Nonextensive Statistical Mechanics

5.4 Many-Body Long-Range-Interacting Hamiltonian Systems 183T0.10.080.060.040.02010 6T BGα = 2 α = 0.610 2 10 3 10 4 t 10 51α = 0.010 5 t α = 0.3c ~N β(α)α = 0.6α = 0.8α = 0.9t c 10 410 3α = 1.1α = 1.5α = 2.0α = 3.0β0.510 210 2 10 3 10 4N00 1 2 3αFig. 5.34 Upper panel: Temperature evolution for α = 2andα = 0.6 and four system sizesN = 100, 400, 1000, 4000. Initial conditions correspond to θ 0 = 0.5, δθ = 0.5, p 0 = 0.3,and δp = 0.05. Fixed constants are a = 0.05 and b = 2. For α = 2 the four curves coincidealmost completely, all having a very fast relaxation to T BG .Forα = 0.6 the same sizes are shown,growing in the direction of the arrow. Left bottom panel: crossover time t c vs. N, showing a powerlawdependence t c ∼ N β(α) with β(α) ≥ 0. Right bottom panel: β vs. α shows that for longrangeinteractions the QS state life-time diverges in the thermodynamic limit. Note that whenα = 0, β= 1, and hence t c ∝ N. Given the nonneglectable error bars due to finite size effects,the relation β = 1 − α is not excluded as possibly being the exact one; more precisely, it isnonunplausible that t c ∝ N 1−α −1(from [359]).1−αattraction at the origin). More precisely, either we shall assume that the elementsof the system (e.g., classical rotors) are localized on a lattice, and the long-rangemanifests itself through a slowly decaying coupling constant, or the elements of thesystem (e.g., point atoms of a gas) are free to move translationally but then a shortdistancestrong repulsion (such as the 1/r 12 potential term in the Lennard–Jonesmodel for a real gas) inhibits them from being too close to each other.As a paradigmatic system along the above lines, we shall focus on the followingmodel of classical planar rotors [177]. The Hamiltonian is assumed to beH = 1 2N∑pi 2 + 1 ∑2i=1i≠ j1 − cos(θ i − θ j )r α ij≡ K + V (α ≥ 0) , (5.44)