One of the most important contributions of humankind into the field of mechanic is perhaps the ability to construct mathematical models that can be used to describe problems arising in this field. Besides the numerous new and successful entropies that have been suggested in this field, and their various applications, for instance the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy that has found application in quite a lot of physical problems, the use of differential and integral operator has been found very important in the field. Due to the limitations of the differential operator based on rate of change, mankind have moved and suggested the use of differential operators based on integral, as they are able to capture memory. One of the must used differential operator based on integral are perhaps the well-known Riemann-Liouville operator or Caputo derivative, these two operators while they are similar when the function is zero at the origin, we must also point out that they are convolution with power law that appears to be the Pareto distribution for which the waiting time distribution is power law, the mean squared displacement is from usual to invariant diffusion and finally the probability distribution is non-Gaussian. As far as statistical mechanic is concerned, this distribution cannot capture all statistical settings. For instance the random walk and Brownian motion cannot be captured with this differential operator. This has been a very big worry amongst researchers, on how to construct a mathematical differential operator that has convenient statistical properties. Very recently, new differential operators were suggested, they are based on the exponential decay and the generalized Mittag-Leffler kernel. On one hand, the exponential kernel is linked to the Poisson distribution; the operator has the ability to capture processes like fatigue, fading memory and so on. Additionally the kernel waiting time distribution is exponential, while its mean squared displacement is from normal to confined diffusion, which already provide new property that the power law does not have. Another interesting properties of this new kernel is the probability distribution that captures the Gaussian and the non-Gaussian with a steady state, a nice property that is observed in many problems in statistical mechanic. On the other hand, the kernel generalized Mittag-Leffler function, which is known nowadays to have created a statistic above the super-statistic, has the properties that can capture the power law, exponential decay law, random walk and Brownian motion. Additionally, its waiting time distribution captures power law and stretched exponential, while its means squared displacement captures normal diffusion and sub-diffusion, finally its probability distribution has already the queuing property from Gaussian to non-Gaussian with no steady state, which is abundantly observed in nature.
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