Long-range interacting classical systems: universality in mixing weakening

Through molecular dynamics, we study the d = 2, 3 classical model of N coupled rotators (inertial XY model) assuming a coupling constant which decays with distance as r ij (α ≥ 0). The total energy < H > is asymptotically ∝ NÑ with Ñ ≡ [N − (α/d)]/[1 − α/d], hence the model is thermodynamically extensive if α/d > 1 and nonextensive otherwise. We numerically show that, for energies above some threshold, the (appropriately scaled) maximum Lyapunov exponent is ∝ N where κ is an universal (one and the same for d = 1, 2 and 3, and all energies) function of α/d, which monotonically decreases from 1/3 to zero when α/d increases from 0 to 1, and identically vanishes above 1. These features are consistent with the nonextensive statistical mechanics scenario, where thermodynamic extensivity is associated with exponential mixing in phase space, whereas weaker (possibly power-law in the present case) mixing emerges at the N → ∞ limit whenever nonextensivity is observed. PACS: 05.20.-y, 05.70.Ce, 05.10.-a Corresponding author; [email protected] [email protected] [email protected] [email protected] In the last few years there has been a noticeable interest in the study of the thermodynamics and statistical mechanics of anomalous systems exhibiting nonextensivity [1, 2]. A system is extensive if its energy and entropy, as functions of intensive internal parameters (e.g. temperature), grow linearly with the size of the system (i.e. with N , the number of its microscopic components). Nonextensivity can be brought into scene by long-range interactions. From the static point of view, nonextensivity means that thermodynamic quantities like the internal or the free energies per particle are not constant with N , but rather diverge in the thermodynamic limit N → ∞. From the dynamical point of view, one might think that long-range interactions induce persistent spatial and temporal correlations leading to the breaking of standard mixing and ergodicity properties, hence to a possible violation of the usual Boltzmann-Gibbs (BG) statistics. Classical Hamiltonian systems with many degrees of freedom are paradigmatic in the discussion of the foundations of equilibrium statistical mechanics. The maximal Lyapunov exponent (MLE) is a well established indicator of chaos. If it is positive, the system will generically be strongly chaotic and will satisfy the standard ergodic hypothesis attained through exponentially quick mixing. If it is instead zero non ergodicity could emerge, giving origin to less-than-exponential (typically power-law) mixing, hence to anomalous thermostatistical behavior. It is then interesting to study maximal Lyapunov exponents in systems with long-range interactions, where nonextensive thermodynamical and dynamical anomalies are expected. In a previous work [3] the maximal Lyapunov exponent of a one-dimensional system of N planar rotators coupled with interactions decaying as the inverse power α of their distances (see model (1) below) has been studied as a function of N . For total energy above some threshold, the maximal Lyapunov exponent is proportional to 1/N, where κ(α) is a function which goes from κ(0) > 0 to zero while α increases from zero to 1 (long range forces) and remains zero for α > 1 (short range forces). In this letter we want to investigate if the exponent κ, describing the weakening of the mixing properties of the dynamics in the long-range-interacting regime, is universal. In particular, we check the hypothesis that κ(α, d) is a universal function of α/d, where d is the dimensionality of space. Let us consider a simple model of N planar rotators (XY spins) placed at the sites i = 1 . . .N of a square and a cubic lattice (i.e., dimension d = 2 and 3). The rotators have unit moment of inertia, angular momentum Li and position specified by the angle θi ∈ [0, 2π]. Rotations occur inside an arbitrary reference plane. The model is described by the following classical hamiltonian in the conjugate canonical pairs {θi, Li}: