the inter-particle force. Using (6), the equation of motion of a test particle becomes…”

“…The discovery of the connection between the polytropic distributions and the S q entropies suggested that the S q -thermostatistics might be relevant for the study of the thermodynamical properties of systems with long-range interactions. Subsequent research generated a growing body of evidence attesting to that (see [1,2,6] and references therein). Indeed, the application of the S q -thermostatistics to the study of the thermodynamics of various systems with long-range interactions has been, over the years, one of the main venues of research related to the S q entropies.…”

Nonadditive Entropies and Complex Systems

“…In the present Special Issue, several approaches have been advanced along those lines. Following the order of appearance, Rodriguez et al [ 29 ] have focused on a classical d -dimensional many-body Hamiltonian with long-range interactions, which numerically appears to exhibit q -Gaussian distributions of velocities, q -exponential distribution of energies, and vanishing maximal Lyapunov exponent in the infinitely-sized limit. Curado et al [ 30 ] focus on a close relationship between the entropy and systems exhibiting power-law frequency of events and behaving similarly to self-organised criticality, like earthquakes, avalanches, and forest fires.…”