A new construction for the statistical theory of the nonextensive systems

We propose a new statistical theory for classical and quantum small systems. It is a generalized scheme of the Boltzmann–Gibbs statistical theory by extending the Boltzmann–Gibbs statistical factor from infinite systems to finite systems based on the microcanonical ensemble distribution function and keeping this factor in all thermodynamic processes. We reconstruct the statistical theory for finite systems by obtaining the expression of the average particle number and the thermodynamic quantities such as entropy and specific heat, in the finite systems. We also explore the discontinuous phase transitions in the interacting classical nanoscale gases without the thermodynamic limit.