Scale-invariant random-walks and optimization of non-extensive entropy

As an illustration of the unfolding of scale invariance in stochastic and s tatistical mechanical systems we consider some properties of random-walks w ith long-tailed step distributions. We analyze how a renormalization group (RG) transformation flows along maximum entropy step distributions under a given constrained moment and ends at a fixed-point distribution that is ide ntified as that for the Weierstrass walk. This walk displays a transition f rom Gaussian to fractal behavior and it is appropriate to employ the genera lized Tsallis entropy for the latter regime. Along the RG flow the entropy decreases monotonically and reaches a minimum at the fixed point. (C) 2001 Elsevier Science Ltd. All rights reserved.