Nonextensive Statistical Mechanics

5.4 Many-Body Long-Range-Interacting Hamiltonian Systems 187Fig. 5.36 (a) Caloric curve: microcanonical ensemble results for N = 10, 000, 100, 000 are comparedwith equilibrium theory in the BG canonical ensemble. The dashed vertical line indicates thecritical energy: Water bag initial conditions (WBIC) and initial m = 1 are used in the numericalsimulations. Temperature is computed from 2〈K (N)〉/N, where〈...〉 denotes time averages aftera short transient time t 0 = 100 (not reported here). The time step used was 0.2 [839–842]. Microcanonicaltime evolution of T , for the energy density u = 0.69 and different sizes. Each curveis an average over typically 100–1000 events (ensemble average). The dot-dashed line representsthe BG canonical temperature T BG = 0.476. The quantity T , which starts from 1.38 (V = 0andK = UN for WBIC), does not relax immediately to the temperature T BG . The system lives in aQSS with a plateau temperature T QSS (N) smaller than the canonically expected value 0.476. Thelifetime of the QSS increases with N, and the value of their temperature converges, as N increases,to the temperature 0.38, reported as a dashed line. Log–log plots for the QSS lifetime (c) andthedifference T QSS (N) − T ∞ (with T ∞ ≡ T QSS (∞)) (d) are reported as functions of N. TheQSSlifetime diverges roughly as N, andT QSS − 0.38 vanishes roughly as 1/N 1/3 (see fit shown as adashed line). Note that from the caloric curve one gets m 2 = T + 1 − 2u = T − 0.38. Therefore,from the behavior reported in panel (d), being T ∞ = 0.38, one gets M QSS ∼ 1/N 1/6 .Resultsare similar when we consider double water bag initial conditions (DWBIC), more precisely initialm = 1 and velocities uniformly distributed within (−p 2 , −p 1 )and(p 2 , p 1 ). In the figure, we reportthe case p 1 = 0.8 andp 2 = 1.51 (from [373]).