Nonextensive Statistical Mechanics

7.1 Physics 243Fig. 7.28 Top: Fits to the (positive and negative) WMAP3 (WMAP with three-year data) CMBtemperature fluctuations data, corresponding to the Q, V, and W co-added maps (after the Kp0cut-sky), in the NUMBER OF PIXELS vs. T/σ ν plots. We show the χ 2 best-fits: Gaussiandistribution (blue curve) with σ Q = 104 K, σ V = 118 K, and σ W = 131 K, respectively,and each nonextensive distribution (red curve) P q with q = 1.04. Bottom: Similar analysis, butnow with the eight DA WMAP3 maps (Q1,...,W4) after applying the Kp0 mask. We plottedthe NUMBER OF PIXELS vs. (T/σ ν ) 2 to enhance the non-Gaussian behavior. To avoid possibleunremoved Galactic foregrounds, we consider only the negative temperature fluctuations. Again,we show the χ 2 best-fits: Gaussian distribution (blue curve) and nonextensive distribution (redcurve) P q ,nowwithq = 1.04 ± 0.01 (from [539]).1f (v) ∝ ( ) κ+1. (7.10)1 + v2κv02With the notation changes κ = (2 − q)/(q − 1) and 1/(κv0 2 ) = (q − 1)β,weimmediatelyidentify eq−βv2 . This appears to be the case of the temperature fluctuations,around the value T ≃ 2.7 K, of the cosmic microwave background radiation of theuniverse: See Figs. 7.27 and 7.28 [539]. Many cosmological theories assume (orimply) this distribution to be Gaussian. As we see in Fig. 7.28, this is not correct,the overall value of the index being q = 1.04 ± 0.01. The Gaussian assumptionis excluded at a 99% confidence level, and does not constitute more than a firstapproach to the problem. Moreover, anisotropy is found between the four universequadrants, the strongest non-Gaussian contribution comes from the South-East universequadrant, where q SE ≃ 1.05.Other astrophysical phenomena might be related with nonextensive concepts aswell. Such is the case of solar flares, whose probability distributions of characteristictimes appear to be of the q-exponential form: see [301].