Statistical mechanical foundations of power-law distributions

The foundations of the Boltzmann-Gibbs (BG) distributions for describing equilibrium statistical mechanics of systems are examined. Broadly, they fall into: (i) probabilistic paaroaches based on the principle of equal a priori probability (counting technique and method of steepest descents), law of large numbers, or the state density considerations and (ii) a variational scheme -- maximum entropy principle (due to Gibbs and Jaynes) subject to certain constraints. A minimum set of requirements on each of these methods are briefly pointed out: in the first approach, the function space and the counting algorithm while in the second, "additivity" property of the entropy with respect to the composition of statistically independent systems. In the past few decades, a large number of systems, which are not necessarily in thermodynamic equilibrium (such as glasses, for example), have been found to display power-law distributions, which are not describable by the above-mentioned methods. In this paper, parallel to all the inquiries underlying the BG program described above are given in a brief form. In particular, in the probabilistic derivations, one employs a different function space and one gives up "additivity" in the variational scheme with a different form for the entropy. The requirement of stability makes the entropy choice to be that proposed by Tsallis. From this a generalized thermodynamic description of the system in a quasi-equilibrium state is derived. A brief account of a unified consistent formalism associated with systems obeying power-law distributions precursor to the exponential form associated with thermodynamic equilibrium of systems is presented here.Comment: 19 pages, no figures. Invited talk at Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, Santa Fe, USA, November 6-9, 200

Exponential Splines of Complex Order

We extend the concept of exponential B-spline to complex orders. This extension contains as special cases the class of exponential splines and also the class of polynomial B-splines of complex order. We derive a time domain representation of a complex exponential B-spline depending on a single parameter and establish a connection to fractional differential operators defined on Lizorkin spaces. Moreover, we prove that complex exponential splines give rise to multiresolution analyses of $L^2(\mathbb{R})$ and define wavelet bases for $L^2(\mathbb{R})$

Mixtures of compound Poisson processes as models of tick-by-tick financial data

A model for the phenomenological description of tick-by-tick share prices in a stock exchange is introduced. It is based on mixtures of compound Poisson processes. Preliminary results based on Monte Carlo simulation show that this model can reproduce various stylized facts.Comment: 12 pages, 6 figures, to appear in a special issue of Chaos, Solitons and Fractal