Nonextensive Statistical Mechanics

182 5 Deterministic Dynamical Foundations of Nonextensive Statistical MechanicsFig. 5.32 Lyapunov exponent dependence on a for different values of α. Fixed constants are N =1024 and b = 2. Initial conditions correspond to θ 0 = 0.5, δθ = 0.5, p 0 = 0.5, and δp = 0.5. Weaveraged over 100 realizations (from [359]).λ M~b 0.6310 –1λ M0.250.2α = 0.0λα = 0.3 M0.15α = 0.60.1α = 0.90.05α = 1.2010 –2α = 1.5 0 0.5 1 1.5 2b10 –1 10 0bFig. 5.33 Lyapunov exponent dependence on b in log–log plot. Fixed constants are N = 1024and a = 0.005. Initial conditions correspond to θ 0 = 0.5, δθ = 0.5, p 0 = 0.5 andδp = 0.5. Weaveraged over 100 realisations. Inset: Same data in linear–linear plot (from [359]).5.4 Many-Body Long-Range-Interacting Hamiltonian SystemsIn this section, we focus on a central question, namely many-body Hamiltoniansystems with interactions that can have a long-range character (i.e., 0 ≤ α/d ≤ 1forclassical systems). To isolate the role of the range of the interaction from any otherinfluence, we shall consider interactions which present no particular difficulty atthe origin (consequently, Newtonian gravitation is excluded since it has a divergent