A Study For The Relationship Between Non-extensive Entropy And Generalized Derivative

Boltzmann-Gibbs statistical mechanics is a very successful many-body theory and has a wide application.It is of great interest to extend the Boltzmann-Gibbs statistical mechanics investigate complex or out-of-equilibrium systems.Recently,non-extensive statistical mechanics has been proposed to generatize the Boltzmann-Gibbs entropy and several different expressions for this non-extensive entropy have been suggested.Some of these expressions have been widely used in the study of various disciplines,and can well describe many practical problems.It is noted that there are a few distinct generalizations of the non-extensive entropy,and it is of great help to study their relationship and try to find a way to unify them.The focus of this paper is to derive a specific formula from the generalized derivative,which can unify several expressions of non-extensive entropy and reveal their relationship with each other.We also explore the possibility of obtaining other expression of the generalized entropy.The main content and results of this paper are as follows:1.The limitations of BG statistics are discussed from the potential energy relationship of long range interaction and the transverse momentum distribution of hadrons in high-energy physics;2.The h-derivative is generalized by adding the variable or modifying the definition field of the variable,and the properties of the generalized h-derivative are explored,and the relation between the q-derivative and the h-derivative is given;3.Using h-derivative and generalized h-derivative to study the form of entropy function,moreover,a two-parameter entropy expression is derived by using the two parameter(h,h')-derivative actiong on the summation of power function,and two-parameter entropy function can unify the various known forms of the non-extensive entropy function.At the same time,some other single parameter entropy functions are derived by using this two-parameter entropy function;4.A two-parameter entropy function is derived by using the generalized two-parameter(?,h)-derivative,and its main properties are discussed;5.The possible entropy function is explored by using the q generalized form of fractal derivative,and the results show that the function of Riemann-Liouville q-fractal derivative is inconvenient as a entropy function,and the function derived from the Caputo q-fractal derivative keeps the basic properties as an entropy function.