Mittag-Leffler functions in superstatistics

•Superstatistics•Fractional calculus and their applications.•Nonlinear equations.•Non-extensive statistical mechanics.•Thermodynamic. Nowadays, there is a series of complexities in biophysics that require a suitable approach to determine the measurable quantity. In this way, the superstatistics has been an important tool to investigate dynamic aspects of particles, organisms and substances immersed in systems with non-homogeneous temperatures (or diffusivity). The superstatistic admits a general Boltzmann factor that depends on the distribution of intensive parameters β=1D (inverse-diffusivity). Each value of β is associated with a local equilibrium in the system. In this work, we investigate the consequences of Mittag-Leffler function on the definition of f(β)-distribution of a complex system. Thus, using the techniques belonging to the fractional calculus with non-singular kernels, we constructed a distribution to β using the Mittag-Leffler function. This function implies distributions with power-law behaviour to high energy values in the context of Cohen-Beck superstatistic. This work aims to present the generalised probabilities distribution in statistical mechanics under a new perspective of the Mittag-Leffler function inspired in Atangana-Baleanu and Prabhakar forms.