The Thermodynamics of a Stochastic Geometry Model with Applications to Non-Extensive Statistics

Abstract

We use the escort distribution instead of the original distribution for calculating the moment generating function and the physical quantities in non-extensive statistical mechanics. According to the associated escort distribution, we obtain the moment generating function for some random variables. In the following, we consider the model of continuum percolation in stochastic geometry and percolation theory which is obtained by connecting the Poisson points with a probability that depends on their relative position. Using a formal expression for the probability of the size of a cluster at the origin provided by Penrose, we derive the q-thermodynamic quantities to evaluate these quantities performance in obtaining the critical point when the percolation occurs. Also, by plotting the q-thermodynamic quantities, we show very interesting fluctuations at the critical point.