Controlling the range of interactions in the classical inertial ferromagnetic Heisenberg model: analysis of metastable states

Abstract

A numerical analysis of a one-dimensional Hamiltonian system, composed by N classical localized Heisenberg rotators on a ring, is presented. A distance r ij between rotators at sites i and j is introduced, such that the corresponding two-body interaction decays with r ij as a power-law, 1/r α ij (α ≥ 0). The index α controls the range of the interactions, in such a way that one recovers both the fully-coupled (i.e., mean-field limit) and nearest-neighbour-interaction models in the particular limits α = 0 and α → ∞, respectively. The dynamics of the model is investigated for energies U below its critical value (U < U c ), with initial conditions corresponding to zero magnetization. The presence of quasi-stationary states (QSSs), whose durations t QSS increase for increasing values of N , is verified for values of α in the range 0 ≤ α < 1, like the ones found for the similar model of XY rotators. Moreover, for a given energy U , our numerical analysis indicates that t QSS ∼ N γ , where the exponent γ decreases for increasing α in the range 0 ≤ α < 1, and particularly, our results suggest that γ → 0 as α → 1. The growth of t QSS with N could be interpreted as a breakdown of ergodicity, which is shown herein to occur for any value of α in this interval.

Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics

“…Indeed, the correctness of the scaling appearing in Eq. (9) for nonstandard systems, i.e., for those with θ = 0, has been profusely verified for several systems in the literature [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]; one of them is going to be discussed in Section 4 below. Furthermore, it has been shown that such scalings preserve important thermodynamical relations such as the Euler and Gibbs-Duhem [21].…”

The World of Long-Range Interactions: A Bird’s Eye View

“…We note that a non-mean-field version of the model was studied in Ref. 29 in the context of the existence of QSSs. 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 0000 0000 0000 0000 1111 1111 1111 1111 000 000 000 000 000 000 000 000 000 000 000 000 000 000 111 111 111 111 111 111 111 111 111 111 111 111 111 111 00 00 11 11 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 0 1 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111…”

Role of dimensionality in complex networks

“…6 , namely the fact that both the index q and the characteristic degree (or “effective temperature”) κ do not depend from ( α A , d ) in an independent manner but only from the ratio α A / d . This nontrivial fact puts the growing d -dimensional geographically located models that have been introduced here for scale-free networks, on similar footing as long-range-interacting many-body classical Hamiltonian systems such as the inertial XY planar rotators 39 40 41 42 (possibly the generic inertial n -vector rotators as well 43 44 ) and Fermi-Pasta-Ulam oscillators, assuming that the strength of the two-body interaction decreases with distance as 1/( distance ) α . Moreover, as first pointed out generically by Gibbs himself 45 , we have the facts that the BG canonical partition function of these classical systems anomalously diverges with size for 0 ≤ α / d ≤ 1 (long-range interactions, e.g., gravitational and dipole-monopole interactions) and converges for α / d > 1 (short-range interactions, e.g., Lennard-Jones interaction), and the internal energy per particle is, in the thermodynamical limit, constant for short-range interactions whereas it diverges like N 1− α / d for long-range interactions, N being the total number of particles.…”