Some possible q-exponential type probability distribution in the non-extensive statistical physics

Abstract

In this paper, we present two exponential type probability distributions which are different from Tsallis's case which we call Type I: one given by p(i) = 1/Z(q) [e(q)(E-i)](-beta) (Type IIA) and another given by p(i) = 1/Z(q) [e(q)(-beta)](Ei) (Type IIIA). Starting with the Boltzman Gibbs entropy, we obtain the different probability distribution by using the Kolmogorov-Nagumo average for the microstate energies. We present the first-order differential equations related to Types I, II and III. For three types of probability distributions, we discuss the quantum harmonic oscillator, two-level problem and the spin-1/2 paramagnet.