Using Gamma Distribution to Obtain Maxwell–R´enyi Statistics and Other Generalized Distributions

A universal method is proposed for performing calculations within the framework of generalized statistics generated by the parametric Tsallis, R´enyi, and Sharma–Mittal entropies. The essence of the approach lies in the use of an auxiliary gamma distribution whose parameters correspond to a particular variant of the statistics. Equations are derived that allow the generalised partition function and the mean energy to be expressed in terms of canonical quantities. The effectiveness of the proposed method is demonstrated using the example of R´enyi statistics. The Maxwell–R´enyi distribution is obtained and its properties are calculated, based on which assumptions about the possible nature of the generalised parameter are formulated.