Nonextensive Statistical Mechanics

126 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanicsf (u 1,u2)10.80.60.40.200.5Bivariate Uniform Distribution, ρ = 000u 2–0.5 –0.5u 1Bivariate Uniform Distribution, ρ = 0.10.5f U1(x), f U2(x)f U sum(x)Marginal Densities: U 1 and U 2 , ρ = 01U 1U 20.50–0.5 00.5xU sum = U 1 + U 110.50–1–0.5 0 0.51xMarginal Densities: U 1 and U 2 , ρ = 0.1f (u 1 ,u 2 )21.510.5f U1(x), f U2(x)1U 10.5U 20–0.5 00.5xU sum = U 1 + U 100.50.5f U sum(x)10.50u 2–0.5 –0.5u 100–1–0.5 0 0.51xBivariate Uniform Distribution, ρ = 0.5Marginal Densities: U 1 and U 2 , ρ = 0.51U 1108f U1(x), f U2(x)0.5U 2f (u 1 ,u 2 )6420–0.5 0x0.5U sum = U 1 + U 1100.5f U sum(x)0.50.50u 20–0.5 –0.5u1Bivariate Uniform Distribution, ρ = 0.90–1–0.5 0 0.51xMarginal Densities: U 1 and U 2 , ρ = 0.91f (u 1 ,u 2 )252015105f U1(x), f U2(x)0.5U 1U 20–0.5 0x0.5U sum = U 1 + U 2100.50.5f U sum(x)0.50u 2 –0.5 –0.5 u 100–1–0.5 0 0.51xFig. 4.8 TMNT model for N = 2 random variables with increasingly large correlation ρ (ρ = 0corresponds to independence). Left: Joint distribution of the two variables. Right up: Marginaldistribution of each of the two variables. Right bottom: Distribution of the sum of the two variables.Notice that, whereas for ρ = 0 phase-space is equally probable, ρ approaching unity concentratesthe probability on only two of the four corners. Notice also that the marginal distribution does notdepend on ρ. From [240].