Scaling transformation of random walk and generalized statistics

Abstract

We use a decimation procedure in order to obtain the dynamical renormalization group transformation (RGT) properties of random walk distribution in a 1+1 lattice. We obtain an equation similar to the Chapman–Kolmogorov equation. First we show the existence of invariants of the RGT, and that the Tsallis distribution R q ( x)=[1+ b( q−1) x 2] 1/(1− q) ( q>1) is a quasi-invariant of the RGT. We obtain the map q′= f( q) from the RGT and show that this map has two fixed points: q=1, attractor, and q=2, repellor, which are the Gaussian and the Lorentzian, respectively. Finally we use those concepts to show that the nonadditivity of the Tsallis entropy needs to be reviewed.