On Nonextensive Statistics

Motivated by the growing evidence of universality and chaos in QFT and string theory, we study the Tsallis non-extensive statistics (with a non-additive q-entropy) of an ensemble of fractal strings and branes of different dimensionalities. Non-equilibrium systems with complex dynamics in stationary states may exhibit large fluctuations of intensive quantities which are described in terms of generalized statistics. Tsallis statistics is a particular representative of such class. The non-extensive entropy and probability distribution of a canonical ensemble of fractal strings and branes is studied in terms of their dimensional spectrum which leads to a natural upper cutoff in energy and establishes a direct correlation among dimensions, energy and temperature. The absolute zero temperature (Kelvin) corresponds to zero dimensions (energy) and an infinite temperature corresponds to infinite dimensions. In the concluding remarks some applications of fractal statistics, quasi-particles, knot theory, quantum groups and number theory are briefly discussed within the framework of fractal strings and branes. It has been known for some time that classical Yang-Mills theories exhibit chaotic behaviour [ 1 ]. Spatially varying non-abelian gauge fields on the lattice have revealed that the gauge field has positive Lyapunov exponent that grows linearly with the energy density. Another signal of chaos emerges in the fluctuation properties of the eigenvalues of the staggered Dirac matrix operator in a lattice SU (3) gauge theory (and in full QCD). Such fluctuations are described by Random Matrix Theory (RMT) [ 1 ]. The nearest-neighbor spacing distribution of the eigenvalues , the distribution of spacings s between two adjacent eigenvalues, agrees with the Wigner distribution for the RMT unitary ensemble : P (s) ∼ s 2 e −4s 2 /π. It has been argued that this could be an indication of quantum chaos. Many special chaotic solutions of Einstein's equations have also been found [ 37 ]. In particular, a perpetual oscillating chaotic behavior in the vicinity of a spacelike singularity has been found that has the character of deterministic chaos. Chaotica like behaviour has been observed also by [38]. Most recently it has been shown by Polyakov and Kogan [3] that in certain superstring backgrounds involving ghost-matter mixing brane-like vertex operators, the world-sheet dilaton beta function becomes stochastic. The Renormalization Group (RG) equation leads to a non-Markovian Fokker-Planck equation whose solutions are described in terms of the Feigenbaum universal constant δ = 4.669.. describing the transition from order to chaos. The appearance of this …